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Microinstabilities
14 Microinstabilities 14.1 CLASSIFICATION OF MICROINSTABILITIES AND THE GARDNER-NEWCOMB THEOREM A microinstability is one which is not derivable from the standard MHD equations. It may occur if the distribution in velocity space is non-maxwellian. On the other hand a macroinstability {fluid instability) , which is due to coordinate space inhomogeneity, may be described not only by fluid theory but also by kinetic theory. Therefore in addition to the classification given in chapter 10 we may divide plasma instabilities into macroinstabilities (whose fluid theory was presented in chapters 11, 12 and 13 and whose kinetic theory was presented in sections 13.7 and will be presented in section 14.7) and into microinstabilities in the restricted sense. The latter may be classified /14.1/ into resonant instabilities and non-resonant instabilities. Resonant instabilities appear as a consequence of a wave-particle interaction occurring when cvfa - u. Since waves have a real frequency and are oscillating, the resonant instabiliTrapping ties are in general overstable, see (3.9). instabilities, the two-stream instability, the bump on tail instability or drift instabilities (c-pfo - Vp) are resonant instabilities. Non-resonant instabilities (also called bunching instabilities /14.9/) do not so much depend on the behavior of the particle distribution function in the neighborhood of a particular speed; pinch instabilities, the firehose in-
stability
or the mirror
instability
are of this type.
All instabilities may however be classified by the properties of the plasma, see section 10.1. One may use the classification scheme of waves (see chapter 9) or a slightly modified Mikhailovskii scheme: 1. Homogeneous collisionless plasma 1.1 unmagnetized (isotropic and anisotropic distribution function) see section 14.2 1.1.1 electrostatic instabilities 1.1.2 electromagnetic instabilities see section 14.3 1.2 magnetized, ^IIS0 1.2.1 electrostatic 459
460
MICROINSTABILITIES
1.2.2 electromagnetic 1.3 magnetized, klBQ 1.3.1 isotropic distribution function see section 14.3 1.3.2 anisotropic distribution function see section 14.4 1.4 magnetized, oblique wave propagation see sections 14.3 and 14.4 2. Inhomogeneous collisionless plasma, see chapter 13 and section 14.5 2.1 unmagnetized 2.2 magnetized, straight field lines 2.3 magnetized, curved field lines 3. Kinetic theory of macroinstabilities and the influence of collisions on microinstabilities see sections 14.7 and 14.6 4. Special effects 4.1 trapping instabilities see section 14.8 4.2 charge excess plasmas see section 14.9 4.3 anomalous transport see sections 12.3 and 14.10 4.4 beam plasma systems see chapter 15 4.5 parametric effects see chapter 17 4.6 nonlinear effects see chapter 19 4.7 laser plasma systems see chapter 20 Another scheme for an inhomogeneous two-dimensional low beta plasma would be that by Coppi /14.7/: 1. MHD modes, independent of I in the sense that for the limit l/a -> °°, where a is given by the characteristic inhomogeneity scale and I represents the order of magnitude distance between a minimum and the successive maximum of the B field along the w ^ere same line of force, ω > ^/^/^ > cthl^' 0} ω bs ^ cths/^' £> ^-s ^Cïe k ° u n c e frequency. 2. Fast kinetic modes with o-f-^/l > ω > o-^-^j/l. c < 3. Slow kinetic modes with ω < Gthl^ thE/^· These three types of modes are drift modes or curvature driven modes. Another scheme given by Coppi for the low frequency, low beta limit is: a.1 hydromagnetic modes b.1 density driven modes a.2 fast kinetic modes b.2 curvature driven modes a.3 slow kinetic modes c.1 wiggly (ballooning) modes d.1 resonant modes c.2 flute-like modes d.2 non-resonant (fluidlike) modes. Coppi gives general dispersion relations for a.2 and a.3. The ordering procedure mentioned on p 386 (volume 1 of this Handbook) may also be used to classify microinstabilities. Another scheme runs as follows /11.82/
461
PLASMA INSTABILITIES
MHD model
ω « ωLI
guiding cenU)«Ü) L J ter plasma Vlasov fluid ωLI model hybrid kinetic model
includes X»r^j gross stab. Λ ^ X»rLI
\>r LI
excludes microinstabilit.
gross stab., finite rL effects
kineeffects
finite r effects finite r^ effects finite r^ effects
microeffects very high ω
full Vlasov model The general theory of microinstabilities is based on the dispersion relations for waves which we discussed in chapter 9. These dispersion relations can be derived from the Vlasov-Maxwell equations (9.151). This system has a manifold of equilibrium solutions some of which we have discussed in section 5.3. Collective phenomena among plasma particles allow these equilibrium distribution functions to relax to more stable solutions (of lower energy states). These are stable states of non-thermodynamic equilibrium. (5.25) (i.e. 8/ o /3u 2 <0) is the sufficient but not necessary condition for any isotropic and monotonically decreasing distribution function of kinetic energy fQ(u2) to be stable. On the other hand, Sf/du > 0 is not sufficient for a plasma to become unstable, see section 14.2. It should also be recalled that the Vlasov theory is valid only for times short compared with the collision time. If collisions become important, the situation may change, see section 14.6. On the other hand, an unstable plasma in which instabilities grow slowly compared with the time between two collisions may be considered Vlasov-stable. The (sufficient but not necessary) stability condition (5.25) is called Bernstein-Newcomb-Gardner-Rosenbluth theorem /14.2/, /9.26/. It predicts stability of an unmagnetized homogeneous plasma with an isotropic equilibrium distribution function decreasing monotonically with velocity. It may also be shown that any isotropic function / 0 , decreasing or not, is stable, if / is a function of (u2 + ufj + u|) . Also all one-dimensional distributions f(u) decreasing monotonically. away from its maximum are stable. The class of functions stable according to Gardner's theorem are all locally isotropic in some frame of reference. Similar results may be obtained from the thermodynam-
462
MICROINSTABILITIES
ic point of view for monotonically decreasing distributions as a consequence of the conservation of energy and the continuity equation in y-phase space, i.e. the Vlasov equation. Gardner's thermodynamical proof is also valid in the presence of self-consistent electric and magnetic fields and is nonlinear. Gardner1s general proof is however of little practical importance since it does not permit proofs of instability. No universal rule is known for finding the constraint which must exist if the system is stable, see section 10.4 and /10.9/, /14.3/. We will use a Lyapunov function of type (3.34) to prove the stability theorem /14.4/. The Lyapunov functional is quadratic in small quantities and is a constant of the motion. When the functional is the sum of positive terms, it follows that the equilibrium must be stable, since the functional, being bounded by its initial value, cannot grow indefinitely. Being a constant of motion, the functional L is independent of time. This can be verified by taking the time derivative of L and using (9.151). As in (3.34) L is the total energy and is given in cgs units by
L = ί { # 2 / 8 π + (B
-BJ2/8T\ o
2
<14·1>
v,
if' and / are equilibrium values, the summation m u {f f )/2 s s s~ o
+
G(fs)]dujdT.
Here BQ 0 over s extends over all species and G is a function of / to be determined. We make a series expansion in powers of the amplitudes / = f0 + f-\ + f2> E = ° + E-\ + E2r B = 9 B O + 5 1 + B 2 , G(f) = G(fQ) + G'(/0)/-, + G'(f0)f2 + + /fG"(/0)/2 + ... and substitute into (14.1). Then, correct to second order, we may write L = ΓίΕ^/δττ + Β^/8-π +
G'(/0))(/1s+/2s)
+ l\[(mu2/2
+
+
f2uG-lf0)/2]dZ}dx
(14.2)
Now, if we choose G such that for each species mu2/2
+ G 1 (f Q )
= 0,
(14.3)
L will be a quadratic functional. For this to occur fQ may depend only on u . If (14.3) is satisfied, the r.h.s of (14.2) is equal to a sum of nonnegative terms provided G"(fQ) > 0. On the other hand, from ~1 , (14.3) we obtain for G" the result - (m/2)(dfQ/du2)
PLASMA INSTABILITIES
463
which is clearly nonnegative as long as d/Q/du^ < 0. Hence all plasmas for which the several f0„ satisfy (5.25) are stable. It also can be shown /14.4/ that even for the non-thermal equilibrium, S = -/£(/)dudx can be interpreted as a generalized entropy and L as an associated free energy, see (10.23). The Lyapunov function approach has also been used for an inhomogeneous two-stream plasma /14.29/. 14.2 MICROINSTABILITIES OF A HOMOGENEOUS UNMAGNETIZED PLASMA When considering what we have said in section 9.7 on Landau damping and the damping rate γ ^ df/dcXr we might conclude that the plasma wave is unstable when there is a "bump" in the distribution function so that inverse Landau 0//Β£)ω/£. > 0 {Landau instability, damping /14.6/). Although this criterion can be understood physically in terms of resonant coupling between the phase velocity of various waves and the speed of particles, it suffers a drawback because it is not invariant under Galilean transformation /14.37/, /4.11/. Particles which go faster than the phase velocity of the wave in a bump region will outnumber those which go slower, and this produces a bump
instability, instability,
trapped trapped
particle instability, trapping ion instability, trapping insta-
bility in moving waves /14.16/, which may be stabilized by dissipation. The particles faster than the wave will be retarded and the wave accelerates those which lag slightly behind it. With a positive slope of the velocity distribution there is a net energy transfer from the plasma to the wave. The necessary condition for instability df/dc > 0 is however not sufficient. For a Maxwellian distribution displaced by w, a longitudinal plasma wave with phase speed c ^ = cQ + w (c0 is the most probable speed of the undi-splaced Maxwellian) is expected to be damped (df/dc < 0 ) , but a wave with > 0) , °ph = °o "" w mi<3ht b e expected to grow (df/dc since in the moving frame there would appear to be more particles faster than the wave than slower. However we will show that single-humped distributions like a displaced one-dimensional Maxwellian distribution are stable. This paradox is resolved by realizing that the slow wave c , = c0 - w has a negative energy in the laboratory frame of reference. There is more energy in the quiescent, drifting Maxwellian
MICROINSTABILITIES
464
distribution than in the presence of an oscillation. Thus the wave gains energy from the resonant particles but loses negative energy and hence damps out/14.58/. The introduction of dissipation (Krook equation, see p 82 of volume 1) may however cause the wave to grow if particles of positive energy are present /14.59/. In strong electrostatic waves a significant number of charged particles can be trapped near the bottom of the moving wave troughs. These particles act coherently as a bunched beam and produce growing waves by a similar mechanism (trapped particle instability
in a strong electrostatic wave, trapped particle stability of large amplitude, bounce resonant
ininsta-
bility) . Other authors /11.21/ define the trapping instability as an electrostatic macroinstability occurring in a toroidal configuration as a result of flute instability among particles trapped between two magnetic mirrors, see section 14.8. Stochastic insta-
bility
/14.26/ (instability
in stochastic
heating
fields) is a term used by Soviet scientists for a special trapping instability in electric fields. The trapped particles have an energy close to the "edges" of the well (produced by an electrostatic wave or by magnetic mirrors). Thus they may oscillate across the wells, up the edges. By stochastic disturbances of the electric field, a random acceleration of these particles then occurs which on the average leads to an increase of the particle energy with time and the ejection of particles from the wells. The motion of the electrostatic waves is the energy source of the instability which might be stabilized by sufficiently small amplitudes of the electric field fluctuations. There are in general no trapping instabilities for transverse waves, because there are no particles going with the phase velocity of these waves, which exceeds the velocity of light. The Nyquist plot discussed in section 3.4 is however an excellent means for finding electrostatic stability conditions in an unmagnetized plasma, since the number of times the plot Re D versus Im D encircles the origin as ω varies from -°° to +<» (conformai mapping of the real ω-axis) gives the number of unstable modes. The dispersion relation (9.153) may be specialized to a collisionless plasma (vs = 0 ) . According to (3.45) it can then be written in the form
465
PLASMA INSTABILITIES
n
1
7f
1 - 1
^Ps(dfso
do
_
SK
= D
+ iD . = 0,
(14.4)
where for real ω, according to (9.161), 2
-L.0O
n - i _ yÜP£ pf 8 ^ ° 2 ωτ
s k
\
/ c=u/k
de
(14 5)
(14.6)
where P indicates again the principal value. On the real ω-axis (which has to be mapped onto the D-plane) , we have D (ω = -°°) = 1, Dr(u = +°°) = . In order that we have an instability/ = exp(2i\in) according to (3.9) and (3.11), we must have Im ω < 0 for modes ^exp(iu)t). According to Table 8 on p 263 (volume 1 of this Handbook), k is real for an absolute instability so that ω/fe is complex. To have an instability it is necessary and sufficient /14.8/ that the plot C of the real ω-axis onto the £-plane encircle the origin of the Z^-plane, see p 47. In order to be able to do so, the plot must cross the real axis in the D-plane, or Dj, = 0 for a particular cm = = ω /k so that from (14.6) it follows
(*L
c
= 0.
(14.7)
m
Since the imaginary part D^ changes its sign when going from D^ < 0 to D^ > 0 we must have dD^/du < 0 and hence from (14.6) > 0
(14.8)
m
must have a minimum at o . Due to (14.7) we or f8 may write the integral in (14.5) for real ω in the form
466
MICROINSTABILITIES
λ P ί ^ Γ 2 —^-7T =P 1 —ΤΤ TT-lflc) J 3 Ö c - ω/fe J σ - ω/k d c L J
'
- f (ω/fc)] dc = J
J
(14.9)
+00
Çf{o) - f(u/k)
=
i
(*-ω/Λ)
+ 00
since
i
^
2
+00
^'
(g)
da = f — ^ ( g ) *
77?
(c
- c
0 )
d g and because the
equilibrium distribution function does not depend on time and on space. Then D - = 0 at % = kcm and
^'r ~= 1 - Ι-^| — ~ ' ^ 2
,
^-^
/7 x 2
This can be regarded as a Kramers-Kronig
dö. (14.10) dispersion
relation /5.9/, /9.5/, /9.55/. Equation (14.10) is then used to obtain a plot of Ζ?(ω) for real ω. The requirement that / s o should have a minimum implies that it has two or more maxima at e-i/^ etc, since /(+00) = /(-00) = 0. The Nyquist plot D (ω) for real ω is then drawn by calculating Dr at ω = kcm, kc^ and kö2· If / has its absolute maximum at c* , f(c^) > f(c) for any c and according to (14.5), (14.4) we have > 0. Dr at kom and ko2 can be positive or negDr(kc\) ative. But since / ( ^ ) > f^°m^ i1: follows from < £>ρ(ω = ko2) · So it is a (neces(14.10) that Dr(um) condition for the appearance of sary and sufficient) an electrostatic absolute instability that the equilibrium distribution function / Q have a minimum at °m
(/' (cm)
=
+
°°f
i
°^
anc
* that (Penrose
(G) - /
/14.8/)
(σ ) 7y
(c-c
criterion
r
de > 0.
(14.11)
m If a distribution satisfies (14.11), it will be unstable only in the range of k in which £ ρ (ω= kc ) <0. This range can be deduced from the Nyquist plot, see Fig. 42. It should be mentioned that stability criteria could also be derived by investigating ensemsee e.g. p 279 in /5.13/. bles of Vlasov plasmas, For v Φ 0 (use of the Krook equation (5.18) instead of the Vlasov equation) the criterion is anal-
467
PLASMA INSTABILITIES
ogous; according to (9.153) a term ^v appears in the denominator and the plot C is shifted slightly. The Penrose criterion will in general have to be applied for each direction of the wave normal, when the distribution function is three-dimensional. Two-humped and multipeaked distribution functions, see Fig. 4 2 , which are unstable according to (14.11), can result from a plasma containing at least two identifiable streams or beams. For this reason the resulting instability is called two-stream instabilibeam plasma instability, twoty , beam instability, humped instability, double stream instability, bunching instability or also longitudinal instability /14.9/. On the other hand a distribution can b e t w o humped and still be stable, see Fig. 4 2 . This indicates that Gardner's theorem (stability of a singlehumped function) is sufficient for stability, but not necessary. T w o beams with distribution functions constitute a good example / 9 . 5 / . If / 1 ( u ) and f2^u) the thermal velocities the directional v e Ί,2 and >:> e values of locities v<\ 2 satisfy \u-v^ ' °*\ I » ΰΛ ?ot hthe values of 2 the distribution functions 'f^ (u) , fntu) will be close o-~ -ri— . ^ c. . c?2 _ the Λ.Λ-- minimum value of / I.v.^ - . v.2 I. » to zero. For and the system of two will also be close to zero and beams will be unstable. If however \v^ -t>2 I « c-\ +
unstable Fig.
42.
Nyquist
marginal plots
for
Z(u/k)
stable from
(14.4)
468
MICROINSTABILITIES
ion velocity distributions however do not necessarily need beams to occur. They may also be produced by the application of short pulses on dc biased grids /14.60/. Examples of stable and unstable distribution functions are also /9.26/ 1. any one-dimensional distribution function with a single maximum like fM according to (5.22) is stable /14.32/, 2. any spherically symmetric distribution function f(u2+^2+ u 2) (isotropic distribution) is stable /14.31?, * 3. an anisotropic distribution function that decreassee (5.25). es monotonically with u 2 is stable, 4. a two-dimensional distribution function f(ux+uy*uz) stable, with one maximum is 5. the Jackson
distribution
function
/14.33/
M ( u - u 0 ) 2 + a 2 + ( u + u 0 ) 2 + a 2 ] is unstable, because it has two sufficiently separated maxima. It should be mentioned that the Penrose criterion systems : If the is valid also for quantum-statistical distribution function is continuous and has a single maximum, the system is stable, see p 325 in /5.9/. If the distribution function has an interval Ac in which no particles are present, f(àc) = 0, ("hole" in phase space), the plasma is always unstable, since - f(kc) > 0 and may exhibit periodic behavior f(c) /14.42/. This instability is weak and limited to a narrow interval ω^/fc (electrostatic plasma with discontinuous velocity
instability of a distribution
/14.30/). Electric fields externally applied are shielded by a plasma. If for any reason there is no such shielding the plasma is micro-unstable. In particular it is unstable when the electric field strength has a maximum inside the plasma /14.31/. If a distribution function is unstable, to reach stability it is necessary to add particles and to fill nearly the entire well in velocity space. On the other hand more ions than electrons are needed at a given velocity to generate a hump and an adjacent minimum, since / - foE + mEfQl/mj /2.12/. Distribution functions
f
so
=
of
m s
the
l\2W^Ts\
displaced
exp
m (u u
s ' os) 2knT B s
type 2
+ exp
describe counterstreaming plasmas
(^ os
m (u+u ) s os 2kOT B s (14.12) = streaming
PLASMA INSTABILITIES
469
velocity). They generate the count er streaming instability , double humped instability, electron instabilbeam instability, electron-proton two ity , electron stream instability, electron-electron instability, double distribution instability, two-eleor on-stream instability, electron-ion instabilities and ion-ion instabilities, electron-ion streaming instability, double electron ion instability, electron-ion two/14.10/. stream instability (= Buneman instability) Electron instability, electron wave instability, or electron plasma wave instability are the names for the electrostatic two-stream instability in which ions do not participate. In the Buneman instability, the ion-ion instability and the electron-proton twostream instability ions do however participate. Otherwise the electromagnetic electron-ion instability is excited by magnetic fields. Shifted distribution functions of type (14.12) may be investigated with the help of an expansion with respect to hermite polynomials /14.17/. Two counterstreaming electron beams with a temperature TE perpendicular to the beam direction may be described /2.12/ by the distribution function foE = (mE/4i\kBTE) exp (~mE (c^ + c\) /2kBTE) · • [& (cy-u0) + δ {cy + uQ)] whereas for the ions fQj = = 6(cl, This distribution is unstable if the electric field E of a wave is Ik. From a Nyquist analysis the instability condition is fc232 + ωρ^Ο - mEu£/kBTE) < 0. Such current driven instabilities are also called pinching instabilities because the pinching of a perturbed current increases the perturbation current. A special case of the counterstreaming electron-electron instability is the finite length instability(finite plasma instability) /14.28/. When two counterstreaming electron beams of finite length are considered, electrostatic growing oscillations with frequencies connected with the transit time of the particles appear. Usually the two beams are considered in a magnetic field. The boundary condition effects (finite length effects) vanish, when the transit time approaches infinity. The mutual velocity of the counterstreaming beams is the energy source of the instability which may be stabilized by increasing the length of the device, changing density and velocity length appropriately. In plasmas also other finite may be found /14.28/. So e.g. electron (plaseffects ma) and ion waves at lower frequency than the plasma frequency have been detected experimentally /14.38/. It seems, however, that this effect may not be due only to the transverse size of the plasma, but also
MICROINSTABILITIES
470
to nonlinear effects and pseudowaves (see also (9,42) may be diagnosed by the exin volume 1 ; pseudowaves perimental fact that their velocity depends on the voltage producing them, whereas the velocity of a (linear) wave is amplitude-independent /5.13/), /14.61/. Another possible explanation is a slight deviation from a Maxwellian distribution function (suprathermal particles) /14.41/· It is of interest that the basic features of the two-stream instabilities can be explained also in orbit theory and in fluid theory /3.1/, /9.26/, /2.10/, /2.3/, /2.12/. If the ions are at rest and two electron beams designated by I = 1,2 move in opposite direaction, then for ui ^ exp(ikz - iut) the linearized continuity equation (n^ « nQ^r u^ « uQ
(14.13)
The equation of motion gives four real roots
-iu>ûii + iku^ -eE/mg. ol nûiI (14.14) The Poisson equation reads ikE
three real roots
= -4πβ(η1 + 9i2) . (14.15)
Elimination of both and & both ui gives the dispersion relation of the two/9.1/, stream instability /9.55/ 2
2
ω ρ1 /(ω - kuQ^) + (14.16) 2 2 1. + ω ρ2 /(ω - kuo2) two real roots The solution of (14.16) P2 -- vt°' i.e ω = ku0+ for oop2 + ωρι is called the fast space charge wave and ω = = kuQ - ωρ* is the slow space charge wave (which ω/u >/i o2 o1 is a negative energy wave, see p 256). Fig. 43. The two-stream When the dispersion reinstability. Plot of l.h.s, lation (14.16) has four of (14.16) versus k real roots, the plasma is
\J
471
PLASMA INSTABILITIES
stable. For two real and two complex roots the plasma is unstable. For equal density and equal speed (U 0 1 = ~^o2 = uo) o f the two electron beams the plasma is unstable for kuQ < /2ωΡΕ. The maximal growth rate γ is obtained when k-m ~~ 0)p/3/2uo and is given by /14.9/ (14.17) ] max
P
When a single low density beam (nQ<\ « n02) °f speed u0 passes through a plasma of density nQ2 then (14.16) gives by ωρ-j = /αωρ 2 ' α = n ol/ n o2 K< ^ 2 2 /,..«. .. ,,.. ,.. ,«. (14.18) 2 ^. ....2 Λ ωρ5,/ω + αωρ£,/(ω - feuQ) = 1 and the growth rate r-
,, 2
γ = /αωρΕ/
/7 2
(b)pE/k
2
1X
1/2
(14.19)
uQ - 1 )
All (electrostatic) bunching instabilities discussed in this section have γ < ω?£' or γ < ω ρ ρ if the electrons are stable (mj ·> °° gives stability) . Oscillations described by (14.18) have a group velocity ^u Q and are therefore called beam oscillations. The real and imaginary parts of the solution of (14.18) have been plotted in Fig. 44 for a «c 1 . /ku„
k mu ο' /ω.^ PE Fig.
44.
kuQ/uFE
Beam instability according to (14.18), (14.19)
MICROINSTABILITIES
472
Dispersion relations of the type (14.16), (14.18) may also be obtained from (14.4) by inserting the two beam distribution function /3.1/, /4.11/ f(c)
= [&(c-uQ)
+ δ(σ+« 0 )]/2.
(14.20)
When the electron component of the plasma moves relative to the ion component with speed uQ we have from (14.16) /15.43/ ü>p£/(ü>-fcu0)2 + ω ^ / ω 2 = 1.
(14.21)
This equation is of fourth degree in ω and has four roots. Two lie close to ±copj and two lie close to kuQ. Longwave oscillations (kuQ < ωρρ) are unstable {Buneman instability /14.11/) with a maximum growth rate γ = ωρχ/3^/3/24/3^1_/3 /9.26/,/15.43/ near the resonance condition ωρ# = ku0 and ω = ωρβ (mjr/mj) V 3 e The alternating current instability /14.35/ is an instability which is produced by a two-stream instability of large amplitude. When the electron beam of density nQ and speed u0 > ctJijr is emitted perpendicular to a cathode and is absorbed at the anode that is separated from the cathode by a distance I, the Pierce instability {space charge instability, ion space charge instabili^ ty /14.12/) is generated. The linearized equations for perturbations K\\UQ read then /9.26/ -toon + un' o -iuu* -Φ" =
+ uQu%
+ nu* o
= 0
= -{e/m)§*
(14.22)
4-nen.
h, û are the perturbed quantities, Φ is the electrostatic potential and ' indicates derivation with respect to z. Since the cathode and anode potentials are fixed and speed and density at the cathode are given constants, the boundary conditions are Φ(0) = = Φ(Ζ) = 0, u% (0) = n(0) = 0. The solution of the system (14.22) yields the dispersion relation of the /9.26/, /14.12/, /10.14/ and a Pierce instability purely imaginary frequency: Re ω = 0, γ i\nl/AuQ. In all beam and two-stream instabilities the energy source of the instability lies in the kinetic energy of monoenergetic beams. Stabilization is possible by a thermal spread comparable to the relative beam velocity since this destroys the coherence.
473
PLASMA INSTABILITIES
The dispersion relations derived so far are valid in the cold plasma approximation, i.e. •th < ω/k or u When the thermal spread of the distrioE > kBTE/mE.
bution function is such that the particle distribution overlaps the region of unstable waves beam temperature effects, i.e. the effect of resonant particles (moving with u = ω/fe) must be considered. Such a distribution function of warm electron beam passing through a Maxwellian electron cloud with a is shown in Fig. 45. Inspeed u\ > ο\^Ε = 2kBTE/mE
unstable region
Fig.
45.
Bump on tail function
distribution
elusion of a pressure term Vp = ykßTVn, γ = Cp/cv, - see (9.49) - into the equation of motion and a Döppler shift by uQ in (9.56) yields the dispersion relation /14.34/ \i = 1,2)
ωP2
"ΡΪ
(ω - ku
„ ) - a- Zc
/
7
\
2
2
7
2
= 1
(14.23)
(ω - ku 2' " ao Here ωρ^ are the stream plasma frequencies or the electron resp. ion plasma frequencies, a| = ykßT^/m^ are the sonic speeds for the two beams or for electrons and ions, see (9.49), (9.50). For nearly cold plasmas (α^ « uQ^) we come back to (14.16). For large thermal speeds an inhibition of the instability is found if α-j + a 2 > uQ^ - u o 2 /2.10/. For a-j + a 2 < uQi - uQ2r the two-stream instability is maintained. Similar results are obtained in kinetic theory
MICROINSTABILITIES
474
from the insertion into (14.4) of an appropriate bump in three dimensions on tail distribution function (14.24) E2
S(c
- > * m {o
E z~Uo)2" ΔΚ
Β
Ε2
J-J
+eXP
x
)6(o ) y
""'"'"
t
L
2
V*2Ε2 J /
"^Β
and / 0 j = 6 (cx) 6 (Ο^) δ (e^) for the ions. Here nE = nE<\ + + nE2 » nE2, TE2 << TE<\ , u0 » 2k-ßT^/m-^. For nearly real ω expansions, using u)£j « o)L, n#i » n^2 anc^ (9.166), give /2.12/, /14.5/, /9.17/ for the gentle bump instability
"i * ωΡΕ(1 + 3 k 2 x L l } and for nE-\kßTE-\ ω
;
s
» nE2mEuo' 71
W
PE1
2
(14 25)
'
^ΌΕλ
-r8X3-exp fe
λ
£Ε1
<<:
"*
(""ii" 1 ) Ν ^
"Z?E1
26)
Since ω^ Φ 0 the bump instability as well as the beam (oscillating) instabilities. instability are periodic When temperature effects are taken into account the beam instability may be called hydro dynamic twostream instability. Various dispersion relations and growth rates have been derived /9.26/, /14.9/, /2.12/ for e.g. 1) a cold beam in a cold plasma, giving (14.18), 2) a slightly heated low density beam in a slightly heated dense plasma, 3) a beam with large tail thermal spread in a dense cold plasma (bump on instability, electron sound instability, electron instability /14.15/, /14.36/), resonant bump on tail 4) a beam with small thermal spread in a dense cold plasma, 5) a cold beam in a dense hot plasma, plasma /14.9/, /9.26/, 6) a relative motemperature effect tion of electrons and ions (Buneman instability in a hot plasma and others as in current carrying plasmas /14.295/), 7) and for the excitation of ion-electron oscillations of a hot plasma by ion beams etc, see also chapter 15. It should be mentioned that in multi-
PLASMA INSTABILITIES
475
invariance of component plasmas the lack of Galileian formulae for the growth rate of longitudinal electrostatic waves has been claimed /14.37/. Furthermore, a beam of energetic electrons (or ions) passing (e.g. parallel to BQ) through a cold plasma can excite ion waves which grow rapidly at the expense of the kinetic energy of the electrons (ionwave instability, ion plasma wave instability, ionwave instability in a magnetized plasma, current driven mode), electron plasma instability /14.13/, driven by the small electric current due to the relative motion between the ions and electrons). The phase velocity of the excited wave is at an angle to the magnetic field, but independent electrostatic oscillations are set up on each flux tube. A large thermal spread smears out and damps the instability, which taps its energy from the electron current along the magnetic field. Experiments on standing ion waves in thermal equilibrium show /14.44/ that two types of unstable ion waves exist: one has a phase velocity corresponding to the minimum of the distribution function which is almost independent of the voltage applied, the second has a phase velocity corresponding to the discontinuity of the distribution function and varies linearly with the square root of the voltage. The two waves may be denoted by two-stream instability and bump instability, respectively. Current instabilities (current driven instabilities) are of various types: on pp 364 and 374 of volume 1 we mentioned a current driven drift insta, on p 374 one finds also the inertia currentbility on pp 374 and 335 one finds convective instability, the nonlinear current driven ion sound wave instability /12.63/, /14.55/, on p 375 the ion wave current /13.34/, and on p 374 the skin current instability driven wave instability. Recently, in addition to the current driven long-wave ion-acoustic instability, a new current driven short-wave ion-acoustic instability at velocities lower than those needed for the excitation of the long-wave ion acoustic wave has been on an ion acoustic detected /14.64/. The modulation wave is shown to be unstable in a direction oblique to that of the wave phase velocity /14.65/. A survey of computer simulation of current-driven microinstabilities in homogeneous collisionless plasmas has been given by Chodura /14.66/. Finally a special current instability (current sheet instability) i.e. instability of the electric current itself is mentioned on p 318, see also /2.12/ p 505. There exists also
MICROINSTABILITIES
476
a current
driven
surface
wave instability
/14.51/,
collisional instabilities excited by currents may be
found in /14·53/, a current
induced
two-stream
insta-
bility has been discussed /14.54/ and nonlinear effects on current instabilities have been discussed in /14.52/. Current^ driven
standing
ion waves have been
used to excite travelling ion waves /14.56/. Standing electron plasma waves have also been reported /14.57/. Current instabilities are generated by a wave propacurgating parallel to the current. Electromagnetic rent instabilities are generated by linear electromagnetic waves in a current-carrying Vlasov plasma; they have lower thresholds than the electrostatic ion acoustic instability /14.62/. In chemically active molecular plasmas circularly polarized electromagnetic waves may cause - even in the absence of external electric fields - a new type of instability, the in-
stability
in active
molecular
plasma /14.63/.
When the plasma is warm there is a heat transfer from the hot electrons to the ions by Coulomb collisions. When T-p > YJTJ with yj being the specific heat ratio of the ion gas 7 this leads to heating of the ions in places where the ion density perturbations have their maxima, thus giving rise to growing ion sound waves (ion
instability,
sound wave instability,
ion sound wave instability
ion
in fully
acoustic
ion-
ized plasma /14.14/). This instability is often termed macroscopic, see section 12.2. The thermal energy of the electron gas is the source of the instability which may be stabilized by viscosity and heat conduction. It resembles the macroscopic acoustic wave in-
stability
in a partially
ionized
plasma,
see section
12.2 and chapter 19. Ion acoustic instabilities may also be excited by strong electromagnetic waves /14.40/, /14.62/. At low density the mean ion velocity is greater than the ion wave velocity in the plasma frame. Therefore it is possible to observe, instead of forward and backward ion waves, fast and slow waves in the laboratory frame (Buzzi /14.14/). Inserting distribution functions of the type (14.12) into (14.4) yields (9.167) and (9.168), if u,QQ = 0. According to (9.167), (9.168) ion waves propagate and are only weakly damped for TE» Tj, Cph= as-isoth>
°thl"j'
but are heavily damped for T% < Tj. When the electrons move relative to the ions with a speed u Q , the Penrose criterion together with (14.12) gives /12.2/, /14.10/ stability for uo
< 1.3(kBTE/mE)U2,
if TE = T ,
(14.27)
477
PLASMA INSTABILITIES and for
U
1 /?
o
<
m
(fe B V I
)
' ±f TEy>
Τ
Γ
(14.28)
If u is large enough (u0 > αβ), this stability criterion is no longer satisfied and the ion acoustic wave grows. The growth rate of the ion acoustic instability is then given in one dimension by /9.69/ ku_ - ωω
/™7Γ if O1/ ^^ o o P.. /3/2 2.2 y Q8/w'
T
w E»
τ
τ- I
(14.29)
The dispersion relation for ion acoustic waves based on a linearized kinetic theory for Maxwellian electrons and ions is /9.69/ 1 + ω ^ [ 1 +i/^zEW(zE)]/k2v*
+
+ ωρχ[1 + i/nraJf/(aJ) ] / f c 2 ^ = 0 , va = (Τα/ιηα) 1 / 2 , α = £ , I . F o r w h e r e za = iß/kvav/2, zE « 1 , ζχ » 1 , W{zE) * 1, W(zj) * ί[λ+λ/2ζ} + j + e x p ( - s ? ) . An e x t e n d e d N y q u i s t t y p e a n a l y s i s / 9 . 1 7 / g i v e s f o r t h e ion wave instability f o r | R e a ) | » | I m ω| and |Re ω/fel » c?* t h e d i s p e r s i o n r e lation ω - ω ρ (1 + τηΕ/2πι].) - iS-nrrij./ (S8mEk +
3,3
λ
+ 3fc 2 fc ß ^[1 + T^/T^] )
expi-rrij./
(mE2k
/2mE
+
2 2 λ )) +
(,/^/^|)exp(-3/2 - Λ"2)
Unless Tj « Tp the waves are heavily damped. hydrodynamic The assumption TE » Tj leads to the approximation. In (14.29) the ion Landau damping term has been neglected, compare with (9.168). In the three-dimensional case one has to replace k(u0~u>r/k) by the instability cone ^ w Q - ω ρ . It is possible /4.11/ to derive from γ = 0 the curve of marginal stability uQ{kc) , where the critical wave number indicates the onset of the instability. A study of ion /14.43/ reveals acoustic waves using Green's function that for TE < 3Tj collective interaction is unimportant. Then the wave properties are determined by phase-mixing of freely streaming ions. Even for fl = 0, for u = Cpfo the waves are damped. Relaxing the usual assumption of constant electron temperature gives the
478
MICROINSTABILITIES
modified ion-aooustic resonant instability /14.45/ in which electron density and temperature are coherently oscillating. The ion background is cold, at rest, and monoenergetic in space and time, but also variable ion-electron temperature ratios have been discussed /14.46/. Using Θ = Tj/TE, y = kaE/u>pE, k = wave number, μ = mE/mT and aE (9.49), ax (9.50), p 188 volume 1, γ^ = γj = 1, from continuity, momentum and Poisson equations in linearized form for a collisionacoustic less plasma the dispersion relation for ion waves 1/0 2ω 2 = ω^[(1 +Ch/2)y + (1 + z/2)]· 1± ,
4y,y [θ+Π+θι/ )] 2 2 2 μ+μθ# +(1+2/ ) is obtained. The range of validity is limited by Landau damping. There exist however undamped ion-acoustic oscillations even in kinetic theory. Using a selfsimilar distribution function Pitayevskii found /14.47/ a new branch of undamped ion acoustic oscillation. He has also investigated the breaking of ion acoustic waves. Ion acoustic waves may also be excit(ion bursts) (Cherenkov ion acoused by pseudowaves tic instability due to ion pseudowaves /14.48/).In turbulent plasmas a new quasi-acoustic oscillation (second sound wave) has been found /14.49/. The ion acoustic instability should not be confused with the acoustic instability in a partially ionized plasma and the acoustic instability nor with the magnetoacoustic instability {magnetic acoustic instability), see pp 318, 335 of volume 1 and chapter 20. Ion acoustic instabilities have been used as a source for travelling ion waves. When a grid is inserted between the hot plate and the opposite cold plate of a Q-machine, a positive dc bias applied to the grid with respect to the hot plate ( 2 - 3 5 volts) drives an electron dc current in the region between the hot plate and the grid. If the current exceeds some critical value, a low frequency ( 3 - 3 0 kHz) current driven standing ion acoustic instability (λ = 2L) appears between hot plate and grid. By changing the distance L between the hot plate and the grid the frequency of the instability is modified. The frequency is inversely proportional to L. Since the ratio of the frequencies for sodium and potassium plasmas is approximately equal to the ratio of the square root of the ion masses ( % a / % ) 1 / 2 at a fixed value of L one may conclude that the oscillation is a standing ion wave. By applying additional external
479
PLASMA INSTABILITIES
signals to the grid, traveling ion waves are excited for various frequencies at a fixed L in the region between grid and cold plate /14.50/. This allows measurement of the dispersion relation outside the instability region. The instabilities described so far have been electrostatic (ΒΛ = 0, curl £<| = 0) and due to charge bunching. All of them had anisotropic distribution functions since isotropic distribution functions are always stable. Particle streaming (beams) represents an anisotropic departure (e.g. Tj_ » T\\) from the (always isotropic) thermodynamic equilibrium. Anisotropy is also a feature of magnetized plasmas, see section 14.4. Besides the electrostatic instabilities in thermally anisotropic field free (B0 = 0) plasmas there exist also electromagnetic modes in isotropic and in anisotropic unmagnetized plasmas. Before specializing to an isotropic unmagnetized plasma we woul
!
= —
τπι
s
\κ· c - ω)
—
-
(14.30)
ÔC
By inserting the perturbed current density into Maxwell1 s equations (9.151) and assuming vanishing space charge and current at equilibrium, one may find the dispersion relation (9.153) (for v = 0) or (14.4) for electrostatic modes, and the dispersion relation (9.175) for electromagnetic modes. Equation (9.175) may also be written in the form /14.18/, /2.12/ c k
V_
kc2
- L>ω„ - L)ωηPs -5 -r—^— de, (14.31) Ps de kc -u) s s J x x where k = kx, ky = k = 0 has been assumed. Equation (14.31) is also valid, when we replace c^ -> e| in the numerator. (14.31) is never satisfied by isotropic fQ with ω = ω ρ + ίω^ and ω^ > 0: These modes can be driven unstable by temperature anisotropies, by beams etc. However, first of all we would like to investigate the electromagnetic stability of an unmagnetized plasma with the isotropic distributions / = fOQ {c£ + + 0* + c^) . in full analogy to the PenroseScriterion = ω
MICROINSTABILITIES
480
(14.11) it can be shown /14.18/, /2.12/, /14.19/, /14.20/, /2.9/, /3.1/, that isotropic distributions and monotonically decreasing distributions, as well fo(-c) as functions symmetric about the origin fQ(c) are also stable to electromagnetic (transverse) perIn full analogy to the turbations (Kahn criterion). method by which the Penrose criterion has been found, the Kahn criterion has been derived from (14.31). To use this technique the unstable roots of (14.31) have to be found. However, the crossing of the real axis in the ω/fe plane at (ω/fe)^ = cm demands now that the distribution function have a relative maximum at cm /14.18/, /14.20/. For instability it is therefore no longer necessary that the distribution function be two-humped. Be cm a relative maximum (^f0s/^ox)cm =0, then Kahn*s instability criterion reads +00 k
a
m-^PB~k s
G
- He J ka - ka s ·* x m
*»„< 0.(14.32)
—oo
The necessary and sufficient criterion for instability of a single-humped anisotropic plasma reads
k2c2 + 2 ω ^ | ^ ( ^ - -^\\ί„τ.+ζτί~τ\ά°<0.
(14.32a)
It also can be shown /14.18/, /14.21/ that in an unmagnetized plasma transverse waves are more unstable than the longitudinal modes. Thus a single-humped anisotropic plasma or a distribution f(oLc^ + acf· + c|) r a < 1 , may be unstable for electromagnetic (and electrostatic) modes. Another example is the bi-maxwelli= const an plasma defined by (7.81). We assume n(x,y) and normalize / such that n = 1. This distribution is stable to electrostatic perturbations, but according to (14.32) it is unstable. This may be found by a Nyquist plot /2.12/. For instability we then have
T
IE/T\\E - 1 > φ 2 / < 4 ·
(14 33)
·
Waves with ?* || 2-axis propagating in the colder direction are unstable. It can be shown /14.18/, that under certain conditions even a very small anisotropy having T± > Tn can produce instability. Mainly the very lonq waves are unstable and the maximum growth rate can be found to be /2.10/, /14.20/, /14.2/
PLASMA INSTABILITIES
481
ym = (32fe5T2/27ïïm52)1/2(1 - Τ 1 / Τ υ ) 3 / 2 ω ρ .
(14.34)
Quite generally, according to Furth /14.18/ it can be stated that the growth rates of the electromagnetic if instabilities are less than uQu>pi?/c or u0upj/c electrons alone are stable. In the general dispersion relation, anisotropic (bi-maxwellian) distribution functions couple longitudinal and transverse waves. The coupling terms have destabilizing effects /14.18/. For a bi-maxwellian plasma {anisotropic velocity distribution function*') (13.41) the dispersion relation for electromagnetic waves (14.31) may be written in the form /2.12/, /14.18/ for waves parallel to the re-axis k2χc2
2 = ω 2 + LΙωΙ JT s\\, - 1) +LΤωPs ζ Μ , Ps (Tsi s Ζ(ςs )Τsi'JTs\\ ' s s (14.35) where ζ^ = u/kxS2kBTs\\/ms and Ζ{ζ8) is given by (9.170). In the limit ζ8 « 1, ω 2 < k$c2, mE « mz one obtains
ω = ik /k
x B2WmE^TEU{TEl/TE\f
-1 * φ 2 / ω Μ >
/Τ
ΕΙ·
(14.36) This mode exists only in an anisotropic plasma. It grows at the expense of the excess electron energy k B^TEi-Tm). When a two beam distribution function of the type (14.20) is used for a first and for a second electron beam or ion beam, electromagnetic counterstreaming instabilities may be found /14.19/. Instead of using kinetic theory one may use^two-fluid theory. For a ^ave f = kx, ky = kz = 0, E = Eyt Ex = Ez = 0, B = κχΕ/ω, crossing two streams of electrons with density n/2 and unperturbed velocity Uy = ±u with stationary ions, equations of motion and of continuity give /2.10/, /3.1/ 2~2 2 22 2 2 k c + o>p(1 +k u /a) ) = GO (14.37) (Weibel instability , transverse wave instability r transverse instability, anisotropic temperature instability) /14.19/. Transverse instability is a rather general name which is used for some particular Some authors call a bi-maxwellian distribution anisotropic /4.11/ p 510 or /13.49/ and /5.14/,whereas others call it isotropic /10.14/ p 38 or /14.17/.
MICROINSTABILITIES
482
electromagnetic modes such as e.g. the Weibel instability. The terminology of these instabilities is not very strict. Lehnert /10.2/ gives the following defi-
nitions: the temperature
anisotropy
transverse
wave
instability /14.39/ is due to the temperature anisotropy T|| - TJL and concerns electromagnetic waves propagating across a magnetic field in a plasma with anisotropic temperature. The transverse wave instability /14.19/ or anisotropio velocity (Weibel instability is due to an anisotropy of the electron instability) velocity distribution in an unmagnetized (or magnetized) plasma. For u ■> 0 we obtain (9.46) from (14.37). The Eccles condition, see section 9.2, is still satisfied for u Φ 0. The dispersion relation (14.37) is shown in Fig. 46. K2
-u
Fig.
2 2,-2 ίύρ/α
46.
Dispersion Weibel
curve of instability
the
The growth rate is given by 2 2 2-1/2 γ = uu^k^ for u « c,
2
mEu
+k c )
= kBTE
w Δ
» u?,mE
,
(14.38)
(strong "temperature"
anisotropy). In kinetic theory /14.22/ one obtains instead of (14.36) the dispersion relation fc252 + co|(1 + + u2Zi2c^}l) = ω 2 , where Z is given by (9.170). Another formula is /3.1/ - compare (14.31) c2k2
+ü>p(l + ^c^tCèf/dc)
(2Î-c - 2ü))"1dS)= ω 2 ,
483
PLASMA INSTABILITIES
where the appearance of c± indicates that the velocity distribution f(~c) is anisotropic even in an unmagnetized plasma. (Then c\ is the component perpendicular to the direction of anisotropy.) Besides the purely transverse (Weibel) instability a coupled transverse-longitudinal mode separating into a quasi-transverse unstable mode and a quasi-longitudinal ion acoustic mode has been found. The ion mode is driven unstable by its interaction with the quasi-transverse mode. The Weibel instability is a microscopic electro-
magnetic anisotropic
temperature
instability
involv-
ing only the electrons. It can be described as the spontaneous formation of local pinches with currents parallel and antiparallel to the axis of maximum thermal velocity. The Weibel instability may be stabilized by imposing a magnetic field perpendicular to the low-temperature direction. Other electromagnetic beam instabilities have also been investigated. An anisotropic ion "beam" with thermal spread 1/2
foi
= (m I / 2 i r Vll , i w I / 2 i r Vlll ) 2
• exp (-mTcIxl2^B^m
(14.39)
2
) expi-mj-Cj.
l/2kBTJi)
in a background of hot electrons
fQE = ^mE/2^BTE)3/2^{-mEa2E/2kBTE)
, 2 ^ (14.40)
is unstable /2.12/, /14.18/ to electromagnetic waves \\x if
T
IL/TI\\ - 1>
Φ2/4Ι
(14 41)
'
with a growth rate ω^ - (^ρχθ^/ο. If the electron background is cold, the same distribution functions are valid, but TE « Tj. Then the electromagnetic wave propagating in the ^-direction is unstable for a certain range of kx. Quite generally, we see that currents in a plasma generate instabilities for waves propagating parallel to the direction of the current. Another practical example of a highly anisotropic electron distribution in a neutralizing ion background is the electrostatic confinement instability occurring in an electrostatic confinement system proposed by Farnsworth /14.23/.
484
MICROINSTABILITIES
14.3 MICROINSTABILITIES OF A HOMOGENEOUS MAGNETIZED PLASMA WITH ISOTROPIC DISTRIBUTION FUNCTION If a homogeneous Vlasov plasma is magnetized it becomes anisotropic but may retain its isotropic distribution function, see p 229. It is a matter of taste whether a distribution function /(ej + cfi ,cz ) symmetric around the direction of the magnetic field, of which a bi-maxwellian (7.81), (13.41) is an example, is called isotropic or anisotropic, see p 481. In this section we intend to discuss instabilities of a homogeneous magnetized plasma with a truly isotropic distribution function. In this situation no new instability appears - only instabilities known from section 14.2 are modified. However, new instabilities appear when an anisotropic distribution function is assumed, see section 14.4. In a magnetized plasma there are longitudinal (electrostatic) waves which can be treated in electrostatic approximation c ■> «> and other waves which are essentially transverse, see section 9.8. The electrostatic instabilities grow more rapidly, especially for low beta /14.67/. These electrostatic (longitudinal) modes are however not influenced by the magnetic field Bo, if they propagate in the direction parallel to ~ÈQ, see p 196. If they do not propagate exactly in the direction of B0, one has a coupling between the longitudinal and transverse waves. This coupling is destabilizing for the longitudinal and stabilizing for the transverse modes /14.17/. For the electrostatic approximation (2 + «>) the coupling goes to zero, but the electrostatic modes are still modified, see p 197 (volume 1 ) . However, for a magnetized plasma there does not seem to exist such a general stability criterion as that of Penrose (14.11) or that of Kahn for electromagnetic modes, see p 480. But for stationary ions a stability condition for the electron distribution function of a magnetized plasma with respect to longitudinal (electrostatic) modes propagating at an angle Θ between 1c and BQ may be derived /9.5/. In the electrostatic approximation c ■> oo, n -> oo according to (9.18), and A -* 0, see (9.35), longitudinal modes are described by A = 0. From (9.33), (9.20) we then have ea,a:sin2e + + e22cos20 = 0. The elements of the dielectric tensor are given by (9.185) or (9.194), pp 231 and 234. For long wavelengths kcpfo « ΩΕ or rL < λ (small Larmor radius, see p 233), -ehe argument kxcx/Q,E of the Bessel function is small. For exact parallel propagation
485
PLASMA INSTABILITIES
(θ = 0/ kx = kI = 0 ) , the argument is zero. So we have for long wavelengths ^ethE <<: ω^ o r f o r P a r a l " lei propagation J n (0) = 0 for n > 0, see p 234. Since this is equivalent to putting n = 0 we obtain from (9.185) the dispersion relation for longitudinal electron
B0
modes
propagating at an angle Θ with respect to
/9.5/, /14.68/, /9.4/ in the form
4cos 2 e + f° roic )
j
+ 00
·ί(
4 g sin 2 e
feilen - ω
•^o(ö||>
fe|.ö|. - ω + Ω„
II
2Ω E (14.42)
Λ> ( °||>
k,.c ..- ω - Ω„>
de
For 5Ό0 = 0 or θ = 0 one comes back t o ( 9 . 1 5 3 ) z I n t r o d u c i n g Ψ1 =2 / ' {c |() c o s,2c 6 + fe|| [ f ç ( ö || + QEk \0 " - /"0 (e11 - Ω^/^||) ] 3ΐη θ/2Ω 5 ϊ one can r e w r i t e (14.4^) i n t h e form +CO
f
*'(c.i)
?
d
° II" ω / ^ il
fen
^ n = - T2 - · II ,x
(14.43)
Comparing this with (14.9) it is easy to see that the necessary and sufficient stability condition for /0(£||) is that Ψ (OH) have a minimum. Thus the elecis stable for all tron distribution function fQ(c\\) ω/k || = c\\ for which Ψ (c ^) has a minimum and the condition + 00
|—-IL-dc
i
II
\\m "
< 0
(14.44)
is satisfied. This is the case for even distribution functions with one maximum. For Θ = 0 (parallel proand pagation) Ψ'(Ο||) -* fo(c\\) (14.44) becomes identical with (14.11) and longitudinal modes propagating by the magnetic field. A along B0 are not influenced magnetic field reduces the manifold of stable distributions: it "destabilizes" longitudinal modes. Stability criteria for fixed k, Θ or üE have also been found /14.69/. Transverse (electromagnetic) modes propagating along BQ (kx = 0, kz * 0, if ky = 0) satisfy the dispersion relation (9.195). Defining the reduced dis-
MICROINSTABILITIES
486 tribution
/ 1 4 . 1 8 / , / 1 4 . 7 0 / , /14.74/
functions
i 00
00
9
v-»
^so(cl'Cll)
3
f
° 21, , foil ° 2TKo«Ld«L
(14.45)
00
Fsi(cu)
=
2^(cl/2)fso(ci,cn)dci o
one obtains the dispersion relation (k = k ) ω2 = a V - ωΐω2
U
all
s i II
( 1 4
.46)
When this is compared to (14.43) one can derive the following necessary and sufficient criterion for instability. For om defined by =
Hs[usFs\\
0
( 1 4
S
"47)
and for +00
2t2
~2v2iT
2 Ρ"ΊιΓ°Ι|",<Ι||'°Ι|'-'"<>ΐ"Ί|' U c — OO
c
ir m
) +Ω
β
>
0
(14.48)
i n s t a b i l i t y e x i s t s , if ZF „ (o - Ω /fe) S " * 2 ÖC m
82F . (c - Ω A ) 1 «1 % s 1> 0 . ( 1 4 . 4 9 ) r. 2. do m
For vanishing magnetic field, (14.48) can be trans(14.32). An isotropic formed into Kahn's criterion distribution function fso(°fi+°^) gives /14.18/ from (14.45), (14.46), (14.47) the single solution cm = 0 and then (14.48) gives the contradiction - 5 2 > 0. We conclude that electromagnetic
BQ are stable
for isotropic
For propagation perpendicular
modes propagating
along
to ^ Q (kz=0r
kx Φ 0 )
distribution functions.
electromagnetic waves with^E||B0 and mixed transverselongitudinal waves with ElBQ are possible, see p 237. When there is no current parallel to the magnetic vanishes, so that field, that is if ffSQc..dc..
487
PLASMA INSTABILITIES + 00
a
cndcn
2
=0
(14.50)
these two types of waves do not couple. This is so because according to (9.190) Όχζ =Ό ζχ = D z = D =0. For isotropic distributions transverse waves are always stable but the longitudinal waves may become unstable for non-monotonic isotropic functions (beam instability, Bernstein instability). For monotonie anisotropic distribution functions the longitudinal waves are stable. The nearly longitudinal waves are the Bernstein waves, the transverse waves are the cyclotron waves. Both of them will be discussed in section 14.4, since electromagnetic modes are heavily influenced by a magnetic field and since electromagnetic instabilities can appear only for anisotropic distribution functions we will postpone the discussion of electromagnetic instabilities to section 14.4 (anisotropic distribution functions). Criteria distinguishing between convective and absolute instabilities have also been found for magnetized plasma /14.92/. A general discussion of modes propagating in an arbitrary direction with respect to the direction of the magnetic field is not possible. For low frequencies, i.e. ω « kc, we may neglect the coupling and disregard the transverse waves. When we neglect the electromagnetic modes we have quasi-electrostatic instabilities satisfying (9.15) and (9.11). For ky = 0 the dispersion relation for quasi-electrostatic waves becomes /9.4/ k2e
+ k 2e
X XX
+ k k (ε + ε
Z ZZ
X Z
XZ
) = 0.
(14.51)
ZX
When the distribution function is isotropic we have to use (9.185) whereas for an anisotropic distribution function the elements of the dielectric tensor have to be taken from (9.194). If one neglects in (9.185), (9.194) the high frequency electron cyclotron resonance terms ηΩΕ - but keeps the ion terms - a further simplification is possible. In many plasmas one has ωρι » Ωτ and since ω < kc was assumed one has for ion waves with ion cyclotron frequency ω - Ω Γ (λ - rLI) 1 ω
ΡΙΚ<
PI
°/rLI
J
LI
(14.52)
MICROINSTABILITIES
488
or with (2.21), (2.40), (2.25) with cth (TE - TT).
ß « 1,
* u^
and (8.3) (14.53)
Thus we conclude that our approximation is valid /14.71/, /14.18/, /11.40/. only for low beta plasmas At low beta, electromagnetic effects increase the magnetic energy so that only electrostatic modes can become unstable and transverse modes may be neglected. Assuming a simple Maxwellian distribution for the ions, and for the electrons a Maxwellian distribution displaced by the relative electron-ion velocity u0 of the form ^exp[(c? + {cu - u0)^)/2kßT] one obtains the dispersion relation of the hydro dynamic twofield. Since stream instability in a uniform magnetic we are concerned with a wave nearly at the resonance with the ion-gyrofrequency, we may neglect all the ion terms except n = -1. The integrations over on and c^ respectively give W-functions. The argument or the ^-function (9.196) is small for electrons, since u f l °thE ^ 1 anc^ large for the ion W-function because otherwise the mode is Landau damped. (In the longwavelength limit correlation damping dominates Landau damping /14.81/). So the ^-functions may be expanded. The real part of the dispersion relation then becomes /14.18/, /11.40/ 2
2
PE
PI
-e
1 1(XT)—£—
= O,
(14.54)
χ Λ 2 c Γ ΐ'ω-Ω. °thE °thl thl 7,2 2 where xj -= k^r^j =_ k^c^^j/Q,^is the argument of the has a modified Bessel function. Since I<| (xj) exp (-xj) maximum of 0.22 at kfrlj - 1.5 one has - for TE^Tj the result that the mode has the frequency
ω = üj
+ ω0.22ΤΕ/ΤΙ
* Ωχ.
(14.55)
An instability ω - Slj/2 has been found also /14.80/. Substitution of ω into the imaginary part of the dispersion relation gives for TF - Τγ the rough instability criterion u
.. ^ 13c., T (14.56) cr%t tnl for the hydrodynamic two-stream instability in a magnetic field. When TE/Tj increases {TE > T£) , the critical relative velocity ucr£t decreases, if the waves propagate exactly parallel to the direction of the magnetic field, the instability threshold is increas-
489
PLASMA INSTABILITIES
ed to ucvit - cthE - 40ethI, but the growth rate is decreased to -Ω^- whereas the growth rate for oblique propagation under angle Θ is larger, i.e. γ ^ oip^sin Θ (uip^cos θ) 1 / 2 /2/Ω^ or γ - ωΡΕ /14.18/, /14.73/. An abundant literature exists on the twostream instability of a magnetized Vlasov plasma /14.72/ since the two-stream instability is probably the most extensively investigated microinstability. Because ω - üj- according to (14.55)fsome authors call this instability, which is due to the intermixing of a weak stream of electrons or ions with a background plasma and the coupling of the slower electron (ion) cyclotron mode with the ion (electron) cyclotron mode
of the plasma, an electromagnetic
cyclotron
wave
in-
stability /10.14/. The other low frequency (ω - 0) wave instability which we mode /14.82/ is an Alfvên discuss also in section 14.4. For laboratory experiments the theory of the instability in finite systems is of importance /14.75/.
The finite
plasma instability
{finite
/14.28/f /9.6/
length instability), produced at a frequency connected with the particle transit time by two oppositely directed electron beams of finite length in a strong magnetic field is a beam instability. Finite plasma effects have also been discussed for cold electrons and hot ions in a magnetized plasma /9.6/, /14.76/. The Vlasov-Poisson system of equations gives for a plasma slab of thickness a in the ^-direction (BQ parallel to z) a transcendental equation for ζ Ξ (ωρ^/ω^ttL - l)-1/2 fe-αζ fe.aC 1 + ccot - ^ — = 0 and 1 - ctan - 4 j — = 0. (14. 57) If we denote the solutions of (14.57) by ζη, for the frequency ω of the instability ω
η
=
ω
Ρ^η
/ (
^
+ 1)
one has (14
·58>
which should be compared with ω 2 = ω^(^ | ( /^ 1 ) 2 /(^^/^ 2 + 1)
(14.59)
for a wave in an infinite plasma. We see that in a finite plasma the wave numbers kn, k\ are discrete. The growth rate of the instability is nearly the same as for an infinite plasma. For a half-infinite plasma contained by a high-frequency electromagnetic field in a distance I from a conducting wall the stability
MICROINSTABILITIES
490
condition cot(j/o)2/52 - k2l) > 0 (14.60) has been given /11.100/. The two-stream instability of a two-component plasma confined in a strong axial uniform magnetic field /10.2/, /14.10/, is also called Buneman instability /13.67/, /14.77/. The Farley instability /14.91/ is a
variation of the ion sound two-stream
instability
in-
duced in a uniform weakly ionized plasma by an electron drift through ions in a direction perpendicular to the magnetic field. The electron drift is often produced by crossed electric and magnetic fields. In the Buneman instability any transverse motion of the charged particles is neglected. Electrons and ions have different drift velocities/ so that uQ Φ 0, and different temperatures; the velocity distribution is Maxwellian. Due to the relative motion uQ a current appears and the Buneman instability is a current driven instability. Bounded and unbouded plasmas have been discussed. Also the computer simulation of current-driven microinstabilities has been given in a review article /14.66/. For bounded plasmas the thresholds given by (14.56) must be replaced by other expressions. As the temperature ratio TE/Tj increases the number of unstable modes in a bounded plasma increases: TE/TI = 1, n = 0,1; TE = 2TJf n = 0,1,2; TE = 6Tj, n = 0,1/2,3. Because of the stronger Landau damping higher modes are difficult to observe. All results are sensitively dependent on both geometry and density profiles, if there are any. Mikhailovskii defines /9.26/ the hydro dynamic Buneman instability '/3 by Tg * Tj as well as by frequency ω= 2^/^pE(m^/mj) /m )1/3f s e e p 492. and growth rate γ = ωρΕ2~4/3/3(m The electron-ion instability is defined as due to the relative motion between ions and electrons in an unmagnetized plasma, see p 469, with growth rates proportional to the derivative of the electron (ion) dis-
tribution. Electron-ion
streaming
instabilities
(elec-
trostatic and electromagnetic mode) are the names applied in a magnetized plasma. The streaming takes place across the magnetic field. This terminology is however not used strictly /14.9/, /14.78/. The electrostatic electron-ion streaming instability
/14.78/ is found for ωρΕ
» ÜE,
TE > Tj, Uç >
ο^Ε.
It is due to the resonant coupling of a Doppler-shifted ion mode and an electron cyclotron mode. Its growth rate is
491
PLASMA INSTABILITIES
γ - (mE/mI)UAüE.
(14.61)
The energy source of the instability is the electron current across the magnetic field. The instability involves the whole electron distribution. It occurs only in sufficiently dense plasmas and is stabilized by the damping of the electron cyclotron and Doppler shifted ion modes. The electromagnetic electron-ion streaming instability
has a monoenergetic cold ion beam directed
across a magnetic field as its energy source. It arises in a large amplitude whistler (collisionless or collision-dominated). The transverse growing electromagnetic oscillations are produced by a coupling between the ion beam and the whistler wave (anisotropic distribution function!). The growth rate of the instability is limited by damping effects and transverse and wave numbers /14.78/, /10.2/. Electron-electron ion-ion instabilities are (electrostatic) two-stream instabilities/ either in an unmagnetized or in a magnetized plasma /14.79/ and arise from electron (ion) distribution anisotropy. There exist no electromagnetic instabilities due to two streams in one species. Current driven instabilities are either field aligned current instabilities (current parallel to BQ as in the Buneman instability or in the cross stream instability, p 300, in which the waves propagate acurrent cross the field) / or they are cross' field driven instabilities (current perpendicular to BQ) . A current parallel to the magnetic field can cause a variety of instabilities /9.26/. For parallel propainstabilgation of the mode (k\ = 0 ) , an ion acoustic For k ι * 0 an ion cycity is excited for w* < o^E. lotron instability (anisotropic distribution, see section 14.4) is excited for u0 < c^^g whereas for u o > °thE the electron-ion two-stream (Buneman) instability may appear. Current driven ion waves (ion wave instability in a magnetized plasma, current driven mode /14.13/) are driven by an electron current along the magnetic field in a plasma with zero ion temperature. For zero temperature the distribution functions are δ-functions. For equal masses of counterstreaming particles moving with ±uQ, ωΡΙ = ωΡΕ, Qj = ΩΕ the dispersion relation for electromagnetic modes in a magnetic plasma is very simple /2.6/ 0 0
0
~Z- 2
c k
o/0)
- ω
ω - ku
+ /cu
2 , 2 /
+ ω Ώ —— Ί
\
O ,
T;
Ρ\ω + kuQ - Ω
θ
+
7
τηι
ω - kuQ + Ω /
\
Λ
= 0.
492
MICROINSTABILITIES
This expression has singularities at ω = ±(ku0 - Ω ) . When 5 2 & 2 /ω£ > 2ku0/ (Ω - kuQ) not all roots for ω are real. Therefore this is the condition for instability. kuQ > Ω is always stable. degenerBesides the normal ion-wave instability a has been reported which is a ate ion-wave instability non-dispersive low-frequency mode (ω « Qj) for T E\\ > TI\\ f ° r Anisotropie distribution function /14.83/. The influence of conducting walls on the ion wave instability in a cold current plasma has also been investigated /14.85/.
If Tj Φ 0, we speak of ion sound instability,
ion
acoustic instability (fully ionized plasma: collisional /14.13/, in a collisionless shock wave /12.64/, p 318, nonlinear current driven mode /12.63/, p 374, in a weakly ionized plasma /14.84/). Ion wave instabilities and ion sound wave instabilities may also grow wavelength due to an ion beam (chapter 15). Long ion-acoustic waves in a magnetized plasma in a gravitational field have been discussed in /4.86/. Mikhailovskii /9.26/ gives the following survey for instabilities driven by a longitudinal current: 1 . u0 > ci.jtE (high current velocity) , weak magnetic (electrostatic field, Qj? < ωρ^- Buneman instability , kz = LùpE/uQ, electron-ion streaming instability) ω - γ - (mE/mj-) '' ωρ^, see also (14.61). 2. u0 > c-f-hEr strong magnetic field, Ω^ » ω-ρΕ elecJ ω - Ω^, k~=flE/u0, tron cyclotron instability 2 4 3 1 3 / JT . There is γ = (/?ωΡ£./2 / ) ((mELupEs±n Q/mz^E) also a second branch described by the dispersion relation 1 = ω ^ / ω 2 + O U ^ C O S ^ / (ω - kzuQ) . 3. u0 < c-f-fo-g, u0 > c-f-foj, TE » Tj. Then one finds high ion acoustic modes. For kj_ = 0, the magfrequency netic field does not affect the instability. These instabilities have a maximum growth rate as in the case BQ = 0. They may γ - (mE/mj) Î/2ωρι, be excited also by runaway electrons. 4. uQ < c^E, u>/kz < ctfoE, uQ > c-f-foi* In contradiction to the ion acoustic modes it is not essential that TE » Tj. Now electrostatic ion cyclotron instabilare found /14.104/. They grow in a plasma ities with TE - Tj if u Q is not much smaller than ο^^Ε. For TE - Tj no ion acoustic instability appears. ion-acoustic in5. Theoretically, also low frequency stabilities could be excited. Their growth rate γ^ is small and proportional to ^foE/^^z· If Ίη be the growth rate of the high frequency ion acoustic mode, one has Yi/yn - Ωχ/ωρρ. The low frequency mode plays a more important role in a collisional
493
PLASMA INSTABILITIES
plasma. Additionally /14.93/, for counter-streaming ion currents along a uniform magnetic field a purely growing instability exists with a growth rate as high as 16 times the ion gyrofrequency. When the streaming ions are only 1 percent of the stationary ions, the growth rate is still 4Qj, but the real part is near the lower hybrid frequency. Finite beta effects (3 > 0.0001) increase the growth rate. For ion beta a 1 three instabilities appear /14.103/: a long wavelength kink-like mode and two instabilities with k > *Vj/ the electromagnetic ion acoustic instability
and the whistler
current
instability.
These instabil-
ities have substantially lower thresholds than the electrostatic ion-acoustic instability, and are favored by increasing ion beta. The whistler current instability heats the ions. Instabilities for ion beta > 1 and kr^j > 1 have also been investigated /14.94/. Instabilities with low phase velocities offer a possibility for ion heating /14.102/. Cross field current driven instabilities form an important family. Theories published may be classified as follows: 1. isotropic distribution function, homogeneous plasma /14.87/ 2. anisotropic distribution function, see section 14.5 3. inhomogeneous plasmas /14.88/, and also non-linear theories. 2
A
It*o
"y
gradient drifts Vrc
Fig.
47.
Cross field instabilities
current
driven
MICROINSTABILITIES
494
We discuss here the isotropic theory. According to Hellberg /14.89/ we assume the geometry configuration as shown in Fig. 47. The basic physical assumptions are 1) high frequencies ω » Ω^, 2) electrons see the ExB drift, ions don't, they are at rest or have straight line orbits and have a Maxwellian distribution, 3) electrons drift on self-consistent magnetized orbits and have a Maxwellian distribution, 4) low beta homogeneous plasma in a homogeneous magfor gradients, if netic field, local approximation any (compare p 366), 5) linearize and integrate along unperturbed orbits (characteristics), compare section 13.4, p 366 6) quasi-electrostatic approximation. When uQ is again the relative motion between electrons and ions the dispersion relation for^electrostatic modes propagating perpendicular to BQ is given /14.89/, /14.87/, /9.26/ by 9
9
1'p /
\
ω- k u
I
_±?>
(3^ 7π η= ι -οο
V2k
IAA r~>\
Here Z
I
=
*/kGthIA'
nE =
Z
( ω
k U
y o'nÇlE)/^kz°thE
"
-v2y? K r y LE = e * I
(14.63) 22 Γ (fer*) n n y LE' to (9.170)/ p 227 is have been used and Z according
the Fried - Conte plasma dispersion
function.
netized
instability
The fol-
lowing regimes have been considered: 1. Hot electrons, cold ions, TE » Tj. For y„ > m^w 0 also the electrons have to be assumed to be hot. 1.1 Low frequencies, n = 0, Zj- » 1 , %E « 1 ΐοη acows, tic instability 1.1.1 kyrLE » 1/ magnetic field plays no role unmagion acoustic
(ion wave)
ω = feajd + fe2x2)-V2f s e e (9.76), Landau damping as T j increases / 1.1.2 kyrrq « 1, long wavelength, magnetized acoustic
\%on wave)
instability
ion
/14.66/, frequency
given by (9.76), dispersion relation
(14.64)
495
PLASMA INSTABILITIES
where ω = ω - kyU . Oscillations are excited at ω - topj. The growth rate is enhanced over the unmagnetized case for this regime, but smaller than for 1.1.1. Growth rate /9.26/ γ = u)/iïs(u - ω/fe) (ω/^α χ ) 2 Γ ο
for s £ 1,
(14.65)
where s = km^2/kz/2mj and kz « fe . As ωρΕ/Ώ,Ε decreases this mode vanishes for ωρΕ = Ω^. 1, 1.2 Cold electrons, cold ions, T« à Tj, zp» 2p » 1f n - 0, fluid mode only, kr^g « 1. This regime gives the (linearized) modified two-stream instability /14.66/. In this instability the Doppler-shifted electron plasma oscillation is nearly perpendicular to B . A negative energy mode (ω * kyUQ - kzu>PE/ky) couples with the lower hybrid mode. The dispersion relation for the linearized modified two-stream instability reads 72
2 1
* ~2
- 7T(
2
2
1
Γ2 + —
(14.66)
k (ω - k u) ω y y ° for kz/ky < (mE/mj)^ / 2 . γ, ω - ω^#. The maximum - Vm^/m-r. As kz/ky ingrowth rate is seen for kz/ky creases, the wave-wave coupling decreases and the modified two-stream instability goes over into the ω
magnetized
ion acoustic
instability.
1.3 kz/ky -*- 0, ω a iïE, kyrrE » 1. Then only a single (resonant) value of n is of interest ω = k uQ - | w I Ω .
(14.67)
This dispersion relation describes the electron
lotron
drift
instability
(beam cyclotron
drift
cyc-
insta-
bility) /14.66/. Here a negative energy Dopplershifted slow electron Bernstein mode couples to ion acoustic modes. 2. Electrons and ions at equal temperatures, TE = Tj. In this regime no unmagnetized ion acoustic instability nor conventional electron cyclotron drift instability appears. > 1 no instability can appear. 2.1 For kyrEE 2.2 By Landau damping the negative energy slow Bernstein mode will be driven unstable and the Bernstein 0.3ct^E. instability appears for uQ > 3. Hot ions, cold electrons, Tτ » TE. 3.1 Now the Bernstein instabil%ty, still with narrow angle appears, but with reduced growth rate γ^Ο.ΟΟδΩ^,
MICROINSTABILITIES
496
depending on the thermal spread of the hot ion stream /9.26/. 3.2 Further, the modified
two-stream
instability
is
present. 3.3 As k2/ky increases, the modified two-stream instability goes over into the electron acoustic insta-
bility
(electron
sound instability)
/14.90/
ω = k u - k a cr (1 + X^fe 2 )" 1/2 , y o
here a SI
z bl
D
(14.68)
'
YkBTT/mE.
(14.69)
The growth rate is large ^ {QjQE) V 2 or in another approximation
^ = (f) 1 \ ^ i w ( 1 + À D f e 2 ) " 3 / 2 / f e 2 c L r
(14 70)
·
The electron acoustic mode has negative energy and is unstable due to ion Landau damping. Mikhailovskii states /9.26/ that the thermal spread of the ions has its greatest effect on the oscillations with maximal / 3 no incos Θ. For cos Θ - 1 and c^j > uQ(mE/mj)^ stability is possible. The largest growth rate occurs for maximal cos Θ - (rn^/rnj) ^/2 for given UQ/C^^J m
^olcthI^'2
a n di s
ωρχ(1 +ω^/Ω|)~1/2. (u0/cthI)
^/2.
Electron acoustic oscillations are excited if u o K cthl* With increasing u0 the electron acoustic instability goes over into a hydrodynamic instability in which the energy spread of the electrons and ions is unimportant. If however uQ » c-f-hj a high frequency kinetic instability is discovered /14.101/. Its growth rate is greater than the maximal growth rate of the hydrodynamic instability. For unmagnetized ions (Im ω » tij, kc+^j » üj) but strongly magnetized electrons (ω «: ÜE, κ ^ ^ « Ω Ρ ' M a x w e H i a n distribution functions and |ωf ^>kc+^I cos ΘΙ the dispersion relation of the new instability reads (in a frame in which the electrons are at rest) ^ 2 /n2 2 1+t*pE/nE-upEcos Λ
-z2 Z 2 Q / 2 _,_ . r E 2 /7 2 2 ^ θ/ω +v/^zEe ω^/k cthE +
^FIU+i/i\zI^(zI)]/k
cthI
= 0,
where z^ = u//2kc l E
« plcos ΘΙ, 2 . = (ω - κ·ΐι)
/}/2kc£}lj,
497
PLASMA INSTABILITIES
W(z)
= e~z
11 + 4= U * dt) ,
see p 372.
In concluding this section we would like to mention some special instabilities like the ion-ion /14.95/ which is of interest cross field instability for theta pinch implosion. The stability of the (bumpy) theta pinch has also been investigated in the framework of the guiding center plasma equations /14.96/. It should also be mentioned that thermodynamic-kinetic methods have been applied to investigate the stability of current-carrying collisionless plasmas /14.105/. There are also some interesting instabilities in relativistic plasmas. The relativistic
strong
electromagnetic
wave instability
/14.97/ is a
nonlinear effect and is generated by a strong electromagnetic wave. Both electrostatic and electromagnetic ) may be excited dependmodes (both for £ll^0 and ing on the strength parameter v = eEQ/mgCU0, such that 1 « v « m-j-fm-j?. The strong electromagnetic wave could be either a wave in the plasma or an external pump wave. Longitudinal modes exist in a frequency band near the plasma frequency and mainly propagate at small angles with respect to the external magnetic field, their phase velocities being greater than that of light. Thus longitudinal wave generation is not favored. Also circularly polarized electromagnetic waves have been studied. The unstable modes pinch the azimuthal plasma current. This new instability seems to have interesting applications in laser-plasma physics and for rotating neutron stars (particle acceleration near -pulsars) . Another microinstability in homogeneous magnetized relativistic plasma is the
resonant
diffusion
instability
in a relativistic
col-
lisionless plasma /14.98/. It is generated when acollisionless relativistic plasma is immersed in a homogeneous magnetic field, in the presence of a cold non-relativistic plasma. Then unstable electrostatic flute modes become coupled with selected particles. A resonance occurs with the associated waves, since the particle mass is no longer the relativistic rest mass. There is then a resonant diffusion of relativistic particles and an energy transfer from the particles to the waves. The kinetic drift motion of the relativistic particles is then the energy source of the instability, which might be stabilized by decreasing the number of relativistic particles compared to that
MICROINSTABILITIES
498
of the background plasma. Other instabilities are the Breit-Darwin plasma /14.99/ or the linear sideband modes eigenmodes /14.100/ which appear at nearby frequencies in a collisionless plasma when a small amplitude monochromatic electron plasma wave is launched. The modes are called linear since the amplitude of these excited sidebands depends linearly on the amplitude of the launched wave. By nearby frequencies one means that the ratio Δν/ν < 0.1, where Δν is the frequency separation from the frequency of the launched wave v Q . The amplitude of these modes exhibits large spatial oscillations while the amplitude of the exciting wave damps monotonically according to the linear Landau theory. These sidebands may also be called -plasma echoes. 14.4 MICROINSTABILITIES OF A HOMOGENEOUS MAGNETIZED PLASMA WITH AN ANISOTROPIC DISTRIBUTION FUNCTION Anisotropie velocity distribution functions may be either symmetric around the direction of the magnetic field, bi-maxwellian (7.81), (13.41) or generalized non-maxwellian distribution functions generated by particle beams, particle injection or particle losses. In several methods used for plasma production, energy is transferred to the electrons by accelerating them in a particular direction. Since in some plasmas the relaxation mechanisms are slow the electron distribution functions will be anisotropic. Thus the plasma acquires an anisotropic temperature even in the absence of a magnetic field. An anisotropic or bi-maxwellian distribution function of electrons provides a coupling mechanism between longitudinal and transverse modes. Such anisotropic distribution functions not only modify known instabilities, see also section 14.3, but they also excite new types of instabilities. To see this we discuss a simple example /2.6/. First we write the dispersion relation (9.195) for electromagnetic waves propagating along BQ for a one species plasma in the form
-oo o
|| II (14.71)
499
PLASMA INSTABILITIES
(see also (14.71a) on p 501). The factor 2π comes from integration around the s-axis and the factor 2 cancels because del,2 _= 2c\dc]_. When we insert into (T4.7V) the beam-like anisotropic distribution function (T,, -> 0) ;f0(öj/S||> = δ ( σ ||)/ 0 (ο 1 ) '
(14.72)
where / Q is an arbitrary equilibrium distribution/ we obtain 2 2 2 ω (ω ± Ω) - ωρω(ω ± Ω) 9 (14.73) S z (oo ± Ω ) ζ
+ ü>p</2
or for a two species plasma /10.14/
-, + 4E/2
,2
+
(ω + Ω^) 2 2 2 ω
1 -
4l/2 (ω - Ω,.)
ωPE ω (ω + Ω„)
ω
2
c
]-
(14.73a)
Ί
ΡΙ 1 ω(ω - Ω ) J '
where
= c )c d (14.74) °°, one ob-
ϊ> //V i'*ll î *A
tains for ions an overstable bility , see (3.9))
wave (oscillating
insta-
^/e/2, ± ωρι<ο-2 > 1/2.,-/^ (14.75) I For isotropic distributions the imaginary part vanishes. Let us now derive a stability criterion for transverse waves propagating along j§ which is simpler than (14.47) - (14.49). For propagation along £ Q longitudinal and transverse waves are decoupled. For one species (14.46) may be written in the form (for transverse waves) ω = Ω
2 ω
~2 7 2
= o k
F (ω/fe. ,C||) ko..
- ω ± Ω
(14.76)
■do,
where
FWk,on)
= [<ω - onKn)FsU
- F;JU|
.(14.77)
MICROINSTABILITIES
500
If damping and growth are present for a given distribution function, ω has to change its sign at some k. Then there exists one value of k where ω is real (curve of marginal or neutral stability). According to (9.161), the integral in (14.76) may be written at such points as
(14.78) So the curve of marginal stability is given by (14.79) It is, however, very difficult to derive general stability criteria /14.164/. It is therefore useful to consider electron resonances (ion motion negligible) and ion resonances separately. For ion resonances in a two-component plasma with distribution functions ( 1 4 8 0 ) Όΐ=^οΙ(*Ϊ+νΦ · foE=^oE^\+aE°\)' the criterion (14.47) - (14.49) for waves propagating
along
B
gives
o
= Ωτ(1 - α~ 1 )Α.
= [SlT/k] m
I
m
I
(14.81)
I
Neglecting the electron terms we find that this satis- cj/k) > 0. Subfies (14.49), since UpjOLjF-r ( [üj/k]m stitution of (14.81) into (14.48) gives the result /1 4. 17/ that under the assumptions Çlj/ajk » cthlt ct^j - cthE, k^c^aj. > Ω 2 |Ωχ(1 - *[/az)/k- tiE/k\ » cthE, the waves are stable if °A
<
^Τ\\
(T
||/T1 "
1)
/Ti'
(14.82)
where cÄ is given by (4.65), p - njmj. This result can also be obtained /2.6/ from (14.79) by inserting (14.80) into (14.71). So for T\\ > Tj_ and small values of B0 we have stability, but the plasma becomes unstable if the magnetic field increases. Instability arises only if Ω* > ω|τ^/ [Τμ (Τ|| /Τ ι - 1 ) ] and then only for fc2 < fc2. p o r rji > rp^ w e have always a transverse instability for k2 < fc2 = {Ω2(1 -T||/Tj_)2 + {electromagnetic + ω|τχ (1 - T||/T_L)/T||}/32, see (14.92) wave instability, transverse electron wave instability , Weibel
instability
/14.116/). In an unmagnetized
501
PLASMA INSTABILITIES
plasma according to (14.33), instability could be generated for a small anisotropy by reducing k: modes with long wavelength are easier to destabilize. In a magnetized plasma k2r^j > 1 is required so that a field-free motion of wavelength 2i\/k persists. 1 is equivalent to k2c2/u^I = kïr^j/ftj > Vßj# k2r2LI> So the transverse mode propawhere ßj = 4-nnkßTj/B^. gating along B0 is unstable if Ω2(1 - Τ||/Τ1)2/ω^χ + Τ1/Τ|| - 1 > g"1.
(14.83)
The circularly polarized mode, propagating parallel is stable for T\\/T± - 1 > β"1 and unto ί , E^ik\\B0 stable for T|| > T\ and is called fire-hose instabili-
ty, whistler instability, ion Alfvên wave instability
Alfvèn wave , garden-hose
instability, instability,
see (9.136), p 214 in volume 1 of this Handbook. From (14.83) we see that low beta or λ > rL stabilizes this electromagnetic instability. The results obtained /10.14/ may also be derived by using (14.71) for many species in the form integrated over e^. This reads n - 7
2
-
D - k c
2
2
-
ω
'2
J.
+ b)pF
[("-fc0||)/og(0||)-fc<0^>/2.a/og(0||)/a0||^ ]
ω - kc.. T Ω Γ
J
l l +
ω
E
+
°\\ (14.71a)
2 ( ( ω - ^ | | > ^ ο ΐ ^ | | > ^ ^ I J ^ ' ^ o J ^ H ^ 3Ö„ Ρΐ] ω - /cou + Ωχ
where -Ω# is for the right and +Ω# for the left hand circularly polarized wave. The singularity ω±Ω-& 0 in the form (14.93), so that even for ω -> O a small anisotropy T. > T results in instability. ^ " The results obtained so far may be specified as follows. When instead of (14.72) we insert / 0 s = 5 ( c ) and expand about ω - 0, we into (14.71) (cold plasma) obtain /9.6/ (9.94), i.e. an Alfvên wave (Alfvèn wave . Without the expansion in ω we obtain instability)
MICROINSTABILITIES
502
(9.90)/ see p 197. In the frequency range iïj < ω « Ω^ we have (9.101)f which means a whistler or helicon
(whistler
instability,
helicon
instability).
To take
thermal corrections into account /14.156/ we find for ω « Ω, = 0 r kcz « Ω by expanding the denominator in (14.71) and by using (4.71), (4.72), (6.68) or Pn = LJ™ \fJ Cuàc, s } os 11 Ml s J the dispersion relation ω
2
'
k
2
=
°A
f—2
(P\ "Pll 2
p . = L\m If -^άο _L s r os 2 s
+ 1
\
P
*
J
l "Pll n
2
+
(14.84)
(14.85)
P A 1+^/δ2\ P^ / see p 214. We see that a hydromagnetic instability
(fire-hose instability, garden-hose hose-instability) occurs, if P H - ρλ > QCA.
instability,
(14.86)
Let us say some words regarding this family of instabilities. The Alfvên-wave instability /14.106/ is defined as an electromagnetic microinstability generated in a cold magnetized plasma when the particle energy is larger in the direction parallel to the magnetic field than in the direction perpendicular to it. The instability is due to the centrifugal force, which acts on the plasma flowing along a curved field line, see Fig. 20, p 213. The curvature of the field lines is produced by a first perturbation and is enhanced by the instability. The whole field pattern oscillates back and forth. There are two modes: the
fast
Alfvên-wave
instability
is due to a large aniso-
tropy involving the whole plasma and the slow Alfvênwave instability which exists at small anisotropies. The slow mode is driven by the tail of the velocity distribution which is in resonance with Qj. The energy source of the instabilities lies in the T.. - T. temperature anisotropy. By making the magnetic field strong compared to the plasma pressure anisotropy the instability (anisotropic pressure instability) may be stabilized. There exists a higher order CGL theory and also a guiding center Vlasov theory of the Alfvênwave instability. The Alfvên-wave instability may be stabilized by finite Larmor radius effects. Resonance effects between helicons and Alfvên waves are known (low density pinch instability, growth rate
PLASMA INSTABILITIES
503
γ <υ c2/3feuy3 /14.112/). The whistler instability /14.107/ (helicon instability) is also an electromagnetic microinstability similar to the Alfvên-wave instability, but it occurs near fij, where adiabatic invariance is destroyed. The whistler instability is produced by electrons exchanging energy with a whistler wave (9.TOO). Since the transverse phase velocity of the whistler mode can be much less than the speed of light, there exist particles which can travel with the wave. This whistler mode is also called pressure driven whistler instability . It is stabilized by relativistic effects and by removing the anisotropy. In the frequency range üj « ω « QE there exists another whistler related mode, the current driven relativistic whistler instability /14.108/. It occurs in a plasma with low-energy thermal particles and with a group of hot relativistic electrons. The latter move in flat helices. The hot electrons further produce an electric current by moving along the magnetic field. The flat helices make possible a resonance between the hot electrons and a whistler wave, which grows. By decreasing the plasma density the frequency range for instability is reduced and stabilization may become possible. A current driven whistler instability (whistler current instability) appears however also in a non-relativistic plasma /14.449/. It can give rise to an anomalous and heats the ions. A current driven Alfresistivity vën instability driven by the resonant Landau and ^ transit time terms for propagation not parallel to B has also been investigated /14.449/. There also exists a surface-wave helicon instability /14.113/. Whistler instability may be destabilized (enhancement of the maximum growth rate) by the presence of cold electromagplasma /14.150/. The same is true for the netic ion cyclotron instability in a mixed warm-cold plasma (warm and cold ions) as well as for the ion whistler instability /14.162/. The fire-hose instability (hose-instability , garden-hose instability, also called transverse wave instability, high-frequency electromagnetic wave instability) /14.109/ may be considered as a whistler instability in a warm plasma. It may be termed macroscopic. For sufficient transverse anisotropy T^/T^ * 1 not only the bounce model pinch instability, a form anisotropy of the fire-hose instability, but also flute instabilities appear e.g. in a cylindrical pinch /14.110/. The fire-hose instability may also be excited by external gravitational forces combined
MICROINSTABILITIES
504
with a magnetic field in a stratified collisionless
plasma (internal
gravitational
instability
/14.111/).
The necessary and sufficient condition for stability against the (electromagnetic) firehose instability (see p 214) is P
l " Pll +
ß2/y
o
> 0/
(14.87)
against the (electromagnetic) mirror later and p 214) is 2 P i + B2/MO
instability
- pj/3p„ > 0,
(see (14.88)
and against the (convective) internal gravitational instability (in an external gravitational potential Φ) is ρ'Φ' + ( Ρ ι + β 2 / 2 μ ο ) · 2 [ ΐ / 3 Ρ | | + (1 - p 1 / 3 p | | ) 2 ?
2
•(2pi+BVy0-Pi/3P||)
-11
'J < o.
(14.89)
The primes i n d i c a t e d e r i v a t i v e s . The growth r a t e s a r e (9.136), i . e . Ύ = [^ 2 (P|| * Pi " S 2 / y o ) / p ] V 2 for t h e firehose
instability
γ = fc[6P|| - p2i/(pi
for the mirror
instability.
(14.90)
and + B2/2yo)]1/2/p1/2(14.91)
Propagation oblique to ~BQ may be described by the superposition of two plane waves exp [i (tût - kzz - kxx)] giving a wave travelling in and exp[i(ut - kzz + kxx)] the direction of 2 , but not a plane wave described by e*v[i(u>t - kzz)] cos kxx /2.6/. A magnetic field subjected to this wave looks like the one shown in Fig. 20, p 213 (mirror instability). The magnetic resonance between such a wave and particles travelling with the phase velocity ω/fe^ along the field lines excites this instability. No instability exists unless there is a minimum in the distribution function (integrated over o±) at some value of cu. If the particles have greater energy E^ than E\\ tney become concentrated in the region between magnetic mirrors thus increasing the particle pressure which expands the magnetic field etc. Let us now turn to electron resonances. In this case either a relativistic theory has to be used or
505
PLASMA INSTABILITIES
based on (14.80) instability in the frequency range ω * QE may occur for αΕ > *\ , e + hirL > cthE\\' in0)2 » 4keQg, b)pF » ^ühn^/m-r (electron cyclotron /14.17/, /14.114/, see stability, Weioel instaoil%ty later). For high frequencies (ω £ Ω# ω > ωρ^) the cold treatment is adequate and shoulâ be carried out relativistically. For low frequencies (ω « üj) we have the hydromagnetic limit, compare the firehose instability. The instability of the two circularly polarized transverse waves propagating along B has also been investigated using the Nyquist technique /14.115/ assuming infinitely massive ions. For isotropic distributions in the plane perpendicular to the magnetic field, growing waves with k < kQ are obtained for TjL > T|| , where kQ is given by
32
* o - 4 s ( ^ - 1)+ (1 -¥-LÎ4-
(14 92)
·
The frequency of instability at threshold k = kQ is ω = ±(1 - Tu/Ti)QE.
(14.93)
When finite ion mass is taken into account /14.67/ for ω ~ Qj the firehose instability appears. Even if there is insufficient anisotropy for this instability, Landau there are other instabilities due to inverse damping of ions satisfying ω - kc\\ - Ωχ. In this case of slight anisotropy the growth is small and given by Y = Ω
χ θ χ ρ[4(ί:" 1 ) ]·
(14 94)
·
All these electromagnetic instabilities (Alfvên instability, firehose instability) are unimportant in low beta plasmas and are of interest mainly for cosmic space plasmas. For plasma devices the electrostatic (longitudinal) Harris instabilities (cyclotron instabilities) are more dangerous. This is however not true in general. Under certain conditions /14.146/ the electromagnetic instability in contrastreaming collisionless anisotropic plasmas may have larger growth rates than does the electrostatic instability. Up to now we discussed electromagnetic waves propagating along BQ. We now turn to electromagnetic £ Q . Electromagnetic waves waves propagating across with λ « rL will be unaffected by the magnetic field. Later on we will discuss longitudinal modes also. When (14.50) is valid, electromagnetic transverse
MICROINSTABILITIES
506
waves {cyclotron waves) with E^BQ and mixed transwaves) verse-longitudinal waves with EIB0 (Bernstein propagate in an uncoupled manner perpendicular to ÎQ /14.117/. For isotropic distributions transverse waves are : always stable, but anisotropic distributions may become unstable. Transverse waves propagating across BQ (fe|| = 0) are described by (9.31), (9.190) which may be written in the form +oo oo
2 „2 ; ω„ I I cuJ (k c./iï ω - c k.1 - 2ÏÏO) L* PsJ J II n x Γ S , n
s
)
-oo o
(14.95) J
OS
dc.. °l "
J
S I
OS
ω I dc.. cl ηΐί
s
_
- ω
J
c
Qs\
\\ dc. // J 1
II
1
compare (9.203). Dispersion relations for wave propagation across the magnetic field in hot plasmas may
be simplified by introducing the magnetoplasma persion function
dis-
π
f {z)
u
=
sin* πωί°°3 ( ω φ ) o
ΘΧρ [
= ΎΖ exp(-2) I (z) n =-oo
n
"a ( 1
+ COS
φ ) ] α φ
=
ω w
n
see /14.165/. It is connected with the Gordeyev integral, see p 513. Deviations from a Maxwellian fQ do not modify the imaginary part of ω determined by (14.95). If all fQQ are Maxwellian, the modes are stable, see (5.25) and section 14.1. Instabilities near ω 2 belonging to n = 0 may appear only for anisotropic distribution. For distribution functions of the type (14.80) the necessary instability condition Υω 2 (1 - a"1) < 0 or Ψ > Tx (14.96) s I I 1 £ Ps may be derived /14.17/. If one assumes λ « rL, kcj_/ü « 1 , and stationary ions the Vlasov equation gives the dispersion relation /14.117/ 1 =
[ α Λ(ω
2
-5 2 ^ 2 )]+ (ω^ 2 <α, 2 >)/[(ω 2 -2 2 ^ 2 )ίΩ 2 -ω 2 )]. (14.97)
507
PLASMA INSTABILITIES
The c o n d i t i o n f o r s t a b i l i t y i s J2 2 2,^2 Ω >ωρ/ο and the growth rate is given by r 2^ 2,-2 η 2 Ί 1/2 γ =
[ωρ<ο„>/ο
- Ω ]
(14.98)
(14.99)
If ion motion is taken into account the dispersion relation is modified and the stability condition is P|| < #2/μ 0 . This criterion agrees with the criterion for non-existence of the firehose instability in the case of P\\ » Pi. Another analysis (Hamasaki /14.117/) shows that no instability of electromagnetic modes propagating perpendicular to BQ occurs if either ßn < 1 h growth rate is given P r i f athEl - cthE\V T The ° mΛVmaximum ■,',1 by i1/2 'thEl 'thE\i J Y = (\ !3"X - μ ' PE 2 2 (14.100) Ö'thEl | C^thEW j1n Also another growth rate has been given /14.117/ namely for λ -*■ 0
fV -
y2~- n|(1 - 1/0 ||E ) r è]}E=^nTUE/B2o.
(14.101)
Comparison with the firehose instability (propagating along BQ) shows that larger values of 3|]# - $]_E are required to excite the new instability. According to (14.101) it grows at a rate less than Ω^ whereas the firehose instability growth rate is less than Ω/. An electromagnetic microinstability of this type
is the anisotropic temperature instability, wave instability, Sharer-Trivelpiece temperature anisotropic transverse wave high frequency velocity space instability
transverse instability, instability, /13.31/,
/14.18/, /14.39/, /14.118/ (see also p 481). It arises when electromagnetic waves propagate across a magnetic field in a plasma with anisotropic temperature. This instability is different from the firehose instability for which the waves propagate along the magnetic field. The present instability has a higher threshold of electron pressure (and needs a greater T\\) to occur than the hose-instability, but leads to larger growth rates (14.100), (14.101). By decreasing T\\/Tior choosing other plasma parameters in a stable range, the instability may be stabilized. This type of instability may arise also in an unmagnetized plas-
MICROINSTABILITIES
508
ma and is then sometimes called transverse
ty /14.118/, transverse
wave instability,
instabili-
Neibel
in-
space
insta-
stability (which occurs however most easily for a wave propagating parallel to the magnetic field). Not only electromagnetic but also electrostatic modes may appear in a magnetoplasma. Perpendicular or oblique to the magnetic field electrostatic electron modes (due to a larger Τμ) may become unstable but also electrostatic ion-ion instabilities exist /14.118/· The mirror instability (see p 213 in volume 1 and p 504 is also a microscopic electromagnetic instability propagating across an inhomogeneous magnetic field and will be discussed in the next section. (The term mirror instability is however also used to designate the instability of mirror devices.) Electrostatic waves propagating oblique to the magnetic field in an anisotropic plasma with ^μ + Ο were first investigated by Harris (Harris instability,
velocity
anisotropy
instability,
velocity
bility, /14.119/). Later his results were generalized for T|| Φ 0. For isotropic electrons instabilities appear near ω - Qj for Tj\ > 8Tj||, whereas electron oscillations become unstable at ω - ηΩ^ for an anisotropic electron distribution if TE± > TE\\. If the electrons are assumed to be cold/ the ion cyclotron instability appears for ωΡΕ > Qj/2 and Tji > 2 ^ j | | /14.17/. Therefore electron heating is stabilizing since larger ^ I J / ^ J N values are needed for instability. The basic (nearly perpendicular) Harris instability has its energy source in the thermal energy of the particle gyration and is an electrostatic microinstability in a homogeneous magnetized anisotropic plasma. Since particle velocities in the direction parallel to the magnetic field are assumed to vanish (Tu ■> 0) and velocities of the transverse ion gyratory motion are assumed to be sharply peaked (e.g. around c*), a strongly anisotropic distribution for electrons fQ(e)
= const*
δ (c\\) 6(c,
- c*)
(14.102)
is assumed. Such distribution functions may be generated by beams or currents which may excite cyclotron instabilities /14.128/, but also strong electromagnetic waves may excite cyclotron instabilities /14.133/. This function has a hump, from which energy can be fed into a growing electrostatic wave at ω - ni]. The mechanism works also in the case of finite ion and electron temperature provided T^/T\\ is large enough. (Stabilization may be obtained by reducing this an-
509
PLASMA INSTABILITIES
experiisotropy) . T^ » Τμ is realized in injection ments. When the distribution (14.102) is inserted into (9.194), a dispersion relation for the Harris instability is obtained /2.10/ in the form 2 -h» O
(khk2)j2n{k
2
n=-oo
2
c*/«z
(ω/Ω^ - n) 2
2
(k l/k )n(J n)
ΏΕ
(ω/Ω^ - n) k^e*
(14.103)
= 1
(for ions at rest). With the Nyquist technique it can be shown that whenfcj_e*/ft> 1.84 there are growing threshold for the waves ω ^ Ω^ if ωρ?? > Ω„ \density Harris instability!. In plasmas of finite length the bouncing of the ions (or electrons) in a magnetic well shifts the frequencies of the unstable modes /14. 124/. The Harris instability has also been observed in an ion cyclotron resonance heated stellarator /14.125/. Of more interest than the Harris instability, for the Soper-Harris distribution function (14.102), is the instability, Timofeev isntability, which takes into account ion and electron temperature effects /14.120/. When in a fully ionized plasma confined in a homogenous magnetic field the perpendicular ion and electron temperatures exceed the longitudinal ones, one
speaks of a cyclotron
instability,
synchrotron
insta-
bility /10.2/, /14.121/. In this case the cyclotron motion can couple to a number of longitudinal or transverse waves such as Langmuir oscillations, Bernstein modes and electromagnetic waves. One may speak of electron cyclotron instability, electron cyclotron resonance instability (hot electrons, cold ions) and of ion cyclotron instability, ion cyclotron resonance instability (hot ions, cold electrons). The ion cycllotron instability may be convective as well as absolute /14.127/. Instabilities may however also be caused by cold electrons (whose density determines the upper hybrid frequency) and by hot electrons (loss-cone distribution, see later). This (convective) instability is called cold upper hybrid instability /14.155/. Also a low frequency flute cyclotron instability is known /14.452/. There are two types of cyclotron instabilities: either there is considerable particle motion across the magnetic field and a humped particle distribution
MICROINSTABILITIES
510
peaked at other than zero energy or there is no hump, in which case the particle motion along the field is essential /11.21/ (two-stream cyclotron instability). In order to investigate instabilities near harmonics of the cyclotron frequencies we start /9.6/, /2.10/, /4.11/, /14.119/ with the dispersion relation for quasi-electrostatic modes (14.51)/ (9.11). Using (9.194) in the form 2
e
kls ,πΩ I s \ ö. 1
= (1
- 4s/("2)6kl
dfJ
os 3c. ^L
«
3Jf
*
v
os li W 3 ο „ / ηr Ω " " H ' ^s
l^të
nω T n ski + kl· ' \\°\\
(14.104) ω
"
'
where Τη8^ι is again given by (9.186) one obtains the Harris dispersion relation X^4s(
2
?rrSfe
J
σ2(&.σ./Ω ) « I l s
ηΩ
+ K.G..
- ω
df
- Λ u
9e
i
df
os
ll9
i
c
*
(14.105)
= 0.
This ε is also called the dielectric response function, compare (9.162) for an unmagnetized plasma. Using the Plemelj formula (9.161) we may decompose ε into its real and imaginary parts J2(k.cjQ
-H
dcJ
n
(fe.c./Ω i l s
LA
)
§ηΩ
s
(14.106)
) δ (ω - fr.,0.. - «Ω ) || Il s
(14.107) II 3c | ( To have i n s t a b i l i t y one can c h o o s e df0/dc\\ or df0/dc\ p o s i t i v e . 3 / 0 / 3 c n > 0 means a two-stream p a r a l l e l t o field, b u t 3 / 0 / 3 ^ i > 0 or k | | 3 / 0 / 3 e | | + the magnetic
511
PLASMA INSTABILITIES
0 give a new situation. + 2niïsdf/dc2\> We may now investigate six special cases of Harris type instabilities. 1. cold electrons, hot ions {T^/m^« Tj/mj). In this case (hydrodynamic approximation /8.13/) we have k\ c-rp ι/Ωο - 0. Since the small electron mass is in the denominator of er (in the plasma frequency) the electrons contribute more to er than do the ions, whose contribution may be neglected. Integration by parts gives zr = 1 - ω 2 kj?/o)2fc2 a n d Zr = o results in = ü)p£>cos θ. Θ is again the ω = ß(fe) = Ω^ = upEKz/k angle between the wave vector and the magnetic field. For small growth rates we may use (9.166). This gives /9.6/ with ε^ from (14.107) the result
^kY^Psc s,n £ 6 (Ω. - fc..£.
2}dcJn(kici/Qs)
-
nü
,ηΩ
) i
(14.108)
so + k *f so] II dc{ dct
df
Ί ^1 A t w o - t e m p e r a t u r e d i s t r i b u t i o n f u n c t i o n of t h e Jf_
so = n s exp[ ^
gives Ύ?
"°Ϊ / α Ϊβ " cN/aNs
]/π
3/2
form
a,,Ils a,I s
/9.26/
= la. V 7 Τ Ί Η Χ ? t- fe î^e /2] xn ( f e Ks / 2 ) 5^2β,η k
(
Ζ'.
•lu* Ω ^—- ηruù Ω fe a
ll lls
a..
\
(14.109) 2ηΩ Il Ils
Ίΐβ.
ais
:[
k a
\\ \\s
}
-
This γ is of the order of (mE/mj-)^ '^ftj. Here a\\E is very small (cold electrons). Another result neglecting Landau damping for η 2 Ω 2 » k2T, k->/w„ may be found b "** in /4.11/. In the derivation of this result (14.109) the formula oo
I
2 2 T-2, v -v _ 1 -s / 2 T , 2 / o x KTn(s2*)e dr = ^e Ι^(β /2),
compare p 23 2 of volume 1, has been used.
(14.110)
MICROINSTABILITIES
512
Ζ^{χ) = /ï\e~x is the imaginary part of the FriedConte-function (9.170) both for small and large argument, but in the latter case only for Im x = 0. A graphical representation of γ^ as a function of tik = ß(fc) /9.6/, /14.119/, /14.120/ shows that near multiples of Ωρ there can be maxima of y, if ωρΕ > ntij. This is an approximate necessary condition for the appearance of the ion cyclotron instability. This instability is generated in the vicinity of Ω η = ηΩ by the coupling of the electron plasma oscillation and the ion Bernstein mode. This can take place only when ωρ# > nüj.This determines a minimum electron density nE > n2£2/4ïïS2mj for the instability. For large n the stabilizing terms 'V/n (aj?j/a?j) in (14.109) are predominant. This gives the condition that for Q(k) - ntijmodes will be unstable only if /14.120/
Γ
||Λΐ B all/all
K
A'
(14 111)
-
This means T\ j > T \\j or that the ion cyclotron motion contains excessive kinetic energy which may be fed into the electron plasma oscillation at Ω^ £ nüj. Finite temperature T,, is stabilizing. It might be worthwhile to mention also that ion cyclotron instabilities at half the ion cyclotron frequency have been observcyclotron ed /14.134/. Furthermore, there exists a thermonuclear instability /14.145/, which is due to a group of fast particles formed in thermonuclear reactions. The growth of this instability is proportional to the square root of the particle density. In a Tokamak this instability is suppressed by the magnetic field corrugation. Another instability in a cold electron hot ion plasma is the cross field current driven ion cyclotron instability /14.147/ in an E*B rotating plasma. It is generated by the coupling of an electron acoustic mode and a Doppler shifted ion cyclotron mode. 2. Hot electrons , hot ions. If the electron temperature increases artÉ if ωρ# < nQj-, an increasing negative bump near Q(k) = Ω^ = 0 grows further and can stabilize modes near the first harmonics of Ωρ. If the electrons become however too hot (TE/mjr » Tj/mj) , acoustic the modes ω - ^TEkßk\\/mj; become unstable (ion instability, magnetoacoustic cyclotron instability /14.458/) at ω < ωρχ. Since the energy of the ionacoustic wave is positive (9 Re ωε/dax > 0) they become unstable if the cyclotron emission of the ions is larger than the Cherenkov absorption of the electrons
513
PLASMA INSTABILITIES
and ions. For hot electrons the density criterion for instaThen the ion acousbility is then n-j- > n2B2/Ai\mj-c2. tic wave is the basic excitation and will couple with the n-th ion Bernstein wave. Soper and Harris /14.120/ have shown that the dividing line between cold and hot electrons is defined by 5 V (14.112) V < (Soper-Harris instability) and that T\\z < TjjVU + γ) / is necessary for instability. where y2 = mpT \\E/mET\\j (for Tj_ = T||/14.122/) Based on the Gordeyev equation and the Gordeyev integral, Timofeev /14.120/ had givinstability) reen a region of instability {Timofeev presented by 1
(14.113) T H WE Hence low T^j and high T^E promote the instability but 2Ti|j < TJ_J is also required for instability. A sufficient condition for stability is therefore T. T < 2T||j. Another scheme is given by Mikhailovskii T
/9.26/ for a plasma with 3.f0j/3öj_ > 0, athil ' 2cth\\Im If k^T^/m^c2 - < ω 2 / Ω 2 < 1 one has the hydro dynamic cyD
clotron m E^mI
ti
1
±1
Jrl
instability, k.. = ^,ηΩ^/ω , γ - ωρι for <ω < an ρχ/^χ 1 d the kinetic cyclotron instabil2
2
9
i t j / fe |j = nQj/Cf-fav, γ = kßTgüj/mjC . j f o r mE/mj < üdpI/Qj < < (mE/mj)^/2. F o r kBTE/mzc2 > 1 , ωρχ/Ω2 < 1 one h a s a 9
2
cyclotron instability, for copj/fij > 1 one has high frequency instability. Of interest also are electroin a plasma static electron cyclotron instabilities consisting of several components having different electron temperatures /14.149/. Under certain conditions a cold component may significantly change the structure of the dispersion relation even if the relative concentration of the cold component is small. The maximum growth rate may be much larger than that in the absence of the cold component. The presence of a hot component may decrease the growth rate. For isotropic electron distribution characterized by TE and an anisotropic bi-maxwellian ion distribution Mikhailovskii /14.459/ gives the following formulae: unmagnetized plasma: for ω « kc^^g, kc+^j, instability occurs if Tmax/Tmin > 1 + c2k2/upi, where
MICROINSTABILITIES
514 T
max
= ™**^LI>
Τ\\Σ)'
=
?min
m i n
^ll'
Τ\\ϊ> ' 9 r o w t h
rate γ - (mE/mj) ' ^thl^PI^' ^ ~ ωρρ/2· F o r a magnetized -plasma several cases have to be considered: (mj/mE) a) Tyj > Tj_j, kz = 0. For ω « QE and g « instability is possible for very large temperature anisotropy, i.e. if TJJJ > IJ_J {mj/m^) 3/2β-1 /2. b) T i r > q , J f fel = 0. Reasonably large magnetic fields affect" ônly^the electron motion. For β * ml/mE and 3 - 1 one has the ion cylcotron instability, γ - Re ω - fij. For kz < ®>E/cthE a n d 1 < ß < mr/m^, perturbations with k■ = 0 have ω - Ωχ. Then the interaction of resonant ions with waves can lead to instability. The growth rate is given by .2 7 2 r2 C k y Yi\m Qj. /2\j Ω z - 1|exp{ -2 2 Ύ = m k Q ιη Ez thE III J K PI z°thE
Αττ~
/ττΩ' k
z°th\\I
c-
H
2 2
Il
z
ωPI f ,v k\ c) T > T fe^ z ' Φ ~0, l * 0. The low-freUif ^ = a0,finite quency U instability"in pressure plasma is the ill ^ e ^re" mirror instability. For 11 j » Ij|j/ ω » ^zcth quency ω is purely imaginary. Tnis is the hydrodynamic mirror instability {probkotron instability) for $11 > T\\l/Tll* F o r Tll > T\\I o n e h a s instability for ω « k„e + i, when <-zcth 1 + k2z\\ + (f ^ ( ΐ 1 τ / ι , I - 1) |χ)/2]Α2 li mirror instability. is satisfied. This is the kinetic 3. #
ΡΕ
>
η
®Έ
or
*-he condition n% >
n^B^/A-nm^c^.
From a
bi-maxwellian electron distribution function for instabilities near ηΩ# (electron-cyclotron instability, electron-electron instability) the necessary condition for instability II E T
ÎE
2n
(14.114)
has been obtained /14.123/. It might be of interest to mention that the electron cyclotron instabilities suppress runaway and thus runaway instability (p 453 in /1.35/). High frequency instabilities /14.126/
PLASMA INSTABILITIES
515
arising from electron anisotropy are called electronelectron instabilities (if there is self-resonance for the electrons)· Ion cyclotron instabilities due to a coupling between the gyratory motion of hot ions and hot electrons (moving along BQ) are called ionelectron instabilities. One speaks also of cyclotron double distribution instabilities, mainly if there exist cold and hot particles of the same species. The excitation of electron plasma waves (electron acoustic waves) in a plasma with anisotropic ion distribution has been treated in /14.140/. In conducting media also acoustic-electric (electroacoustic) waves are known /14.456/. 4. Cold electrons, cold ions. In this case both the ion and electron distribution functions are represented by ό-functions of the type (14.102). Instead of (14.103) a dispersion relation for hydrodynamic modes in an anisotropic plasma may be derived from fluid theory. It reads /14.114/, /9.26/ for cos Θ = 1 (£lB0)
A similar result may be derived from the general dispersion relation by assuming ot^^E » c-^hWE' n e 9 l e c t ~ ing the ion contribution to the permittivity, assuming Ιω - nQ,E\ » k c^||F (neglect of resonant particles) , and by assuming K.rr^ < 1 (long wavelength approximation) . For Tu/T^ < f<-]_r^E one has nonresonant instability /9.26/. For short wavelengths also ^irLE > 1) a dispersion relation and an instability has been derived/9.26/. condition T\\/T^ < (kj_rLE^2:n)'''] and ions. The 5. Relative drift between electrons instability occurring in this situation is the twostream instability in a magnetized plasma, two-stream cyclotron instability, two-stream ion-cyclotron instability, two-streaming cyclotron instability /14.129/. In many cases authors assume a simple isotropic (Maxwellian) distribution for the ions and a displaced Maxwellian electron distribution, see p 488, /9.6/, /14.129/, /2.10/. One then obtains the hyd.rodynamic in a uniform magnetic field two-stream instability ion cyclotron instabil(electrostatic current driven ity)
and the formulae
(14.54) - (14.55). For Re ω > ηΩΣ
growing modes exist. For waves travelling at a large angle to the magnetic field the instability can be excited by an electron drift uQ along the field. When
MICROINSTABILITIES
516
the relative velocity uQ > ω/k^ the energy of the drifting electrons is fed into the Bernstein mode and a growing Bernstein mode appears. Its growth rate must overcompensate ion Landau damping. Since the mode propagates almost across the magnetic field, u must be large enough to exceed the phase velocity ω/feji along the field. If the relative motion of electrons and ions is due to different angular velocities in a rotating plasma, one speaks of the cyclotron resonance instability (in rotating plasma) /14.131/. A new almost electrostatic temperature anisotropy instability
has been found for a bi-maxwellian plasma in a
uniform magnetic field. Instability exists for 3M = > 0.591 when Γμ = 0 and ions are neglect= Ai\nkßTn/B2 ed /14.160/. The growth rate is -Ω^ and the polarization is almost electrostatic for almost perpendicular propagation. The instability is obtainable only from the complete dispersion relation. A drift of Maxwellian electrons (streaming along the magnetic field relative to Maxwellian ions) generates in a current carrying magnetized Vlasov plasma a whistler
current
instability
/14.449/ (ion beta >1,
^rLI > 1)· T his electromagnetic current instability has lower threshold than the electrostatic ion acoustic instability /14.161/. 6. Loss
cone
distribution.
In a plasma confined
between magnetic mirrors only those particles located in a certain domain of phase space, i.e. outside the loss cone, see p 56 of volume 1, can be trapped. The resulting loss cone distribution function is clearly not Maxwellian and is partly anisotropic. It vanishes within the loss cone Θ < θ 0 or for c^_ < I o\\ I cot θ , 4 / (
,e.=const)
O
m
I
Loss-cone distribution Fig,
48.
X
x
Double humped distribution Loss-cone
instability
517
PLASMA INSTABILITIES
but is isotropic for c outside the cone. The loss cone angle dc is defined by cos Qc = (B {s) /B (s0) ) " 1 /2. This anisotropic distribution function gives raise to the (electrostatic) loss cone instability, Dory-Guest instability, Dory-Guest-Harris instability, instability
of
mirror
traps.
Since the distribution function
f0 of cx integrated over Cy and cz is double-humped having a central minimum due to the lost particles, see Fig. 48, the appearance of this instability is understandable. For low plasma densities it is electrostatic and has frequencies which are harmonics of the ion cyclotron frequency. For finite beta an electromagnetic loss cone instability appears, see later. When we integrate an equilibrium distribution ,cz) such that function fQ(cx,c +00
f
•Ό
(Q
) = fff (c JJ
#
J o
— 00
,c
,c
x' y' z ^
)dc
dc..
y II u
and calculate its derivative at c~ = c we obtain ' X o /4.11/ + 0°
dfo (c x ) J ns v' * àc ^
/· /· ' o df Γ Γ "1 , 2 o2, 1/2 3c. O P"1J
C
hl· J
_
J œ
dc. ( c. - c )
J
1
;r c o ^ο ~ 1 o (14.115) may become positive for where cj = c~, + Cy. Now f'Q(e0) cQ < cm and we expect instability for ω/fej^ < cm. To investigate the loss cone instability, distribution functions of the Dory-Guest-Harris type fso{cl'cz)
^ ^exp[-cJ/aJ-^/a^]
(14.116)
have been investigated (I a positive integer). In the simplest case one might assume k^ = 0 (flute model) and one obtains from (14.105) the dispersion relation /9.6/ 2
Σ k~ s,n -^f ωρ
2
where
c
C
nti
VUJ SH
/ n i = °r '"V (ω - ηΩ ) '
. (JL £ » j Ä j a ;
(14.117)
(14.118) le. de. n\ Ω / y 1 1 s (14.117) has unstable roots for distributions of the type (14.116). Since it would be too tedious to investigate all sn
MICROINSTABILITIES
518
possible solutions of (14.105) for fen * 0, we make several approximations /2.12/, /4.11/. First we assume that the electrons are isotropic and cold, or ω » k\\ctfoE. (There exist, however, also investigations of the electrostatic electron cyclotron loss cone in/14.163/). Second, wavelengths longer than stability the electron gyration radius will be considered (kr^g«
1 or Ω# » ^l°lE'
small
Larmor
radius
limit).
Due to this assumption only the n = 0 term remains for electrons. Due to the first assumption we may expand 1
- fe, Then t h e d i s p e r s i o n 2 ,2 ΡΕΚ\\ 2 ω
ω
_
v2
K
k 1 \l°\\ = — + —V 1 - + .. . ω
relation
- Ιω ρζ 2 dc
(14.105)
+ 00
4 iί
n
(14.119)
fnti
de
reads 3/
oJ
de,
nj)oi
■ fc,iMvvy -o. Il 3c,
(14.120)
ηΩ Γ + k.| c.. - ω
w h e r e de = 2i\c]_dc±dc\\ , s e e ( 9 . 1 8 0 ) h a s b e e n u s e d . S i n c e ωρ^ » o)pj s e e ( 2 . 2 1 ) , t h e u s e o f Ω^ = ωρ^/cii/ki s e e p 511 a n d o f ( 9 . 1 6 1 ) g i v e / 2 . 1 2 / II ^ = ω^—p + k
ω
(Ω^ -
^ oJ
ηΩχ)·
3ÖM
2 2 . v 2ÏÏ ω t i
0
3/,
oJ + ηΩ X . 2 I j
ω
ΡΕ
fT2/*l°l\ J
* Ίΐο
• 2
(14.121)
e.|= (Ω, -nüj)
/k.
To be able to find instabilities Ω^ ηΩ should I! wecondition have o)p£ > ηΩρ which results in the density k\\
y =
k
2
2n2
ω ρ π nη Ω uT Ç , . 3 / 2 J n\ k\\(kBT £ I/mI)
2(Κ1°ΐ\Γοΐ\ dc±. Ω Γ Λ de. I 0 . 1 = 0 I I
(14.122)
519
PLASMA INSTABILITIES
The sign of this integral depends on c^ and its location relative to those c^, for which Jj^kj^i/Œj) is small, γ is positive and we have (the fastest growing) instability, when kxc > nüT (14.123)
Im
I
(because J^ has its maximum nearfcj_£j_s ηΩχ and when > k\_c \/Ωτ this maximum is within the positive kj_cm/iïj domain of (df/dcj)e., =0)· Consequently the rc-th harmonic may be viewed as having the same effect as a beam moving along the magnetic field with the velocity nQ,j/k±. It follows from this analogy that a plasma with an anisotropic distribution function can generate the same instabilities as a plasma with particle beams. The maximum growth rate is given by 2 γl « max
* ζ . n 2ttn2 T
The instability for which k.c.
loss
cone
instability.
(14.124) = nQj is the
resonant
A more refined theory of the loss cone instability distinguishes between the maser instability (convective loss cone instability propagating almost perpendicularly to the magnetic field ωρι » üj, k\\ < k\) and an absolute flute-type instability k^cy » iïj existing in the presence of a radial density gradient
(drift velocity space instability, loss cone flute instability instability, drift cone f Mikhailovskii instability, drift loss cone instability, gradient
loss cone instability /13.25/, /14.138/, /14.459/), requiring df0/dc± > 0 in inhomogeneous magnetized plasmas. This instability is caused by the coupling of an electrostatic ion cyclotron wave with an electromagnetic ion cyclotron wave. The effect of magnetic field inhomogeneity reduces the growth rate, but does not stabilize. Since the instability is highly resonant (ω - ηΩ « Ω) it might be possible to stabilize it by detuning the resonance. In a uniform plasma the loss cone mode is convective. To grow it needs a plasma length of the order of 400 rL /14.157/. The mode may, however, be reflected from plasma boundaries or from magnetic field gradients (negative energy loss /14.158/). At high densities the concone instability vective loss cone mode becomes absolute /14.159/. This mode may be stabilized by magnetic field gradients. The negative energy dissipative mode is a loss
MICROINSTABILITIES
520
cone ion Bernstein wave and becomes unstable if there is a source of dissipation in the plasma (electron Landau damping). Mikhailovskii /14.459/ defines the maser instability as unstable extraordinary electromagnetic waves with kz = 0 (perpendicular propagation) and E1BQ. From the relativistic dispersion relation ε22 = n^ (for ωρ « kc) the equation for the electromagnetic waves becomes 4π£ 2 η Γ ΐ^ c,J|23F/3p. ~2 7 2 c k , o -, -, >^ In *l _ r, Λ 1 + " —T ^ l^l^l^ll *2,-2.1/2-°· o M ω y n=-°° ω-ηΩ(1-£ /ο ) Here pu, p. are the relativistic momenta associated to the particle speed c and F is normalized such that Jp_LFdp|dp|| = 1. Then for spherically symmetric F and dF/dp - F/p the growth rate is given by 2 γ - ωρ/Ω.
This is the relativistic
synchrotron
instability.
There exists also a current driven high frequency loss cone flute instability /14.138/. Also electromagnetic (absolute) loss cone instabilities in homogeneous high beta or mirror devices have been found /14.137/, whereas electromagnetic waves propagating across the field become more stable in the presence of loss cone distributions. There exists, however, a whistler loss cone instability /14.142/ (γ-Ω^/2Ω^/2/β = = ω^υ/β for kc - ωρ^). Other authors who investigated the instabilities of finite pressure plasma with hot electrons and cold ions found that electromagnetic waves which correspond in isotropic plasma to whistler, fast magnetosonic waves and slow ordinary waves become unstable in the presence of the loss cone and are driven by the electron pressure anisotropy /14.143/. For a high beta plasma (g » 1) with electron temperature low compared with the transverse ion -> O, the high frequency loss cone energy TE/mj instability is^described for a transverse ion velocity distribution ό-function around c* by the dispersion relation /14.459/ 2 2 2 ωΩ„ Ω cos Θ . mw H ω (1 + q) I (k.c -ω ) ' where q = u 2 E / c 2 k 2 . Then γ * ω2Η/[k.c*
+k.c*
(u2E/c2k2)].
PLASMA INSTABILITIES
521
This growth rate has a maximum when k\ - ωρ^/2. The maximum is γ - ω^υ/ΖΙχ, ω Μ is given by (9.113). In the limit ω » k\c* the dispersion relation describes the whistler loss cone instability. For k » ωΡΕ/ο one has the electrostatic approximation and γ * nj,/2fl1./2, kyrLI = (mj/m£.)V2r kzrLI * 1 . For fej_ « ωρ#/ ßj 1 , where ^ 3j = c^/c\. β Also an electromagnetic ion cyclotron loss cone instability in counterstreaming ion beams has been - mj/mE at zero frequency and the found for T\\J/T±J first several ion cyclotron harmonics /14.148/. This is the case when in addition to containing hot ions > 0 one assumes that the characterized by dfi/dcj_ plasma contains cold ions with density n^ = anQ « nQ. We then have a double-humped distribution with a small fraction of cold ions. Assuming kz = 0, r JjI
522
MICROINSTABILITIES
tion instability. Its energy source is the gyratory motion of relativistic electrons. In the relativistic cyclotron instability one has κ·Β = 0, whereas κ·5Φ0 is required for the nonrelativistic case /10.2/, /14.132/. Furthermore, a loss cone type distribution function for the hot electrons is necessary. A decrease in the reflection coefficient of a wave packet at the boundaries decreases *-tie growth rate of the instability. Devices radiating at nüE from relativistic electrons have been called electron cyclotron maser. Other authors look for the plasma wave synchrotron instability, a longitudinal relativistic unstable mode with an approximate dispersion relation zzz - 0 and having a phase velocity close to that of light in an anisotropic magnetized plasma /14.135/. This instability appears together with the transverse wave synchrotron instability. The growth rate of the longitudinal waves exceeds, however, the growth rate of the transverse instability. It should also be mentioned that quite generally also non-relativistic electromagnetic (transverse) cyclotron instabilities are cited in the literature. They are nothing other than unstable electromagnetic waves propagating perpendicularly to a uniform magnetic field. Ring-type and loss cone type distribution functions may make these waves unstable for either zero frequency or near the cyclotron harmonics /14.136/. Although kinetic instability is very often synonymous with microinstability, this term has also been used to designate unstable cyclotron modes excited in a thetatron discharge, propagating obliquely to the magnetic field and due to an anisotropic electron distribution /14.139/, see also pp231, 238, 266 and 335 of volume 1 of this Handbook. The term is also used to designate the Bernstein instability {Bernstein wave instability) /14.140/. The term Bernstein wave instability may be somewhat misleading. If Bernstein waves are excited nonlinearly (as can happen by decay of waves) they can also become linearly (and nonlinearly) unstable in the form of anisotropic temperature insta, cyclotron instability and two-stream instabilbility ity
. On the other hand, the ion cyclotron instability
may be regarded as an instability of the obliquely propagating ion Bernstein wave generated by electrons drifting along the magnetic field. There exist also other temperature anisotropy instabilities. Recently, a lower hybrid temperature anisotropy instability in a uniform plasma /14.144/ has
PLASMA INSTABILITIES
523
been found. (The ordinary lower hybrid instability is excited in non-uniform plasmas, see section 14.5). The new mode is unstable for T^ = 0 and Tj = Tg for almost perpendicular propagation. This instability is in addition to the hose-instability for parallel propagation and to the Forslund ion instability with zero real part and to another electron mode. The latter two grow also fastest for almost perpendicular propagation. In anisotropic magnetized plasmas a self-trapping instability of MHD waves along the magnetic field and of cyclotron waves has been reported /14.152/. In an anisotropic magnetized plasma a microscopic self-gravitational instability appears, whose energy source is the self-gravitation energy /14.151/. The rotation of the magnetized anisotropic plasma modifies the instability criterion when the magnetic field and the rotation have a stabilizing effect on the condensation process. The Hall effect has a destabilizing influence /14.448/. High plasma pressures may stabilize the instability. The kinetic theory (Vlasov-Maxwell equations) of surface electromagnetic waves in a bounded hot plasma may be found in /14.457/. 14.5 MICROINSTABILITIES IN AN INHOMOGENEOUS PLASMA When a plasma is inhomogeneous, instabilities known from homogeneous plasma appear and may be modified. But new modes are also generated. This is especially true for a magnetized plasma, see chapter 13. First of all we will consider an unmagnetized inhomogeneous plasma. The question of stability for inhomogeneous plasma with non-monotonically decreasing distribution functions is difficult because no variational principle has been found. (For a plasma in an external non-uniform magnetic field a variational principle has been given, see later). A necessary and sufficient instability energy criterion for a collisionless plasma in a nonuniform magnetic field has been given /14.226/, but the stability against electrostatic perturbations of an inhomogeneous plasma has also been investigated by other methods and stability criteria have been derived /14.211/. Also differential equations in wavenumberspace have been derived /14.212/ {k-space tecknique) . Local Nyquist diagram analysis may also be found /14.213/. For a homogeneous plasma the equation for the electric potential in Laplace-Fourier space is algebraic, but for an inhomogeneous plasma
MICROINSTABILITIES
524
one has an integral equation. Usually the integrodifferential equation in #-space may be reduced even for an inhomogeneous plasma by the eigenmode ansatz §(x,t) ^ exp(iudt) Φ (x,ω) for the electric potential Φ with the assumption r^ « k~^ . The differential equation for Φ(χ,ω) is solved by the WKB approximation. If the plasma is described by a water bag model, then the stability of special configurations can be discussed by using the Nyquist technique /14.166/ or the normal mode analysis. The water bag model assumes that the phase space particles occupy a constant volume. This single water bag model (one bag in phase space) is equivalent to the closed moment description by the one-dimensional continuity equation (7.2), equation of motion (7.11), Poisson equation (2.3) and water the adiabatic law (4.87). At t - 0 an electron bag
distribution
function
is given by f^{x,c,0)
=A =
= const > 0 within a certain region, see Fig. 49, and fE(x,c,0) = 0 outside. Subsequently, fg{x,c,t) evolves according to 3/ -A _ e g a / g = o + aat dx m^ de
(14.125)
and
|f = -^e{nE
- nQ)
(14.126)
/=0
a
f=0
Fig.
49.
Water
bag
model
(x,0)
525
PLASMA INSTABILITIES
These equations describe the incompressible motion of the electron phase space fluid in phase space. When distort to new values the boundaries £+(#,0)/ ο_(χ,0) c+(x,t), c-(x,t) the electron density remains constant between the boundaries and vanishes outside: f
= A
o_{x,t)
for
< o <
c+(x,t)
rs 4- ·* = 0 outside. According to (6.4), (6.12) we have nE(x,t)
= \fE(x,e,t)dc
(14.127)
= A(o^(x,t)
e_(x,t))
-
(14.128) -f-00
vE(x,t)
+00
= \cfElx,c,t)dc/\fEdc
= (14.129)
1
= -^ {e+(x,
t)
+ o_ (x, t) )
+00
P E = m E\
(o - vE(x,t))
fE(x,c,t)do
= (mE/12)A(c+
= -
cj
3
(14.130)
.
Since the electron heat current (6.17) vanishes in the single water bag model we have adiabatic behavior and may write (4.87) in the form PE(x,t)
= p Q n" 3 n|(x,t).
(14.131)
Using (14.127) - (14.130) we may also derive by the moments method from (14.125) the continuity equation (7.2) in the form
ΎΓ + έ ( V s 1
=
°'
(14.132)
the equation of motion (7.11) in the form dV
and
dv
E
E
e
-rjr
+ v^-—
= - —
at
E dx
do a/
at
_
+ o, +,-
mF da a
1
E
! n
wmjr
'" = -—E
dx
m^
^E
τ^
oX
(14.133) (14.134) '
MICROINSTABILITIES
526
as well as Poisson1s equation (2.3) in the form (14.126) or dE
-r— = -4πβ(η 7Ρ -η ) = -4πβ (Αο^ - Ac E
OX
o
+
-
-n ). o
(14.135) On the basis of similar equations· a stability criterion has been derived for special configurations of inhomogeneous plasmas /14.166/. The equations may also be used to investigate electron plasma waves in warm unmagnetized inhomogeneous plasmas, since chapter 13 has demonstrated that the use of the Vlasov equation for inhomogeneous plasmas is cumbersome. If we assume that ions are stationary and if we linearize V n
V
=
E
E =
η
η
+
ο
ν
+
Eo
=
Ελ
ν
ΕΛ '
=
?E
E =
ΕΛ '
+
?Eo E
o
+
?ΕΛ = ^ 1 E
^ =
Ε
λ
(14.136) then for a given frequency ω = const and for excitation ^exp(ÎQ)t) by a grid the equations (14.132), (14.133) and (14.126) become for the space dependent parts n
. .
(x)
Ei
ννΛ (x) Ελ
n
=
o
dy
- ^ ST βΕ
=
λ
Ελ
- Ί^ α
2 -.
Ε
r
^ p tü)
E
ν
E1
η ^ω
o
dn
o
(14.137)
-3F' άη
Ελ
-τ^,
(14.138)
άχ
where (9.49) has been used, and dE -dF = '^^EV
(14.139)
If we differentiate this equation and substitute for dnE<\/dx into (14.138), we obtain 2 2 -eEa„ d ΕΛ (14.140) vwAx) = r1 + -A s_ ■ E
Substitution of ηΕλ gives the result άΕΛ 1
3
1=
4πβ da:
ax
o
from (14.139) into (14.137) n
di;^
£ω
dx
2
u
dn
ίω
dx'
El _ _£1
2
(14
141)
527
PLASMA INSTABILITIES
Differentiation of (14.140) and substitution of and vE^(x) from (14.140) into (14.141) gives άνΕ^/άχ after an integration the dispersion tron plasma waves in an unmagnetized plasma 9 ά Ε
ω2Ε
= ωρ ( x ) ^ - al
relation for elecinhomogeneous
γ,
(14.142)
άχ
where (2.19) and din η0/άχ « k have been used. ωρΕ = const, For homogeneous plasma aE = const, Ey (x) ^ exp(ikx) one obtains the Böhm-Groß dispersion relation (9.62). For din n/dx « k density gradients enter into (14.142) only through ωρΕ(χ) and may nQ(x) be given arbitrarily. A very similar dispersion relation has been derived by H. Motz using the VlasovPoisson system of equations and assuming the Miller force
(4.29) ("ponderomotive
for E 0 * 0 (hf containment). The Motz
dispersion
2
ω ^
relation
2
= u
pE(x)E^
force11)
of a hf field
reads /14.167/
a2 2 - — | A-L*{x)E
(χ)γ
(14.143)
ω da; where now the density gradient is determined by EQ 2 ωρΕ(χ)
2 = cop5,(0)exp
e
E 2
o
4mEu kBT9
(14.144)
Similar calculations have been made for linear acoustic waves in a density gradient /14.168/. A linearization (14.136) of the five ion and electron equations (14.132), (14.133), (14.126) results, for the assumptions EQ = 0, TE » Tj (ion thermal speed neglected with respect to the ion acoustic speed), Vp# » Vpj, mE/mj - 0, ω <^ωρτ, such that nE<\ ,x i n '" t,-h"-e -e q u a t i o-*n s ( 1'14. ΕΛ -" ^J1 4.137) * in
nj\9
ν
T 1( *{x) *;π > il
= "Ter -— = ^Ζ %u>mT da:
(14.145)
instead of (14.138) and in ηΙλ(χ)
= nQ[eV(x)/kBTE]
(14.146)
instead of (14.139). Calculation of V(x) from (14.146), and of Vj^ (14.137) with substitution into (14.145) gives
from
MICROINSTABILITIES
528
r!+ *<*>ii + kln = °>
(14 147)
·
άχ
where k% = u>2mj/k Τ^ = ω 2 /α| is the square of the wave number of the ion acoustic wave in a uniform plasma, h(x) = [1 /nQ (x)][dnQ(x)
/dx]
= λ/Lix)
and n(x) =
= rij^ix) /nQ{x) . Equation (14.147) may now be solved (leading to the for a number of cases like nQ = const case of a uniform plasma, or nQ = nQ(0)exp(αχ), leading to the Parkinson-Schindler dispersion relation /14.169/ (similar to the gravitational instability) k2 - ak - k^ = 0
(14.148)
of ion acoustic waves in a gravity supported plasma etc. Experimental situations often have nQ(x) = nQ(0) •exp(-x2/a2) resulting in 2 an 2x an , 7 2 (14.149) ö " ~~ô X" + kn = 0, -, 2 2 dx o da; a which may be solved using the confluent hypergeometric function /14.168/. As Budden has shown /14.170/ such an equation has interesting properties. Complete absorption of a plasma wave may take place but also a Budden tunneling factor exists. In general, equations of type (14.147) have to be solved with the WKB method (Wentzel-Kramers-Brillouin method), see p 371 (method of geometric optics, /9.13/, /9.4/). An asymptotic theory has been given for waves propagating in spatially and temporally dispersive, slightly inhomogeneous, and slowly varying media. The resulting eikohas been given /14.178/. For longitudinal equation nal waves (14.150) has been obtained. Geometric-optical methods have also been used for longitudinal oscillations in nonhomogeneous magnetized plasmas /14.234/. Another method for treating electrostatic integral modes in (cold) inhomogeneous plasmas is the equation
method /14.176/.
The theory described so far had a TE/Tj » 1 so that Landau damping could not be an important process. For TE = Tj- Landau damping as well as growth occurs (see p 477) in the relative amplitude n~/nQ, when the mode travels into more tenuous regions. In spite of heavy
Landau damping for Tg - Tj ion acoustic
waves can be
generated in a Q-macnine by electrostatic grid excitation. Calculations and experiments showed /14.171/ that in an exponential density variation, growth oc-
529
PLASMA INSTABILITIES
curs for λ > 2π£. For λ < 2T\L the ion acoustic mode is Landau damped in the same way as in a plasma of uniform density. (conversion
Inhomogeneities
layers)
may have vari-
ous effects: conversion of electromagnetic waves to electrostatic waves may occur /14.172/, see also p 212, which in their turn may produce energetic electrons /14.173/. This is due to the fact that waves of constant frequency ω satisfying (14.142) convect to lower and lower phase velocity as they propagate outward into regions of lower plasma density. The waves always convect outward, for if a wave is initially propagating inward, it will be reflected at the plasma cut-off ω = ω ρ / see p 184 (volume 1). For E^ ^ exp[ik(x)x], (14.142) may be written in the form ω 2 = UpE(x)
Since ω = const, = const de„7_
EÎL
dx
+ a2Ek2(x).
(14.150)
(3.40) gives with (14.150) and aE = =
J7 .
ω dk
7 2 dx
k
=
4πβ <2_,ω-η„ α η 7
""* "pti"PE 72 2
k
~"Ε
dx '
M41R1Ï u
ilDlj
α„τη„
If Cp » aE (no Landau damping), the high phase-velocity waves go to lower phase velocity quickly. Electromagnetic waves incident on a plasma at an angle to the density gradient can also be absorbed resonantly by linear conversion into the electron plasma wave if the incident wave is polarized in the plane of incidence /14.174/. Sources of electromagnetic waves are located outside the plasma or in low density regions. The plasma is inhomogeneous and resonances (k ■> *>, see p 184) appear in the high density regions inside the plasma. If waves are not reflected before they reach their resonance layer (conversion layer) the plasma is said to be accessible for this wave. Tunneling may occur if reflecting and resonating layer occur close together. A rough estimate of accessibility
(accessibility
con-
dition) for the perpendicular ion cyclotron resonance (9.119) may give the answer: the electric field in this resonance layer will be less attenuated by evanescence in a plasma than it would be in the absence of plasma /9.4/. The accessibility condition for the lower ion-electron hybrid resonance may be derived from the full cold-plasma dispersion relation. The hybrid resonance occurs for n 4 + 0, n^ -> °°. For n see (9.18).
530
MICROINSTABILITIES
Accessibility will require n| > 2. A ^nore precise sufficient accessibility condition is /9.4/ n\ > + 2, where D and P are given on p 183. For > \2D2/P\ the Buchsbaum two-ion hybrid resonance the same accessibility condition holds in the form n| > 2(1 + where y Q = k^c2(Ω^ - ω 2 )/ω 2 Ω?. + y0mE/mI), In concluding this part on modes in mhomogeneous unmagnetized plasma we would like to mention some special modes occurring in inhomogeneous unmagnetized plasmas: There has been reported a longitudinal instability in an inhomogeneous -plasma due to periodic alteration of one of the plasma parameters in time. The theory leads to a generalized Hill's equation. This instability /14.175/ in which subharmonics also develop should be termed parameter instability and will be discussed in chapter 17. Electrostatic marginal instability of a one-dimensional inhomogeneous finite plasma has been investigated in /14.177/. Whereas df/dE < O is sufficient for stability, sufficient conditions for instability if df/dE ^ 0 are derived. The spectrum of Langmuir oscillations in an inhomogeneous plasma has also been given /14.181/. In the geometric-optical approximation some calculations on cavitons have been done. A caviton or wave trap is a plasma cavity filled with electromagnetic high frequency (ω » ωρ^) radiation whose radiation pressure is balanced by the pressure of the surrounding plasma with arbitrary density profile /14.179/. Mainly in inhomogeneous plasmas multi-stage instability may appear /10.2/, /14.180/. In this case a certain plasma instability can affect the growth rate of other instabilities if the amplitude of the former becomes large enough to change the equilibrium plasma properties. In such a way conditions for the growth of other instabilities may then become realized which cannot be satisfied, if the original equilibrium remains unaffected. This is a quasilinear effect. Finally, we would like to mention the ambiplasma instability suggested by Lehnert /10.2/. The possible existence of an ambiplasma consisting of a mixture of matter and antimatter containing light and heavy particles of both charges leads to the assumption of new types of inertia and charge separation mechanisms thus, leading to new modes. Magnetized inhomogeneous plasmas exhibit drift instabilities which we discussed in chapter 13. There are, however, other new instabilities, and old instabilities are modified by the inhomogeneities. An inhomogeneity does not by itself necessarily lead to
531
PLASMA INSTABILITIES
instabilities. One must also have a mechanism for transforming energy of the steady-state motion of particles into energy of the instability, see e.g. chapter 13. (Examples are the bump driven drift mode, driven by a bump in the distribution function /14.453/ or the drift-superheating instability /14.496/.) Let us first discuss some general results /14.17/, /1.20/ (p 501), /1.29/ (Vol. 2, p 259), /9.26/, /11.6/, /10.4/, /14.299/ for magnetized inhomogeneous plasmas. To start with we assume that all modes are purely electrostatic, which means ß « 1 according to (14.53). To simplify matters a plane slab of a nonuniform plasma with a magnetic field ÎQ = £0e„(1 +εχ), a gravitational force gmex and a density gradient in the x direction will be assumed. The equilibrium distribution function fQ satisfies the Vlasov equation
Ofo/a* = o) u*B
m
o , s -> V f = 0 (14.152) + —ge + m— os e * x uJ os sL c s and may be constructed from the constants of motion, see p 368. The equilibrium distribution is (for small ε) M V J/
f
os
= nos(ms/2^5T)3/2exp(-msW2/2V' +
gxmjk^) (14
•Γΐ - ε·χ- ε-u / ß l , L
s
s y'
·153)
si'
see (13.40), p 368 of volume 1. Since we have assumed here a gravitational force, ε' is now given by -(din n
/dx)
=
e
s
"
gm /k T
s B
(14.154)
x=0
and not by ε^ =-din fQS/dx, see p 368. The introduction of gravity g permits simulation of curved magnet ic field lines, compare (4.21), (4.95). Equation ^14.154)^results from (14.153) and (9.1d), where 0 = leQfufosdu. From quasineutrality of the equilibs e'j = ε^. and le jfosdu rium? = 0 one obtain 2
s
SJ
* us
s
(14.155) s B s εΒ /4π = sJ nsw s (e'Lr/m - g), so that for g = 0 ε1 > ε (14.156) β « 2ε/ε ι / holds, see (13.44). Our calculations are valid for a slightly inhomogeneous plasma or for waves
MICROINSTABILITIES
532 k » din n/dx,
g « ttj.0
h
.
(14.157)
Physically, this means that the waves are low frequency waves (ω « üj), and that the ion drift velocities resulting from g and BQ are small in comparison to the thermal ion speed c^j. T ^ i s means that only the electron drift vE, see (13.1), in the perturbed electric field E>| vE = -eE^/mjiïj.
(14.158)
is of importance (here eBQ/c = mjüj has been used) gradient drift eu^/2QE and the and that the field gravity drift -g/QE, may be neglected for electrons. Linearization / = / 0 + f<\ , etc see p 369, leads to the dispersion relation for electrostatic drift waves (13.46). Before discussing this dispersion relation we derive another relation. Insertion of (14.158) into the electron continuity equation /7.24/ yields in first order (neglecting nQdvE^/dx) dt
= -v„-^
(14.159)
E dx
or for harmonic perturbations ^exp(-iü)t) ^ r^2. (14.160) nFl = 5Ί m Ω τ dx The drift of the ions due to gravity and due to the electric field E* (given also by (14.158) but with the other sign) gives in first order for the ion continuity equation 3ητ1
I\ dt
U s i n g rij^
9^τΛ
_ J o _ _£_ ~ ~VE dx Ωχ
% e x p ( - i o ) £ + ik η
^ητΛ
ΙΛ Zy '
(14.161)
y) we o b t a i n
-ieE« 9nr τΛ = Ô M J—JT, Γ - ^ 2 · 11 m Ω ω ( 1 - gk /Ω^ω) 8#
(14.162)
This was a macroscopic calculation. In a microscopic theory nlQ has to be replaced by +oo
oo
/K.U. n
Io
2 dv
+ ^ z$ l oiin)fol(uL'uz)n0ldul— oo
u j2
<14·163>
o
Quasineutrality eEnEo
+ eznIO
= 0, neglect of cyclo-
533
PLASMA INSTABILITIES
tron resonances and of all electron terms gives from the Poisson equation the dispersion relation k
= ω
PI. 2
ÙLÎ Ω
ωPI
1
d l n
\j
1 n
Io
dx
rslk
21
U.
8(F
y 1
lo»lo) dx (14.164)
u « du - / + 00
where Fz (u±) = 2π / fr (u,,w eral than (13.4 6) ■°° Using " ' ~~
)dur
This is more gene-
ou
f
_ 2 , λ -x J (a#)e n
o I (#) = i
, -α / 2 _ d# = e I (a 2 /2) 7 (14.165)
(ix) , n where I is the modified Bessel function, see pp 232 and 238, one obtains from (14.164) for a Maxwellian distribution function the dispersion relation /14·17/ # /9.26/ 2
k
y
=
ωPI n2 2 Ω ^
J
2
7
(.-.Io,„ - ι ) + - £ *
31η η Jo dx
(14.166)
-e-*IoU)
= 0.
y
y i 2 2 H e r e z = fc/2 a n d = 2 r 2 i s t h e averaged ion gyration radius, rL i s t h e u s u a l i o n g y r a t i o n r a d i u s . U s i n g y = ω 2 / Ω 2 , V = -(c%hj/nl04g)CènIo/dx), Φ(s) = = 1 -<=> z~LQ{z) one o b t a i n s from ( 1 4 . 1 6 6 ) t h e stability criwrion for electrostatic microscopic modes in an inhomogeneous plasma — + Φ 2/
\
4v
4νΦ
yl νΦ.
(14.167)
For rLI + 0 (macroscopic limit) we have s -* 0, > 0, so that (14.167) cannot be satisfied. For ν Φ Ο
534
MICROINSTABILITIES
the macroscopic theory results always in instability. In the microscopic theory (rL * 0) this is not the case. One speaks therefore of finite Larmor radius stabilization. After multiplication of (14.167) with y2 one may write /14.17/
(y - y+) (y - yj (y - y+) (y -yj
< o for 1 - o + ι/4ν) νΦ > o (14.168) > o for 1 - o + ι/4ν) νΦ < o,
where =
^1 - Φ(1 + 1 / 4 v ) / 2 ± /Ϊ^~Φ
+
2Φ(νΦ(1 + 1 / 4 v ) 2
-
(14.169)
1)
are the curves of marginal stability. For an infinite plasma unstable modes always exist. For a bounded plasma a minimum zm exists making stable modes possible. The curves from (14.169) are shown in Fig. 50, following Mikhailovskii /9.26/ It should also be mentioned that a cut-off at has been found for drift waves in an (ckBTE/eB)k^nt0 inhomogeneous magnetic plasma /14.232/. i,y
100stable 10-
1 -
unstable
io-1-
io- 2
stable
S£-
1—
—»
10
Fig.
50.
Curves according
of marginal to (14.169)
—i
100 stability
*»
535
PLASMA INSTABILITIES
The theory presented here is valid only for z = krL « 1. Without this restriction the dispersion relation for electrostatic modes in slightly inhomogeneous plasma reads / 1 . 2 0 /
- s,nΣ
-z I
w
we - 1 i ω + ηΩ - gk /Ω
ωPs
2 2 Ω rT s Ls -z
y
coe Ί U ) ω + ηΩ -ngk s/Ω
ττ?Ω ω s
-s U ) we + ï/(p)ω + ηΩ - k g/ü s
y*'
i \
B s kOT k B s yt mû ω
s
3In n, os 3# m
s^
IVs
(14.170)
3 In n os 3x
where ω + ηΩs - k .y* g /Ω ' s (14.171) 1/2 fc (2 /w ) || Ve e and W is given by Ζ' (-ρ)/2, where Z(p) is the plasma dispersion function (9.170), p 227. For k\\ = 0 , ω « Ω / n = 0, w e obtain the flute instability and finite Larmor radius effects. Due to W(°°) = 0 w e obtain (14.166). If If in (14.166) w e assume Zj- « 1 (rL -* 0) ω~ » Ω | w e obtain / 9 . 2 6 / , / 1 . 2 0 / for n ( nQ (x) dln n 0 fc feDTT din n r ol = 0 (14.172) 2 2 7 £ J OJ 9dx dx rrijtij ω - ω—*and the flute stability condition (see p 277) in the form 2 2 2 - k ΚΤΔΤ din n T ± y B I OJ (14.173) g 4 2n2 dx P =
m ΛΙ j .
and the growth rate (11.26) for rL ■+ 0. For ω « Ω n = 0, °thE y ωΑ|| > cthl' w e o b t a i n from_(14. 170) the drift Then (14.170) instability (Wz - 0 , WE = -1 +ix/H). reduces to dispersion relations of the type (13.61) , zE < 1, or the like /1 .20/. For ω = Ω τ =fet^feu= 0 , ζχ»λ ω « ÇlE, n = 0 for electrons and only n = -1 for ions, one obtains the Mikhailovskii-Timofeev instability , drift ion cyclotron instability, drift cyclotron re-
MICROINSTABILITIES
536 Qonance instability. For Tß = Tj, lim e - 2 I 0 ( 3 ) = ( 2 π ^ ) " 1 / 2 (14.170)
/4.11/
η% = n j , reduces / 1 . 2 0 / ,
to 1
(1 +fe yVD/(u)("
1 + ω - Ω^ 2 πζ J
\
1
7
2,Λ2 ^ y D
2 λ LE (14.174)
compare (13.77). Equation (14.174) indicates two waves: a drift wave ω = -kyVp and an ion cyclotron wave. If k is chosen properly, instability is possible. The stability condition is m
,2 B
n^T\2
r,dln
2
o i
da:
A-nnm a
(14.175)
) ·
We now return to (14.166). For z « 1, y » 1 one obtains the theory by Rosenbluth, Krall and Rostoker /14.182/. This theory may be derived from (14.152), (14.153), (14.154) etc which lead to (13.46). Expanding the Bessel functions and neglecting λ^ « ν^, this dispersion relation reads kz[
[[Va " ('k„TB s ω
b
Y ~ ω-kg/ü
-
■
(14.176)
2
k zk„Tjmntll\
S
^)
B s' + kezkT}T D
s
S
s. ■dz
/mû S
= 0.
S
A Nyquist analysis shows that there is one unstable mode if ε' > msg/kßT > 0. The frequency of the mode is given by ω
ke'IkBTI = 2
rrijüj
t
mIîlI
)
~A3ZE
. (14.177)
This should be compared with the macroscopic result ω„ = iSge', see (11.26). Now the system is stable if kzlk„T/mTSlT o
l
> 2ω„ l
n
(14.178)
(finite Larmor radius stabilization). The case z « 1, vy « 1 in (14.167) has been discussed by Kadomtsev /14.183/. If FlQ(u\), in (14.164) is a Maxwellian distribuiton with a temperature gradient χ = 3In Tj/dx then (14.164) becomes /14.17/
537
PLASMA INSTABILITIES
4. ^
Μ
"-Tin
"F1
dln n
°
J
(14.179) -
gky/üj
ω
where Q = 1 -χ^(1 - I ^ I Q ) [din n^/da;]" 1 . An analysis analogous to (14.167) - (14.169) gives (14.180) -.1 -Φ(0 + 1/4ν+ (1 -0)/Φ) ±/Q(1 - Q ) a 2 2Φ(νΦ(0 + 1/4ν + (1 + 0 ) / Φ ) - 1) A detailed discussion of these curves of marginal stability shows that a temperature gradient parallel to the density gradient stabilizes modes of short wavelengths (temperature gradient stabilization) . The larger x(dln nQ/dx)~î is, the longer may be the wavelengths where this stabilization is effective. Up to now we have considered stabilizing effects of finite Larmor radius in the frame of the Vlasov theory. The same results may be obtained in a guiding center plasma theory /11.6/, /14.184/ or even in a macroscopic anisotropic MHD theory, in which the stress tensor (6.10) is calculated from a local anisotropic Maxwell distribution. For a magnetic field in the ^-direction one has to write /14.184/ ±
(14.181)
where pi is defined by the usual Π-,*, Π 2 2' an<^ v k =· Comparison with (6.21) indicates that the additional terms describe viscous effects. Therefore (14.182) is called gyroviscosity (transverse ion viscosity). In this fluid theory one has to use (7.42) instead of (4.82) . If the Hilbert prescription (see p 91) is followed with fM in (6.1) in the form
MICROINSTABILITIES
538
3/2 ffie
m
t+ ^Ι = ns(x't)\2vk„Ttift))
s
^
e X
I1 [v -u(x,t)]2\ m Pr2 s"T^l¥7^—l
ΰ * (14.183) {local Maxwell distribution), then one obtains /14.187/ from the Boltzmann equation (5.8) and (5.33), (5.34) , (6.6) u-v = c, (6.10), (5.36), (6.17) the set of MHD^ equations in the form (6.5), (6.7) and (6.16) with Π not given by (6.21) or (6.22) but by Π = p - r]S
XX
^
XX
yy
zz Π
ffi/
n
X2
Π yz
1 +
= p
^ Ω 2 τ 2 [ yy
'Τ~Τ6Ζ2ΓΊ\ιS 22
üü 1 + Τ9Γ Ω i"T LTJT L
=Π
-η
yx
1 +
= n =Π
zy
Λ 1
+^ti2TT2T{S
9 I I
[«
4^2 2|Sxy 9ΩΙτΙ 4 22
SX
9 I I
A3
5
, 1602τ 2. +-9"Ω L yz
+
yy
z/z/
zz + S
3 I I yz ) -4^
22y
T
S
3 TI IT yz
3ΩΙτΐΛ:ζ
(14.184)
|a I x I xyj - ±Ωττ T(S -5 ) 3 I I yy zz
where η is given by (7.78), τ χ = v j 1 = (5/2/1) •{mj/m-g)^ '^τ E, compare (2.33), τ^ = 4 v ^ , v^ £see (2.27). 5^^ is given by dv
'feZ
k
dv
l
2
(14.185)
In the derivation of (14.184) it has been assumed that the magnetic field is oriented along the #-axis. Staof instabilities can be obtained by the bilization gyroviscosity terms, see e.g. /14.463/. Although in general finite Larmor radius effects orbit instability stabilize, there exists a finite {finite
Larmor
radius
instability
/14.185/).
Thermal
energy of the plasma in the form of drift motions is the energy source of this instability. The inhomogeneity of the magnetic field in toroidal devices causes periodic guiding center drifts {bananas , see p 162) which yield particle displacements larger than rL, see (12.87). Since this is due to col-
539
PLASMA INSTABILITIES
lisions, we will discuss the finite orbit instability in section 14.6. We would only like to mention here that the periodic motion of guiding centers around the field lines leads to another collisionless viscosity guiding center gyroviscosity (due to finite orbits) in the same way as the motion of charged particles in Larmor orbits leads to a magnetic viscosity (gyroviscosity). When discussing finite Larmor radius effects we neglected the diamagnetic current, which is necessary to produce the pressure gradient. Pfirsch /14.17/ and others /14.186/ have shown that the diamagnetic current acts like a destabilizing gravitation and produces drift instability. To see this one has to take into account the relative motion between electrons and ions which is due to the pressure gradient and the gravity. From (13.1) , (2.41), (2.69) we find the electron drift in the presence of density and temperature gradients to be
t>niP = Here again DE eBQ/c
k^T ,~Ί m 31n n T\ B din T ol) (ΛΛ 1 Q £ v r-— — + 1. (14.186) m ti \ dx Sx > T T = m-j-Q-r has been used. Insertion into
the full electron continuity equation results in -ΐωη^ 1 = ~ί1<νΌΕηΕλ
- vEn'EQ{x)
(14.187)
instead of (14.160). So we have ieE^ η
Ελ
- ω/τ? βχ(1 -kvDE/u)
dnEo
~~ΖΊΓ
(14.188)
which replaces (14.160). From the full ion continuity equation -iomj- = -Vjpnj0
- giknj^/Qj-
- Vpj-iknj^
obtains instead of (14.162) the result η
ΙΛ
=
-ieE*
dnT
^ Ω ω ( 1 - grfc/Ω ω - kvDI/u)
Τ^Γ'
one then
(1 4 β 1 8 9 )
This shows that the diamagnetic current -Vpj = +vDE acts like a destabilizing gravity. In accordance with Low's theorem (p 267) drift instabilities are new instabilities and are not found for g = 0, V-Q^ = 0. Since drift instabilities have been discussed in chapter 13, we make here only some remarks on recent literature. When the temperature gradient (perpendicular to B0) comes from the ion temperature one speaks of the ion temperature gradient instability, tempera-
MICROINSTABILITIES
540
ture
gradient
driven
ion acoustic
instability
/14.188/.
When density gradients, temperature gradients and collisions are involved, the growth rate formula may become lengthy. Sweetman gave /15.6/
1/2 *
γ
=
π ' ω k
^ " \\°m
+
2ν π ω* E
*YA^VywSV
(a + e + ß 2 )fe^K^(l + ^ ) - | ε ^ ( 1 + β ) + - ^ i r d + ß)
ak V
l DE Here the first line describes collisionless and the second line describes collisional effects. The first term in each line describes density gradient effects, the second term temperature gradient effects and the last term describes current effects, ω* is given by (13.57). The current driven collisional mode is described by γ - V£
magnetic field the drift ion (electron) cyclotron instability, anisotropie cyclotron instability /14.189/ plays an important role. Ion viscosity and heat conduction clearly have an influence on drift instabilities in an inhomogeneous (weakly collisional) high beta plasma /14.190/. Linear instability of a qua^istatic guiding-center plasma due to non-uniform E*B drift is investigated in /14.191/. Low frequency instabilities in magnetized plasma with non-uniformly compressed flow (div $% Φ 0, ι>Ε electric field drift velocity (4.6)) are considered in /14.192/.
Magnetic
drift
instabilities
due to V£ have been
discussed in several papers^ the plasma-wall interaction through uncompensated B-ripple convection in toroidal devices has been investigated /14.193/ (drift loss cone instability), electrostatic long wavelength instability in a perpendicular shock due to 5-drifts is discussed and reviewed /14.194/, instabilities of drift magnetosonic waves due to the magnetic drift resonance in finite pressure plasma are considered /14.195/ and V£ drift instability in a Tokamak has been investigated /14.196/ as well as low frequency instabilities in a V£, Vn plasma /14.225/. A minimum drift velocity for instability in current-carrying non-maxwellian plasma has been found /14.225/. In spite of their small relative number the high energy fusion products can excite low-frequency drift modes in a uniform magnetic field, a-particles may enhance or reduce the stability in the kinetic or hydrodynam-
541
PLASMA INSTABILITIES
ic regime of density gradient modes /14.461/. In paper /14.196/ a new instability/ the charge separation gradient B drift instability in a torus has been found. A new drift like instability has been detected in a density modulated plasma for λ^ > rLj /14.197/). Cri(density modulated drift instability tical density and temperature gradients required for the onset of drift instability as a function of shear and other plasma parameters are reviewed in /14.189/· Digital spectral analysis techniques have been applied for the experimental drift wave study /14.199/. and of radial elecAlso the influence of non-uniform on drift instabilities has been studied tric fields /14.228/. The two fluid high beta microscopic nonlocal drift instability in a collisionless theta-pinch has been investigated in /14.460/. Several papers have been concerned with the drift mirror instability (drift cone instability, gradient loss cone instability, drift cyclotron loss cone instability (DCLC-mode) , ion cyclotron drift loss cone instability, also Dory-Guest-Harris instability in inhomogeneous plasmas /14.200/, /14.459/. If σ/_|/3/_|_>0, the gradient loss cone instability can be excited. Its growth rate depends on the density gradient. The instability appears mainly for 3 - 0, but can also appear for finite beta. In an inhomogeneous plasma (and the plasma in experimental devices is inhomoge(14.116) generates a neous) a loss cone distribution drift loss cone instability. The main theoretical problem with this instability is its apparent absence in high density mirror machines when u>pE * ΩΕ. Coexisting cold plasma seems to have an influence on the growth rate. For ÎQ{y) = BQ (1 - e<\y)ez , nQ = n0(1 -
lls
2u
\\s4src2ü2s-
=
Hasegawa / 1 4 . 2 0 0 / d e r i v e s the dispersion relation for the drift mirror instability in the form (but see a l so (13.80)) ω =
Wj-ife||W||J/2Ä!T||J/(2'iJß1J)
k2
(14.191)
4n+I(3Is-3|ls)/2)+1+Ißis(1-Tis/T||s) ■ K. .
S
S
MICROINSTABILITIES
542
where ω_ is a modified drift frequency given by ω
*=
Kk u
ï f\s/Çls'
(14.192)
compare (13.57). In a two-^cqmponent ion plasma {h = hot, w = warm ions) with E*B drift the DCLC mode has a more complicated dispersion relation /15.6/, /14.459/ ω ω ΡΕ 1 + - ^ + 2 PE 2-2 n 7
PI , 2
ωPE
utiEk
ω
rp
ΡΙ
7vZ
1
-L
(14.193a)
)-o. ( 2IL·^ 3
Σ ω -ω„Ω T\dciciJî(-itr)
U "-■ -o ^ ""l where r>p is the plasma radius and F(o1) =
2 ft
1
2
+ -5 exp I -
expl-
p«
Stability is possible if ,fec
= ^ , d ClJn(lÇ7 'a o
2
Ω I /ïîkc, c 'h ph w \ "u°J , For large m i r r o r m a c h i n e s isfe_j_l a r g e , Qj/k - βρ^ and Δ ~ Cp^/2c? is n e c e s s a r y for stability. For an ion-electron plasma, for β ->· 0, and ωρ^» Ω 2 the equations n-jj = n-j^, neglect of the electron thermal motion, the hydrodynamic equations of continuity and motion for the electrons and the ion Vlasov equation give in electrostatic approximation /14.459/ the dispersion relation Kfc Ω m i ] 2 J + s ; uJ2{k c /ÏI ) 3 / 1 ο T T ^ I ^ I = 0.(14.193b) 1 +-^H +-i-4l Σ ω-ηΩτ 3 e, 1 1 7 2 7 2J *-^_ Here a g a i n κ = 31η η0/ΰχ.
For k^i/üj
»
1 this
re-
543
PLASMA INSTABILITIES
suits in γ - Re ϋ =ίfijand kr^ - (mj/m^) 1 /^. For finite 3 > (m^/mj)^/^ (non-electrostatic case k < ωρ#/2) and k,c./Ωj » 1 the relation /14.459/ 2 ωπ„ Kk iï^ m-r +~ (14.193c) 1 ~2 7 2 L^ ω - η Ω τ 72 73 3 ti
LI
has been obtained. Then instability is possible for K > 2m^pß/mjo or for g - 1 if KrLI ^ 2 (m^/mj) 1 / 2 /14.459/. The growth rate is γ - Ω ^ 1 / 4 ( 2 π ) " 1 / 2 •(^j3j)"1/4· For 3 > 1, in addition to considering Vn 0 also V5 Q has to be taken into account. The electrostatic and also high frequency instability is due to resonant coupling between drift waves and cyclotron motion in an inhomogeneous plasma with anisotropic loss cone ion distribution. Stabilization by finite beta seems to be possible /14.25/. In some experimental devices 10 4 x the threshold of the instability is reached without the instability appearing. The effect of nonuniformity on anisotropic distributions (e.g. the drift cone instability) may also be (13.73). Midescribed by the drift kinetic equation khailovskii discussed recently the drift mirror instability in a plasma of finite pressure with hot electrons /14.201/. Electrostatic as well as electromagnetic drift instabilities can exist in non-uniform plasmas with loss cone distribution functions in inhomogeneous magnetic fields. The same is true for anti-loss-cone (inverse loss cone, reverse loss cone) distributions/ which are found in toroidal devices and in the magnetosphere /14.202/ and which look like f(u^,u.)
' a.a..)
= 2ηο(π Γ
2, 2
(u„/a..) 2,
(14.193)
2]
exp L" M ||/ a || " w . / a . J . In an anti-loss-cone plasma a new high frequency
ti-loss-cone-ins
tability
(anti-loss-cone
drift
an-
insta-
bility) occurs with a growth rate ^0.1Ω^ /14.491/. Drift mirror instability (instability of a mirror machine) and mirror instability (due to local mirror/ see Fig. 20, p 213) should not be confused. The drift mirror instability (PVC f Ss f vT 1 Bb 1 fE 1 LM'tRpA 1 I) (see p 253 for the meaning of these symbols) is a low beta electrostatic flute-like particle resonant drift in-
MICROINSTABILITIES
544
stability in an inhomogeneous plasma with a loss cone distribution. The mirror instability (PV'C'SsT'Bbf'ELM1t'R'A*I) is an electromagnetic high beta nonresonant instability. It is due to the magnetic mirror action concentrating charged particles in the weak field regions of an inhomogeneous plasma. The increasing transverse plasma pressure in these regions pushes the field lines apart and increases the mirror ratio. At sufficiently large temperature anisotropies and strong pressure gradients a growing mode is generated, whose energy source lies in the T|-T|l anisotropy. By making the magnetic field strong compared to the plasma pressure anisotropy, the instability can be stabilized, see (14.88). The instability condition for the mirror instability may also be written /14.203/ (for a homogeneous plasma) 1 + Eßi8(1 - Ti8/T]l8)
< 0.
(14.194)
S
In an inhomogeneous anisotropic plasma also electromagnetic waves may become unstable /14.204/. When they propagate perpendicular to an external magnetic field, B (x) = BQC\ +ex)
-3/2 - 2 - 1 Γ 2, 2 = ft π α , α,, e x p - u . / a . os is \\s ^ L 1 I s
os
1
0
1-
0
0
0
2/2 1 "U../au II' II eJ
\ i
( 1 4
1 9 5 )
\\
( ε ' - δ . u./a, - δ,, u../a.. ) x + u / Ω )}. is Γ Is | | s W \\sl \ y' sl\ \ s
For wave propagation perpendicular to BQ the dispersion relation sçlits into ordinary and extraordinary mode (K = k~ey, E-jllB, see chapter 9 ) . The dispersion relation in local approximation contains: 1. drift instabilities for ω ρ < nüj « 9,j-r c2k2 » ω 2 , ω^ » kzu\/2tiE. 1.1 resonant electron instability ωρ > 0 1.2 ion drift cyclotron instability ωρ < 2ΩJ 1.2.1 resonant ion instability γ -* 0 1.2.2 off-resonance drift cyclotron instability
2. temperature-anisotropy instability for Ω Ι <<: ω ρ <:> keul/2üE.
545
PLASMA INSTABILITIES
Another general dispersion relation for high frequency electromagnetic drift waves in the range Qj « ω « Ω# arising from an electron stream drifting in a density gradient in a magnetized plasma may be found in /14.205/. Waves in a magnetized inhomogeneous high beta plasma are treated in /14.206/. Helicon waves in a non-uniform plasma are discussed in /14.230/. Reflection and absorption of electromagnetic waves in a hot magnetized inhomogeneous plasma are investigated in /9.4/. The negative pressure instability of the ordinary mode is discussed in /11.60/, /14.207/. In a finite 3 (>m^/mj) inhomogeneous plasma the filamentation instability of high frequency electromagnetic waves may occur /14.208/. Both instabilities_are nonlinear. "Or sufficiently long parallel wavelengths the transverse modulation of the electromagnetic wave gets coupled to drift-Alfvên waves and leads to a modula-
tion
filamentation
instability.
The filamentation in-
stability is a parametric effect and will be discussed in chapter 17. When waves have frequency near the
hybrid frequency the hybrid instability, nance instability, lower hybrid drift
hybrid instability
resomay
appear /14.209/. A theoretical model valid for the of a low beta plasma sheath region (thickness of rLj) has been given for the lower hybrid instability which can be stabilized by high beta effects. The instability may be understood from the fact that the sheath consists essentially of a homogeneous ion beam moving along the sheath at a constant velocity, and of electrons of zero temperature /10.2/. The beam is perpendicular to the magnetic field. It represents the macroscopic flow of matter due to the pressure gradient and Larmor motion at the boundary of the confined plasma. Due to resonant coupling between the ion beam and electrostatic oscillations at the hybrid frequency , a growing wave develops. This mode is similar to the loss cone instability, but the driving force is here due to the anisotropy in the ion velocity produced by the ion diamagnetic current. The beam velocity is the energy source of the instability. When the beam couples to non-electrostatic oscillations an instability appears also in high beta plasma. In addition parametric excitation of the upper and lower hybrid instability by high-frequency electric field is possible. Radio frequency energy can be carried towards the lower hybrid sheath by means of electron plasma waves. In the presence of magnetic field gradients , the lower hybrid wave can be absorbed through
MICROINSTABILITIES
546
collisionless damping at cyclotron resonance. In the low drift velocity regime (i.e. of the order of the ion thermal velocity) and for finite beta a magnetic gradient drift-wave resonance can be important and is stabilizing or destabilizing. Substantial plasma heating and a large anomalous resistivity can occur /14.209/. Plasma inhomogeneities also influence other instabilities. For the E*B-drift instabilities, cross field current instability a temperature gradient increases the growth rate, but Vrc and VB have little effect /14.20/. For sufficiently high beta however, VT is stabilizing (temperature gradient stabilization). A local plasma inhomogeneity may itself also propagate. The spread of the plasma nonuniformity is determined by the greatest diffusion coefficient values along and across the magnetic field. The ion acoustic instability and also instability due to the relative motion of electrons and ions in a magnetic field /14.235/ are also modified by non-uniformities. For a magnetic field BQ directed along the 3-axis, a low beta plasma equilibrium depending on x and perturbations ^expdkyy + kzz ) of the equilibrium one obtains /11.100/ distribution functions fos(x,u) from the Vlasov equation in drift approximation (ω «
Qj)
fJ
= — T 5 Ί / - ^ + — E - τ ^ . (14.196) ω + k ult x Ώ ** ™ * a,, i z II For E - -^Φ the dispersion relation can be obtainThe ed from div j * ikzUz = 0, j z = \eQ\uzfQduz. transverse current components are 3 suppressed by the magnetic field and do not enter into the stability criterion. Then the dispersion relation assumes the form s
u
[
(ok
df
e
df
21
* U-f- 4 ^ + — 4
,
d
" -0.(14.197)
ω + k u \B k dx m du I z z z \ o z s For Maxwellian ion and electron distributions with ^-dependent temperature one obtains
(dT /dx)/emTB * 0 (14.198) ω 3 - 3k2k cknT z y D o o i o {din T0/dx) . In a which holds for k /kz » eB/cy/k^T0mT toroidal plasma confinement device the current-driven ion acoustic instability exists only in the collision-
547
PLASMA INSTABILITIES
> al regime where veff ^bE' the bouncing frequency of the electrons (14.306). In the trapped electron bana< ω ££) t h e i nsta bility is inhibited na regime (veff /14.445/. This inhibition demonstrates that in the banana regime the entire Ohmic heating current is carried only by freely circulating electrons. This behavior of the current driven ion acoustic instability is of relevance because its anomalous resistivity may be useful in plasma heating and may explain the anomalous skin effect in the penetration of Ohmic heating current in large toroidal plasmas /14.446/. Fission driven ion acoustic instabilities in a uranium gas core reactor of finite geometry in inhomogeneous plasmas have also been investigated /14.447/. Bernstein modes also become modified in an inhomogeneous plasma /14.224/; they may become unstable {electron cyclotron drift instability) by interaction with ion acoustic waves. In the presence of weak inhomogeneities the propagation parallel to the drift velocity of the w-th Bernstein mode is described by the dispersion relation (Sanderson /14.224/) 2
7
T
K
/
TP
*
*_ζ.(_ω
v
fe
+ ! +
( ö . 7 » UT)
th
J
E
. Ζ(ξ)
^
+
+ /72Ω-,
E
E Z> (ξ)
°th
uE /Ta
I
2a
=
(14.199) ω
kuT/2*
Linear electrostatic instability of a low beta plasma in a magnetic field has been studied for modulated distribution functions /14.225/ of the type /
= nQf(u
. ,u..) Pi + 2acos(a[x
+ u /Ω^.])].
(14.200)
Periodic magnetic fields have also been considered /14.231/. Ion-acoustic instabilities in magnetic pulses /14.227/ as well as instabilities in helical magnetic fields /14.233/ have been investigated. Besides the instabilities which are modified by plasma inhomogeneities and besides the drift instabilities there exist other instabilities which are excited only in an inhomogeneous plasma. One example is the Kelvin-Helmhotz instability, which occurs only in an inhomogeneous plasma. This instability, which is due to discontinuities of the tangential velocity of a streaming plasma (see pp 74 and 284) has several
MICROINSTABILITIES
548
modes. The macroscopic mode has been discussed in volume 1 of this Handbook, see p 297. There exist, however, two microscopic modes too: these are the microscopic electrostatic collisionless Kelvin Helm/11.49/ and the electromagnetic holtz instability collisionless Kelvin Eelmholtz instability /14.125/ in anisotropic plasma, see volume 1. Velocity gradients may not only generate the classical (magnetohydrodynamic) Kelvin-Helmholtz instability, but also transverse Kelvin-Helmholtz modes in Q-machines and proton heating neutral sheet instabilities in space plasmas, e.g. in the aurora. High frequency instabilities may be suppressed by velocity gradients, low frequency instabilities are responsible for anomalous resistivity. For ω/Ω^- « (w^/wj) ~' 2 one has the classical KH mode, for ω/Qj » (m^/mj)^/2 a n e w mode driven by άν/dx (shear mode) appears /14.462/. This mode is collisionless and may be described by an electron-ion two-component Vlasov theory. In a Q-machine centrifugal force, density gradient and sheared velocity act together and produce a radial electric field, which is stronger at the plasma boundary. If ujr(r) = -mcE/rB frequency depending on the rais the local rotation dial coordinate r and ω# its average over space, the dispersion relation reads /9.59/ ω = | ^ + ;[οω^(1 + 2/(fc*)) + + 2~2 / (k2L)] 1 / 2 ) / ( 1
+2/(k2l)).
Here δω^ = ω# - ω#. The first term describes the effect of the shear of velocity, the second term describes the centrifugal effect. When we ignore the centrifugal effect and the shear effect, but consider a Gaussian density profile, we obtain ω = ω - ω /2 + ii όω - ω
/4J / .
Here ω* = k V-Q is the drift frequency (diamagnetic . For <5ω2 < ω*2/4 we have a stabilizing effrequency) fect of the finite Larmor radius (here also called density gradient stabilization). For a Bickley jet profile one has ω(ρ) = a)Qsech2 (r - rQ), δω£ = ω|^ αχ /16, / 4 , the stabilization condition is ω^ χ< <ω^> = ω 2 maX <2ω* Another instability due to inhomogeneity of the magnetic field is the tearing instability. The macroscopic collisionless tearing instability has been
549
PLASMA INSTABILITIES
discussed on p 314, the macroscopic collisional (resistive) tearing instability has been discussed on
p 313/ and the collisionless microscopic tearing instability (sheet pinch instability) on p 380 /14.216/. By use of a variational principle based on the Vlasov
equation a plane equilibrium distribution function fos(x,u) = f0s(H0,P0), expressed as functions of the constants of motion H0 (energy) and P 0 (momentum), has been investigated for a magnetic field BQ = = [0, 0, B0(x)]. An energy integral investigation based on the Vlasov equation results in a stability condition and the fact that the real part of the frequency is determined by the gradient of the external magnetic field. A linearized Maxwell-Vlasov investigation for ions and electrons gives the growth rate /13.53/ for a high beta plasma * ~- Gthl('
-
k2 k2
' n)/2d·
( 1 4
·
2 0 1 )
where d is the thickness of the plasma slab and km the marginal wave number ^λ/ά, compare p 313. A study by Galeev /14.216/ showed that the presence of a magnetic field component normal to a diffusive neutral sheet affects the electron orbits near electhe sheet and results in stabilization of the
tron
tearing
instability.
Development of
ion-tearing
instability with a given wavelength is possible only in a restricted range of values of the normal magnetic field component. As with the macroscopic tearing mode due to inertia, electron pressure gradient or Hall effect in Ohm's law, it is a matter of taste whether one calls the macroscopic rippling mode due to the same effects collisional or collisionless, compare p 314. The re-
sistivity
gradient
instability
or rippling
instabili-
ty which is due to inhomogenity in electric conductivity may be resistive (collisional) or may be due to electron inertia, electron pressure gradient or to the Hall effect and may then be called collisionless
microscopic rippling instability /13.53/ or less macroscopic rippling /10.2/ instability
collision(p 316)
since both fluid theory and kinetic theory may describe it. For drift tearing instability see /14.493/. is an electrostatic The negative mass instability collisionless microinstability which can occur only in an inhomogeneous magnetic field. It is found also in particle accelerators /14.217/. When an ion beam, neutralized by an electron
MICROINSTABILITIES
550
background, circulates around the axis of an axisymmetric inhomogeneous magnetic field, then a small (e.g. sinusoidal) perturbation of the space charge density results in particle acceleration (or deceleration) . The particles ahead of the potential bump will be speeded up,and those behind it will be slowed down. Now the average gyration frequency of the particles may depend on their energy, and it may happen that for those slowed down the angular velocity increases. This is the case when the magnetic field strength decreases in the radial outward direction (negative radial gradient). For slowed down particles which increase their angular momentum the angular acceleration is opposite to the force, as if these particles had a negative mass. The result of this behavior of particles is that they tend to move toward the angular position of the potential bump and that the bunching and the perturbation grow. So the negative mass instability leads to an azimuthal clumping of intense beams rotating in magnetic fields. One may see the mechanism of this instability also the following way: if energy is fed into the azimuthal particle motion, it will be brought to an orbit of increasing diameter which has a longer period of revolution. Thus a force in the positive azimuthal direction produces azimuthal bunching of particles which therefore behave as if they have a negative mass producing a displacement in the opposite direction to an applied force. Three known effects give rise to a dependence of the gyration frequency averaged over the period for a particle to bounce between local magnetic mirrors. These effects are: 1) relativistic dynamic effects (E total energy) Ω = Ω (1 - E/c2m)
(14.202)
2) radial variation of magnetic field machine)
(e.g. mirror
Ω = Ω (1 + b\\BQB' /2ü^vBm)
(14.203) %
for u]_ » Un, y is given by (4.14), B = r is the radial coordinate 3) variation of the axial magnetic field
negative
mass
Ω * Ωο Π
instability)
+ (E - \IBQ) /2]iBQ]
.
dB/àr, (modified (14.204)
551
PLASMA INSTABILITIES
Relativistic dynamics and axial magnetic field variations are always destabilizing while the radial magnetic field variation has a stabilizing or destabilizing effect depending on whether the magnetic field has a radial well or hill. By use of a canonical formalism and the relativistic Hamiltonian, a dispersion relation has been derived by Landau and Neil /14.127/. An azimuthal magnetic field tends to stabilize the negative mass instability. It should be pointed out that Maxwell was the first to investigate the mechanism underlying the negative mass instability in his essay on the stability on Saturn's rings /14.218/. He pointed out that whenever the frequency of rotation of particles in a ring interacting via repulsive (attractive) forces is a decreasing function of their energy then the ring is unstable (stable).
Work on the mass-conjugate
instabilities
/14.21/
showed that the negative mass instability and the two stream instability are a little related effects in the limit of applicability of the Vlasov equation. To each and every positive mass system there corresponds a negative mass system which behaves identically except that particle excesses in one case are to be identified with particle deficiencies in the other. The two systems are called mass-conjugate. It seems that the so-called second instability in the AS device is a modified negative mass instability /14.217/. A background cold plasma as well as a thermal spread can stabilize the lower azimuthal modes of the negative mass instability. Mikhailovskii presents the following theory of the negative mass instability /9.26/. If the bounce period τ = φάΐ/u^ or (4.45) is short compared to γ~1, where γ is the growth of an instability, then the ion frequency Qj in a dispersion relation must be replaced by its average over the bouncing period τ
<Ω > = Hü 1
ij
(l)—. 1
(14.205)
U ||
Here is a function of ui , u\\Q taken at some fixed point in space. For a field of e.g. the form = £ 00 (1 +z^/l2), where I is a characteristic B0(z) length, one obtains from (14.205) the result <Ωχ> = Ωχ(0) (l + ^ 0 / 2 ^ 0 ) ·
(14.206)
For low density, ωρΕ « ÜE the electrons may be neg-
MICROINSTABILITIES
552
lected and the dispersion relation becomes for monoenergetic ions (F ^ δ-function) and for ω ^ n
(14.207)
One finds that for 91n/2u| < 0 one has instability. Lehnert suggested /10.2/, /14.220/ that even small differences between the Lorentz contraction of ions and electrons in a slightly relativistic quasineutral inhomogeneous plasma could lead to significant charge separation effects and thus produce a relativistic
space charge
instability.
Other instabilities occurring only in inhomogeneous
plasmas are e.g. the critical
velocity
instability
/14.221/ suggested for a spatially non-uniform electric field in an inhomogeneous magnetized plasma or
the nonequilibrium
plasma instability
/14.222/ which
has been detected in capacitive HF discharges. The pressure gradient and its associated drifts and magnetic curvature effects are the energy source of the
collective
electrostatic
instability
in a two-dimen-
sional field /14.223/. This instability occurs in a low-beta plasma confined in a two-dimensional configuration, such a configuration being the linear analog of a multipole device. The macroscopic field geometry of the system is taken into account, leading to integral equations involving the coordinate along the magnetic field lines. There are periodic particle orbits of both the transit and trapped particle type. There exist several modes depending upon whether the action integrals associated with the motion of ions and electrons along the magnetic field lines are conserved or not: with the bounce frequencies oo^j and ω^^/ there are slow ("magnetohydrodynamic ") modes ω » ω^Ε y> ωΐ>1' fast kinetic modes oo^j « ω « ω^^ and slow kinetic modes ω « a)£j « ω ^ . Stabilization may be possible by a proper choice of field geometry such as minimum average B configurations. Landau damping, finite Larmor radius effects etc. Ordinarily in nonhomogeneous plasmas finite length
effects stability
are important. Thus the negative-energy inin inhomogeneous mirror geometry may devel-
op /14.236/ (there are also other types of negative
553
PLASMA INSTABILITIES
energy instability). Negative energy waves are excited in mirror machines by a removal of energy through the variation of plasma density and magnetic field strength along the magnetic field-lines. At the center of the configuration there develops a negative energy wave which transforms at the ends of the machine into an outgoing positive energy electron-plasma oscillation. Since this outgoing wave carries off energy from the interior, it serves as a dissipation mechanism, driving the negative-energy wave unstable, see p 2 56. The present mode is absolute, is related to the loss cone instability and takes its energy from the deviation from a Maxwellian distribution. It is stabilized by Landau damping (for high density). A similar instability may develop in sheared fields. Other finite length effects in mirror machines or in slab geometry may be found in /14.237/. Another instability of this type is the negative energy instability in shock waves /14.238/. When a collisionless shock wave propagates perpendicular to the magnetic field, particle drifts arise from the magnetic field gradient, the electric field, and the plasma density gradient of the shock. When a negative energy Bernstein wave comes into resonance with an ion acoustic wave or when resonant ions absorb energy from the negative energy wave, an instability is produced. It takes its energy from the drift motion of the particles and may be stabilized by Tj- ~ Tg, since the ion acoustic resonance can only take place when Tj « TE. Effects of cylindrical geometry (including finite Larmor radius stabilization /14.239/) as well as toroidal geometries /14.240/ have also been considered. Mikhailovskii /9.26/ discusses homogeneously magnetized plasma with nonuniform density variation along its length. For electrostatic modes he presents for the electric potential Φ the dispersion relation 2 -, _d_ (Λ ωPE \ άφ -k2 ax V " 2 ) άχ
ω
2
.2
™Ρΐω
ΡΕ
Φ =0(14.208) 2 2 X 3/2 ω E * l o~ ' " compare (14.142). When the plasma is uniform for \x\ < 1/2, and its density drops to zero at \x\ = 1/2 + a) , the solution of (14.208) becomes Φ - cos(kzz where a an arbitrary constant. For ωρπ» ω and Φ 1 (1/2) = 0 one has the dispersion relation 1+
n2
/72
U
MICROINSTABILITIES
554
2 2/
ιΛΛ
<ω
+>?h/™ il
"
= 0. (14.209)
a Lu2o-a2)3/2 2
α\
Similar results have been obtained for inhomogeneously magnetized plasmas (adiabatic traps, multipole traps, stellarators) /9.26/, Vol 2, p 271 ff. There exists a formal analogy between the Penrose criterion (14.11) and the necessary and sufficient condition for the instability of a weakly magnetized (fij « ^pj) inhomogeneous plasma with a monotonie velocity profile v0(x) . It reads/9.26/ for low frequency long-wavelength perturbations of a plasma with constant density X ry
PjJ
Jn £
,(u ,(x). -"*u ,(x) ,2 > °.
#,.0
0
0
X
{TT)-°\ax I λ ' X=X
Various geometric effects as e.g. the influence of gaps on the electromagnetic wave excitation or of the form of the vessel have been investigated /14.241// /14.485/ (Tokamak microinstabilities) . 14.6 COLLISIONAL MICROINSTABILITIES Collisions may provide a damping mechanism for instabilities and may modify known instabilities. On the other hand, collisions may also excite new instabilities. Collisions of charged particles with neutral molecules in a partially ionized gas modify the particle drifts in such a way that a relative velocity between the two charged species appears and a twostream instability may develop. For the cyclotron oscillations collisions provide a new source of damping but also a small destabilizing relative drift. Since the frequency v of Coulomb collisions depends on temperature and density as nT"~3/2 according to (2.28), collisions modify instabilities mainly in dense cold plasmas, see p 389. To see this more clearly we consider (9.153). The partial integration leading to (9.154) was made under the assumption v = const. When we obtain allowing vs = vs(cs) 2
*
fee 8 = - Es - p s *J / o eκω/k — - c -tv—^2k)ά%,' -«> sx s ___^ ? f
-1 - i/k·
dv
(14 210)
'
555
PLASMA INSTABILITIES
On the other hand the dispersion relation for electromagnetic waves with k± = 0 in a weakly ionized magnetized plasma reads /14.459/ ~2 7 2
ww^3f
3
1 - 2 * + Ρ \ - ί £
2
?
äc
(14.210a)
3 J ^ c ω - Ω + ^v (c) o
and gives for a low-density plasma ω = ok and 2
R«"Â
c v (g) L(Ü> - Ω ) 2 + v 2 (a)J
de.
(14.211)
For monoenergetic electron distribution function fo - δ(o - cQ), the electromagnetic wave becomes unstable if {d/dc[a^v(a)]}c - CQ < 0. So for an instability the electron collision frequency has to fall more rapidly than
s
(c)
=
ao
one finds by a numerical evaluation /14.242/ that electrostatic instability may appear for a plasma with rapidly decreasing collision frequency (h ~ 3 ) . Observations of this instability have been reported /14.243/. Since this instability does not need a magnetic field it may be looked on as a bremsstrahlung effect rather than a cyclotron effect (bremsstrahlung
instability
/14.244/).
The full electromagnetic dispersion relation for a dense Krook plasma (5.18) with strong current may be found in /14.245/. A more refined discussion of the effects of collisions has been made on the basis of the Landau collision term (5.10) /9.26/. Since the electron velocities are large compared to the ion velocities the collision integral in (5.10) may be simplified. If there are no external fields in the plasma the establishment of Maxwellian distribution functions takes place within the relaxation time (2.34)(2.36), where (2.36) is the slowest process. Fast ions (beam) with a speed u come into equilibrium with the plasma in a time of the order 3/2 -3/2 · (14.212) V a Uo XII°th
MICROINSTABILITIES
556
For Tjj/ TEE s e e P 2 2 · A collisionless approximation (p 220) is correct only for time intervals much less than binary collision times (2.34) - (2.36), (14.212). That means that the instability with the growth rate γ must satisfy τ > γ~1 that it might be considered as collisionless. Let us discuss some typical instabilities. The hydrodynamic two stream instability has γ - ωρ#, see p489. From τ * TEE (2.35), (2.19), (2.9) equation (2.15) follows immediately as the necessary condition. Therefore this instability is not sensitive to the frequency of collision processes. For an electron beam with and of low densmall thermal spread c-tfo < u0n\'^/n^'^ sity n-j (ni/n0 < 1) penetrating into a plasma of density nQ the condition that collisions may be neglected becomes 2 n«u
o
-γ^^Ε»
°th
1·
(14.213)
The cross section of Coulomb collisions decreases with increasing relative velocity of the colliding charged particles. As a consequence, a group of electrons will run away at increasing velocity when the applied electric field is chosen above the critical value (7.62). This run-away effect /14.464/ may cause the run-away instability {electron run-away instabilelectron instability) /14.465/ since ity , run-away the run-away electrons form an electron beam (compare section 15.2). Such run-away electrons and run-away instability are generated in Tokamak machines and have also been investigated in the relativistic region (suprathermal range). Whereas a non-relativistic theory predicts that an exponentially small number of electrons run away also for fields less than the (7.64), an exact relativistic theory Dreicer field finds that below that critical electric field runaway is smaller for TE » mc^. For large thermal spread collisions are unimportant. In a fully ionized gas the slowing down of electrons as a consequence of collisions with ions and the acceleration in the electric field are balanced is where ^crit 9 i v e n by (7.62). In this for E < Eorttr case a stationary state is possible for frequencies ω > V EE' I n a weakly ionized plasma collisions between charged particles and neutral particles are more frequent than the collisions between charged parti-
PLASMA INSTABILITIES
557
clés, see (5.16), p 81. Then in a three fluid theory a friction term (compare the last term on the r.h.s. in (7.33)) or Langevin force, see p 17, appears which influences stability. In order to show how a collisional background plasma affects the growth rate of the two-stream instability a general method for obtaining the curves of marginal stability of cold streams has been presented /14.249/. The region of absolute instability is shown to be reduced whenever v Φ 0. For v = 0 the plasma background does not change the domain of absolute instability but reduces the growth rate. solid Collisions play a very important role in state -plasmas. Several collision induced instabilities are known. Streaming instabilities in solids described by a hydrodynamic model become resistive and collision-modified. But also new collision-induced instabilities as e.g. in a drifting electron-hole plasma situated in a perpendicular magnetic field may also appear /14.246/. In addition collision-induced surface wave instability can occur in composite magnetic and semiconducting structures. Vural /1.68/ gives the following classification of collision induced instabilities in solids 1. Collision-induced single-stream transverse wave instabilities 1.1 transverse wave collision-induced instability in nonmagnetic semiconductors 1.2 transverse wave collision-induced instability in magnetic semiconductors and in composite structures 2. Collision-induced two-stream instabilities in electron-hole plasmas 2.1 collision-induced instabilities in drifted electron-hole plasmas due to wave excitation 2.2 collision-induced instabilities in drifted electron-hole plasmas due to circularly polarized wave excitation.
In concluding this short introduction we would like to mention that for an inhomogeneous plasma with a periodic equilibrium potential a like-particle collision operator may be derived in the guiding center approximation /14.247/. That paper discusses how collision operators of the type (5.10), (5.11), (5.14) are modified by strong magnetic fields /14.248/ or by non-uniform particle motion /14.247/. Before we discuss the modification of known instabilities by collisions we will discuss the new colli-
MICROINSTABILITIES
558
sion induced instabilities.
The finite
orbit
instability
{finite
Larmor
radius
instability) /14.185/ which we had mentioned on p 538 is due to collisional guiding center displacement in . In an inhomogeneous magnetic field {"mixing effect") toroidal systems the diffusion of trapped particles enhances the influence of collisions by producing displacements larger than those by classical diffu-
sion {"transverse
collisionless
viscosity")
. This
leads to a drift dissipative instability occurring at smaller collision frequencies than the usual instabilities of this type. For small shear (BA « B0) and long wavelength {k ■> 0 is more dangerous) and u\\t «I, I from B{r ,z) = B0[1 - r {cos {2z /1) - h) /R] , t time and kzc^yl <<: ω an<^ τΙΙ = œr λ# « I, Pogutse /14.185/ gives a dispersion relation for the finite orbit instability, which may be considered as a dis-
sipative
trapped
particle
instability.
The maximum
growth rate is reached for ω - ω#, ω| being the electron drift frequency given by ω|» = kv^g or by ω* = -kckBTEn*Q{r)
/nQeBr
(14.214)
compare (13.2). For development of the instability the system must be finite but large enough, i.e. certain conditions must be satisfied. If they do not hold, the longitudinal ion motion becomes important and the instability vanishes. And electron dissipation should be strong enough. Otherwise electrons can reach equilibrium along the field lines, which would lead to a Maxwellian distribution and to the vanishing of diffusion. Systems with small shear or large field modulation cannot achieve the largest growth rates. For small shear a minimum B-configuration does not provide stabilization. The joint action of large shear and minimum B reduces the growth rate appreciably. /14.250/ is another The entropy wave instability instability occurring only in collisional plasmas. When a low beta uniformly magnetized plasma with density and temperature gradients perpendicular to the magnetic field is permeated by electrostatic low-frequency waves k\\ - 0, then for din T/dx « din n/dx a new collisional overstable mode appears which propagates in the ion drift direction. It is due to the combined effect of ion-ion collisional viscosity and heat transport. These entropy waves have neither magnetic field nor overall pressure disturbances, with
PLASMA INSTABILITIES
559
temperature perturbations Tji - -T#i. The energy source of this instability lies in the thermal energy of the plasma. The instability vanishes at low plasma densities. It has very small longitudinal wave numbers and its transverse phase velocity depends essentially on the particle drift in the magnetic field. Another form of the entropy wave instability appears when effects of the order of ω/vjj in the excitation of electron waves are taken into account and when din T/dx « din n/dx or Bin T/dln p > 7/10+3/4 is not satisfied. The terms of order ω/vjj correspond to the (collisional) gyrorelaxation effect. Entropy waves become unstable due to this effect in a certain range instaof 81n n0/81n pQ and for VT 0 = 0. The entropy bility is a nonisothermal effect which is eliminated by parallel ion pressure at high densities. In a partially ionized plasma the collisions of charged particles with neutral particles may excite another type of instability: the electron-neutral atom collision instability and the ion-neutral atom /14.251/. Timofeev has shown collision instability that in an inhomogeneous weakly ionized plasma in a homogeneous external magnetic field a drift instability can arise due to the effects of collisions between charged particles and neutral atoms. (On the other hand drift instabilities can be quenched by ion-neutral atom collisions in a Q-machine /13.11/.) The electron-neutral atom collision instability is due to the dissipative effects of collisions in a plasma of non-uniform density, nE - nj, TE » Tj. The neutral drag instability (p 334) occurring also in a partially ionized magnetized plasma may be considered as a difference effect of the electron-neutral atom collision instability and the ion-neutral atom collision instability. When electric field and density gradient are parallel plasma particles drift in the ExB direction, and because the ion-neutral collisions are more frequent than the electron-neutral collisions {Vjfl/Qj » vEßj/Q,E) the ions lag behind the electrons. The resulting difference in drift velocities gives rise to long wavelength oscillations (ω < üj, ω < ωρ) parallel to the magnetic field. /14.242/ Clearly also the ionisation instability and the recombination instability are instabilities which are due to collisions. A collision induced instability of transverse waves propagating along B in weakly ionized gases (as well as in fully ionized plasmas) due to the velocity de-
MICROINSTABILITIES
560
pendence of the collision frequency has also been discussed /14.253/. Ion-neutral atom collisions may also excite another dissipative instability in a weakly ionized plasma.
The collisions induce supersonic
ion motion
instabil-
ity /14.254/. Here hydrodynamic supersonic flow with friction between charged and neutral components in a plasma is analyzed. If the plasma velocity with respect to the neutrals exceeds the ion-acoustic velocity, an instability develops. In the initial stage the growth rate is proportional to the frequency of ion-neutral collisions Vj^. In the absence of such collisions the instability disappears. Let us now turn to the collisional modifications of instabilities which also exist in a collisionless plasma. A relatively simple example is the effect of Krook model collisions on the electron-electron and electron-ion two-stream instability /14.255/. Here the electron Krook equation (5.18) is solved numerically for a uniform neutralizing Maxwellian ion background. A first order linearized (/# = / 0 + /1) analytical treatment using Laplace-Fourier transforms yields a complicated dispersion relation analogous to (9.153). Such a dispersion relation predicts also collisional damping of Langmuir waves /9.30/, /14.256/. In the investigation of the electron-ion two-stream instability with the Krook model a gradual heating of the electron stream due to Coulomb interaction with the ions at rest was found. It was shown that, as the kinetic energy of the electron stream is thermalized, the resulting enhanced Landau damping is opposite to and comparable with the growth of the instability. The Krook equation is also useful for the investi-
gation of the collisional
current
driven
ion wave
in-
stability /14.257/. Ion waves propagating nearly across the magnetic field in a weakly ionized plasma with cross-field electron streaming uoE become unstable due to electron-neutral collisions in the parallel equation of motion for the electrons and due to ion-neutral collisions. According to p 82 this equation of electron motion (magnetic field in the s-direction) reads for a Krook plasma with electron inertia neglected e
9Φ
1
Έ1 ΎΓ E
^EkBTE
- lOi
E O
9
^1
§5- -
V V
E E^z
a
n
°
MA
01„
(14.215)
compare (9.53). The use of a fluid equation for the
561
PLASMA INSTABILITIES
parallel motion of electrons is justified since v# > (ω - ku0]r) is assumed. Φ-| is the electric potential,rc#-|the electron density and v*z the velocity of the linearized perturbation ^expK (£·? - ω£)] , γ^ = 1 or 3 depending on whether the electrons are isothermal or adiabatic, see (9.51). Neglecting the perpendicular electron motion (in contrast to the analysis of field aligned kz = 0 resistive instabilities where the perpendicular electron motion is essential) and using the electron continuity equation, one obtains n
S1 / n o
=
Îk e
l ^^mEVE{{u-t'toE)
+ Î
^ E W -
( 1 4
·
2 1 6 1
Since vj » 9,j will be assumed, the ions are essentially unmagnetized and the linearized Krook equation may be used in the form df
n
- ot ^ -
-
e
a/
oi
n
n
/
I
+ u-V/V, — νΦ,-τ-^ = - νI Γ \ /l\Γ 1 - —n /J f tolT . I J I\ - m T n I 1I duT o ' Again with the assumption of exponential dependence for the perturbations, /J-J may be calculated. Integration in velocity space and use of the quasineutrality approximation nE^ = nj-| (valid for lowfrequency modes) gives the dispersion relation /14.257/ 2
-1
Γ ^ + ΐν
° thi*- L
k2 2
+ iuFE(kz/k
kG t h I + hI
^
ω+ ^ Ί Γ K k o V K O
*t h Ä II ^^
)vvE ) [(ω \{u -%>u -t*u„voE )y)
χ
+
^IJ" kG
th.I ™m
^I\\
[kG
thI —thi
2 + + ik„ywikk„Tjm 2yEkBwT\E/mEj
+
'-*.( 1 4 > 2 1 7 ) = 0.
For \(Ù + ivj/kctjtfl » 1 t h e F r i e d Conte f u n c t i o n Z may be expanded and f o r s m a l l γ t h e r e s u l t / 1 4 . 2 5 7 / ω - kaq r b (14.218)
γ = -1 ^ ν Γ Τ - KmEl2mx) ( A ^ i v ^ f l -
uQE/aJ
2VJJ is obtained, where α^ is given by (9.51). Here T^»Tj has been assumed in order to avoid the large Landau damping that occurs for TE * Tj. VJJ is the ion viscosity term and the second term arises from electron ion collisions. We see that UQE > aß is destabilizing {supersonic electron motion instability). In contrast to results for fully ionized gases /14.258/, where electron-ion collisions lead to instability, elec-
MICROINSTABILITIES
562
tron-neutral collisions in the electron parallel motion lead to instability in a weakly ionized gas. When perturbations in TE and in Tj are also taken into account, then in the linearized Krook equation for unmagnetized ions (14.217) the r.h.s. term has to be replaced by a more complicated term containing additional terms depending on TJ^/TJQ. Then a temperature relaxation mechanism is found which has a destabilizing effect on ion waves propagating nearly across the magnetic field in the presence of streaming electrons. Experiments on the effect of collisions on the current driven ion-wave instability /14.261/ confirmed an effect on the onset of the instability: the critiparacal wave number kQ decreases, when the collision meter ^pjTj increases. The minimum drift velocity for the instability is called the critical drift velocity vc corresponding to the critical wave number and is defined by [dvD(k)/dk]^
= 0.
The effect of ion-neutral collisions and of more sophisticated collision operators on the ion sound wave have also been investigated /14.292/. Boundary effects, thermal conduction and relativistic effects have been included /14.262/. Relativistic treatment of the electrons reduces the growth rate (relativis-
tic
stabilization).
When oscillations with frequencies low compared with the collision frequencies are to be investigated, it is necessary to solve the Boltzmann equation. Braginskii's method /14.263/ with Maxwellian distribution functions (7.1) yields after some algebra (moments method, see p 92) a system of equations of the type (7.2), (7.11), (7.16). For vanishing magnetic field Braginskii writes these equations in the following form 9n -TT- + d i v ( n v ) = 0 at
SS
(14.219)
dv + ^ m n -ττ^- = -Vp - ν · Π + e n E + R s sdt s *s s s s s dT 3 n 2 sdT
+
PsdLv
% = "
d i v
*e"
(14.220)
t V(V#V+«s(14-221)
s
where now the transport RE = -O.MmEnE\>E(vE-v
phenomena
are given by
) -0.71n VT^ = - £ χ (14.222)
563
PLASMA INSTABILITIES
compare (7.33), for friction, and qE = Ο.ΊληΕΤΕ(νΕ for electron heat
electron thermal
- v χ)
ouvrent,
- K^VT^,
(14.223)
compare (6.18), where the
conductivity
is given by
κ^ = 3^6nEkBTE/mEvE, compare
(6.20), Qj =
and
-
(14.224)
(7.76b) .
Furthermore
K
KT =
j
V
ef = v w
V
-QI' I Q
for the dissipation The stress tensor
3.9nTk BTI/mIvI
3m
=
(14.225)
EnEkB
(14.226)
function. is given by ,(*vk'kl
_,_
II
2Χ
,.
->
(14.227)
where η1 = 0, compare (6.23), electron viscosity has been neglected (see p 129) and η χ = 0.96nTknTTT_ 1
1
compare (7.78). For the ion
D 1
acoustic
(14.228) 1
instability
(TE »
?j)r
γ - œpj/m^/mj ) (see p 477) the condition for applica-
bility of the collisionless
approximation
ωτ^, « 1, or γτ^ « 1,
λ
ΙΕη0ΐ71Ε/ηιΙ>>
is 1
' 04.229)
since the electrons are slowed down during τΕ. Now in Jhe refined Braginskii theory a static electric field EQ gives rise to a relative motion u0 between ions and electrons in a fully ionized plasma which instead of (7.62) is now expressed by = 0.51eE
£ o
/m^vr o' E El
(14.230)
according to (14.220), (14.222). This relative motion can excite an ion-acoustic instability. Under the assumption k\D « 1, nE^ = nj-| , ToE = TQl, with neglect of electron inertia, linearization of the equations yield for frequencies in the range τ E k^T Q /m E » ω » » τ 7 ^ 2 Τ 0 /^ χ the result /9.26/ ω^ = k(8k
T /3m ) ^
/ 2
(14.231)
MICROINSTABILITIES
564 and
γ =
m ku £-(4-—2) -|(0#96 ωρ ; 3 mITE\2
+
3 3β 8
k^k
T τ 5_o^ f ^
(14 .232)
where uQ is given by (14.230). Equations (14.231), (14.232) now replace (13.218). Ion acoustic modes are In contrast to a colliagain excited for UQE > c^j. sionless plasma, instability is possible even for ToE = T 0 J· T ^ e results so far are valid for a fully ionized plasma. For a weakly ionized plasma collisions between the charged particles and neutrals have to be taken into account and R according to (14.222) has to be modified. Then one obtains the result that and that for modes grow for u > (kßTE/mj)^/^ < k(kBTE/mj)*'2 one has ω ρ = k {kBTE/mI) 1/2, vEmE/mI and for longer wavelengths (vEmE/mj > γ - mEvE/mI compare > k (kßTE/mj) î'2) ω ^ = kuQ, γ = k u^rrij/m^, (14.2187. Ion acoustic modes dominate the Bernstein modes for homogeneous and inhomogeneous plasmas in the re< v/ü « 1 gime of small collision frequencies (krE)~^ for high frequencies /14.259/. This leads to a collisional destruction of Bernstein waves in the high frequency regime in which the modified Gordeyev integral may be reduced to the plasma dispersion function /14.260/. Since drift instabilities have been extensively discussed in chapter 13, we content ourselves with some additional remarks. In /14.264/ the structure of
the oollisional
drift
instability
has been examined
in several density regimes. For kz\E > 1, where λ^ is the electron mean free path, the drift mode is isothermal and should be treated by a kinetic theory. In the finite heat conduction regime kz\F < 1 the threshold is reduced at low densities (vj_ < 0.1νμ) and increased at high densities (v_^ > 0 . 1 V M ) . In the energy transfer limit (k2XE < (mE/mj) ^'2) tne instability is adiabatic in a fully ionized plasma and isothermal in a weakly ionized plasma. The current driven oolli-
sional drift instability (and the hydromagnetio current driven Alfvên wave instability) have been studi-
ed /14.265/ on the basis of the Braginskii equations (14.219) - (14.226) written for a magnetized plasma. The strong destabilizing effect of an axial current has been demonstrated theoretically as well as experimentally in a Q-machine. By use of the Krook model /14.266/ a dispersion relation for the drift dissipa-
PLASMA INSTABILITIES
565
tive instability in a non-uniform electric field has been derived. In /14.267/ the effect of longitudinal electron heat conductivity and electron-ion heat exchange on the drift instability in a strongly collisional finite beta plasma has been considered. The dissipative effects considered result in broader regions of instability than those found when these effects are not taken into account. The stability theory of drift waves has been extended to include dissipative contributions from electrons in various collielectrons and sional regimes and also from blocked inverted gradient profiles (VrcVT < 0) /14.268/, as well as to include the effect of collisions on impurity modes /14.269/. Inverted profiles, which are realizable by introducing cold gas, show in a certain collisional regime a substantial gain in stability over normal profiles (VnVT > 0 ) . Cross-field gradients of guiding-center drift velocities excite /14.270/ several high-frequency (ω > ßj) electrostatic velocity gradient instabilities {longitudinal electron drift velocity gradient instability, cross field velocity shear instability, cross-field two-stream inguidstability with ion velocity shear, collisional ing center drift instability, etc). These instabilities can enhance electron-electron and electron-ion collisions, which in turn destroy the velocity gradient. In pulsed turbulent heating experiments this effect may appear as anomalously rapid current penetration {anomalous skin effect). The theory of collisional drift waves in two-ion(low beta) plasmas /14.271/ shows that a high species ionic charge and a TE > Tj have destabilizing effects. In /14.27 2/ the influence of ion-neutral and electronneutral collisions on the drift dissipative instability in an afterglow plasma is investigated. An interesting effect has been found /14.273/: while the ordinary electrostatic drift wave is stabilized by either high beta effects or an admixture of cold plasma, a compressional drift instability is shown to be excited under the same conditions. When the cold plasma density exceeds a certain threshold, the compressional Alfvên wave is destabilizing either by inverse transit time damping of ions or by inverse Landau damping associated with resonant particles diffudrifting due to magnetic field gradients. The /14.274/ appears even if the sional drift instability conditions for drift dissipative instability are not fulfilled. Clearly, all theoretical predictions de-
MICROINSTABILITIES
566
pend on the model which is used: when the effect of electron-ion collisions on drift instability is investigated it is found /14.27 5/ that the Krook model gives for small collision frequency a destabilizing effect, whereas the Fokker-Planck model indicates stabilization. For large collision frequencies, the growth rates increase for collisional drift waves, but the growth rate obtained from Fokker-Planck model is not nearly as high as in the Krook model. Growth rates tend to vanish in both models for very large collision frequencies. Another mode which is also modified by collision is the whistler wave. The collisional whistler instability is an electromagnetic microscopic mode /14.276/ which draws its energy from the deviation from the Maxwellian distribution due to an electron peak. In a partially ionized collisional magnetized plasma electrons are released, e.g. by photoionization. The generated monoenergetic peak in the electron distribution function is assumed to survive until it is destroyed by an instability. Due to a phase shift and the feedback due to collisions, the whistler wave splits into two branches, one of which becomes unstable under certain conditions. When frequent collisions destroy the non-maxwellian peak of the distribution function before the instability sets in, then the growth rate is zero. Collisional cyclotron frequency phenomena like the gyro-relaxation effect or ion cyclotron resonance heating are important in plasma physics, see chapter 21. Here we will be concerned with the collision-
al Alfvén
cyclotron
resonance
instability
/14.277/
and the collisional cyclotron instability /14.278/. The first instability is of interest for mirror devices. Various problems have been discussed. The stability of electromagnetic ion cyclotron waves propagating obliquely to a parallel current in a fully ionized collisional plasma has been investigated. It was shown that when a certain threshold current is exceeded, instability results through parallel resistivity, thermal conductivity etc. /14.279/. Longitudinal (electrostatic) ion cyclotron instabilities in a magnetized collisional plasma are discussed in /14.280/. Of physical interest may be the collisional cyclotron instability in a Ramsauer effect ionized gas /14.283/. A collisional Weibel instability is treated in /14.281/. The guiding center kinetic equation with
567
PLASMA INSTABILITIES
Fokker-Planck collision term is used in /14.282/ to
study the microscopic oollisional
tearing
instability.
It also has been shown that toroidal effects can stabilize the tearing mode, whereas contraction of the current channel is destabilizing. An electric field
driven
collisional
flute
instability
has been report-
ed in /14.284/. A Fokker-Planck type investigation of plasma oscillations in a constant electric field is presented in /14.285/. If the solution of the kinetic is written in the form equation df/dt = {df/dt)0 f (e ,\i,r ,Q ,t) , where ε the kinetic energy, μ the magnetic moment, the drift kinetic equation (13.73) may be written in the.form df/dt +3//3Θ·Θ + d f / d e · έ = Here Θ = u\\Be/rB, ε = -eEu\\BjB, ufj = = iîf/*t)ç.
= 2 (ε - \iB) /m. Then pitch sion)
angle
scattering
is described by the Fokker-Planck term
{\y-dvffu-
where ν(ε) = (<ε>/ε)3/2<ν># if collisions are included into the Bohm-Gross dispersion relation (9.62) reads /14.301/ (9.62a) where γ^ = 1 or 3, compare (9.158). 14.7 KINETIC THEORY OF MACROSCOPIC INSTABILITIES Macroscopic instabilities have been defined as those instabilities which can be described by a simple fluid model (p 251). Since the fluid equations (chapter 6) can be derived from kinetic theory (chapter 5) it must also be possible to investigate macroscopic instabilities within the frame of a kinetic theory. Macroinstabilities are such slow processes that the displacement current may be neglected, and the gyration radii of the charged particles of both signs may be assumed to be zero. Hence the discussion of macroinstabilities is usually based on magnetohydrodynamics (chapter 11) or, for a collisionless plasma, on the drift approximation of the Vlasov equation (quasi-magnetohydrodynamics, see pp 88, 104 and 381). For plasma motion across the magnetic field, the drift approximation may equivalently be replaced by the CGL-theory, a dissipationless magnetohydrodynamic description with anisotropic pressure (see pp 103 and 272) .
MICROINSTABILITIES
568
The comparison theorem (see p 271 and /11.40/) . < 6WnrT, (14.233) δ^ < δ^ CGL9
quas%
which says that a system is stable both in quasi-magnetohydrodynamics and in CGL-theory when it is stable according to (11.22) in ideal magnetohydrodynamics, warns us that stability in CGL theory does not assure quasi-magnetohydrodynamic or magnetohydrodynamic stability. On the other hand /1.20/, if collisions are taken into account in kinetic theory, one has 6W ηη > &W (14.234) coll
so that stability is not destroyed by collisions. What now is the collisional domain in which low frequency (ω « Ω) modes may be described by ideal magnetohydrodynamics (chapter 12), and when do these modes have to be described by appropriate low frequency kinetic methods? In order that we may neglect resistivity and assume an ideal plasma, according to (6.51) we should have \i0avQl » 1. Furthermore, to be able to assume that the plasma may be considered as a fluid, (2.56) or v » ω must be valid, see also (14.212). With ω - vQ/l, a given by (7.43) we obtain 2 e n„v l/m„ » v » v /I, E o E o
or
_„ (14.235) 9 3x10 nl » 1 (cgs units) for the condition that ideal MHD may be used for low frequency modes (ω « v, ω « Ω) propagating in a plasma of characteristic length I and velocity i>0. In a typical fusion plasma (14.235) is not satisfied. Since we have discussed MHD theory in chapter 11 we will here investigate the collisionless microscopic theory of low frequency (ω « v) phenomena. To do this we consider the order of magnitude of the terms in the Vlasov equation (5.6). They are written for 3 = 1 (MKS system). The terms whose magnitudes are to be compared are ω ^
v
0/1"
°thl^
'
eE mc
/ thI'
and
e#/m = Ω. (14.236)
For slow motions vQ < c+^j- and the second term e thl/T- y> ω · T h e ra"tio of the third term to the second / (ctj2j/l) . These terms are of the same is (eE/mcthI) - ^BT ~ eE^ ' a n d w e assume this to be order if mo\}lI the case, see also p 14. When the last three terms are
569
PLASMA INSTABILITIES
approximately equal and larger than the first, we have with v0 < c-thjr £ » λ^, ω « Ω, eB/m = Ω » ω or rL « I (14.237) (MHD-ordering, see p 101). For finite Larmor radius magnetohydrodynamics (drift approximation, adiabatic theory, quasimagnetohydrodynamics) one has the low frequency ordering ω/Ω « r2L/l2«
1
(14.238)
(finite Larmor radius ordering). A third possibility is the CGL-theory (double adiabatic theory) comprising ω ^c+hk-r infrequent collisions and large magnetic fields (expansions in powers of 1/5) In order to derive some general results /1.99/ we now reduce the Vlasov equation (5.6). For 3 = 1 and drift velocity u = c + vD, where V-Q is the electric defined by (4.6), (4.82) we obtain
|| + s 7/+ s.v, - ^ . f dt
D
dt
(14.239)
33
£ - (îDn+5).vyDnde || + mH..-ML II 83|, + m[5χ5]·|Ι de = 0.
Here (dc/'èt)u = -ivO/ît, (dc/dx)u = -ZvD/dx and vD according to (4.82) (so that E^ may be eliminated) have been used. We now introduce cyclindrical coordinates (9.180) and define D
= TZ + £ n # v + ^ιιΤ^-/ dt
D
(14.240)
II OX..
where x\\ is the coordinate in the direction of the magnetic field. Since the plasma and magnetic field are both inhomogeneous, the unit vector î = Î/B along B is not constant (as in chapter 9 ) . Therefore in c,. = (
c. = ~c - c..î
(14.241)
b has to be differentiated \£hen the transformation (9.180) is made. Since £· (df/dc^) = 0, (14.239) then becomes
-1·
-1
"
(14i.242)
MICROINSTABILITIES
570
where β\ = (cicos φ,
2 1
-y
->
2
m OX,.
2
and 3/3ci = 2cj_3/3e|. The condition for single valuedness in φ (see p 230), (integration over φ from 0 to 2π) plus an expansion in powers of rL/l (Eu «E\, E\\ - TjEi/lt - see also (4.13)) gives in rirst^order Ej_ - ~E, t>£ = (Ε*Ϊ>)/Β - v the equivalence of (4.82) and (6.70). Use of (6.43) and of
at'1
=
= -—'dt
~*'νΎ - °\\-2' P
(14·244>
- compare p 73 -, finally results in
2
s + $.^2l II
c v
2-
(14.245) +c
hl)M- _ y JB. _i£= o
ΊΙ- 2p/ 3 o M
u
m 8x„ 8c,,
instead of (14.242), since / depends on ej only through μ, so that dy = -yd£/£, and 2 3//3cJ = 3//53y, compare with (13.73). For a stationary (3/3t = 0) and static (v = 0) plasma one has a !£° + % . . ^ - ϋ ^ ^ 2= ο Il dx,, m II 3c.. m dx.. 3c..
(14.246)
with the equilibrium solution
fQ - f 0 ( » . K ) .
x
= I e 2 + M-£J^^ 2 4 7 )
This solution may be verified either by inserting it into (14.246) or by showing that y and K are concharacterstants of the equation of motion (Lagrange istics equations) associated with the partial differential equation (14.246). The characteristics equations of P(x,t,c,f)ft
+ Q(x,t,c,f)fx
-Tlx.t.cf) are 76.3/, /14.286/
+ R(x,t,c,f)fc
= ( 1 4
·
2 4 8 )
571
PLASMA INSTABILITIES dt P
=
dx _ de Q R
df Te
(14.249)
Therefore the equation of motion associated with (14.246) is (compare (4.15)) m- d*
-H
35\dx dx,,'dt
μ eE „ - m
(14.250)
and has the solution K according t o ( 1 4 . 2 4 7 ) . I f v does not vanish but is small, we s o l v e ( 1 4 . 2 4 5 ) by the linearization / = / o + / v £|| = £||o + £v 5 = 5 0 + s1 and obtain for f^ 85 -
_3_ 3 * + c,H3as. = —o. ■ v / 0 -
Il 3x ,\mJ
111
Λ
3e„
dx„ - ^ B II
dt
(14.251)
,) - *ç]
9/, O
Here 3/0/3* = 0 and (3fo/3J0 (3*/3ÖN) = 3/0/3tf|| have been used. The structure of (14.251) is very similar to that of (9.151). We may use for its solution the method of integration along the unperturbed particle trajectories c\\, see section 13.4. This gives with ν^ = ξ, see p 271, f
Λ
(x,c..
, t)
= f.
{x
ΊΙο
f-vf
,0)
(14.252) ΊΙ 3tf..\m 3a:, J
||1
II
m 1/
II
P
9/. 2 J 3X
compare with (13.30). If ξ η is the projection of ξ on the normal to the drift surface ψ = const, the term containing f*Vf may be written Jdt'ξ η 3/ 0 /3# n = = /dt'f·νψ3/0/3ψ. If νψ can be neglected, then the disturbances are determined only by ξ (magnetohydrodynamic
limit).
In the low
beta
limit
,BQ
being a
vacuum field has a minimum energy. Low energy modes do not change the magnetic field and they are therefore electrostatic modes. If J = §>c\\dx\ = i>cf[dxu/c , is introduced, we obtain from (14.247) dx. J =
2(K-
\xB/m) + 2eJE
dx
/m
(14.253)
MICROINSTABILITIES
572
On the other hand, if f<\ varies slowly in comparison with orbital periods τ we may average^over the fast orbital variations. We then have for / = f^(x^\,c^,t) ~ fîteoiCWoit) o n a drift surface (νψ = 0) the result da:, dt
f
Tjdt
τϊΊΗ
— {\ m J ~ N 1 îa?NM \ 'H3a?
(14.254)
8
Λ>
2% + ox II rnM II p 2 ZK
By use of (14.253) it then can be shown that and that (8.142) follows. The MHD energy f = f(K,\i,J) principle and a comparison theorem may now be derived. For low frequency instabilities with weak electric field, the electric drift is small. Then after some algebra and a second order expansion the Vlasov equain /7.5/, /1 .99/ tion results for f(K,\i)
If + V v '
+c
n^
=
°'
(14 255)
·
where now V
D
=
ΊΓ + T^V
+
-Ω-<μ>·
(14.256)
Ωρ Then in the denominator of the dispersion relation resonance, see p 370). So the ω ± κ·νρ appears (drift modification introduced into the mag netohydrodynamic by a refined kinetic treatment (orbital eftheory denominators fects) is the appearance of resonant /14.480/. Another which may produce drift instability Larmor radius stabilizamodification is the finite tion, In chapters 11 and 12 one finds that macroinstabilities are generated by pressure anisotropy (Pu * PD 9 by external forces (gravitation in connection with density gradient, Coriolis and centrifugal forces) and by gradients like Vp, ν·Β, In, VT, Vv, Vn, \7Φ. Thus mag ne tohydro dynamic instabilities are fast gradient instabilities (see p 350) and are due to inhomogeneities. They are therefore closely related to drift instabilities /14.495/. After these general remarks we will look more flute closely at some special macroinstabilities. The (k u = 0) may be discussed on the basis of instability (14.166) /1.20/, /4.11/, /11.11/ or of a canonical flute treatment /14.287/. For the gravitational
573
PLASMA INSTABILITIES
(11.26) the well-known result (p 277) is recovered that for instability the gravitational field must be directed toward the direction in which the plasma density decreases. This can be shown explicitely in a kinetic theory as follows. Consider a plasma slab suspended against gravity by a magnetic field along x· Since the constants of motion are K = mu /2 + mgz and py = muy + eAy/c, and since y is a cyclic variable, f0(K,py) the equilibrium distribution function / 0 = for A = -BQz is given by (see p 368) / \3/2 2 fr, = oUTπ / exp[-a(u + 2gz)] ·[ 1 - ε ' (z -y u /Ω) 1, ° ° (14.257) where a = m/lkWI and pnl/m = unj - Qz has been used. Now n
n = \f
du = n (1 - ε'z)exp[-2agz]
(14.258)
gives , ε' ~ , , din n -ÔT = 02ag + T ^ 7 r 7 - 2ag + ε' which goes over into ε' for g = 0, see p 3 68. For perturbations kz » ε and low frequencies this yields a dispersion relation in local approximation which reads (n = 0) /9.26/, /1.20/ for ions 2
fc =JL compare
e
~
2a
k r
H - 1 + «V
(13.54)
L
£
'e
17 2 2 2 L
Here ω = k
8ε = Im ε = γ-τ—
T\k
y
k D v,
y i (lk2r2)-z(
rT L
V
ν^. F i n a l l y , 7
ε' exp
)} ,
z
th' (14.259) from (13.56)
22
JL
D
z
th
72 2
(14.260)
If the plasma is supported by the magnetic field against gravity, then (dnQ/d;s) < 0 f ε1 > 0 and we have growth of the instability, ε is the (dimensionless) dielectric function. When the flutes are due to thecurvature of the magnetic field, we may replace mg -* F, see (4.21) , (4.95) (pp 58, 73) and get (4.96) for abrupt change of density, or /4.11/, /11.11/
-r-
±n n 2ΚΒ(ΐΕ + ΤΙ) dx rrtjp
(12.261)
MICROINSTABILITIES
574
for continuous change of density. According to (11.30) a flute instability can develop if the field curvature is convex outward. For a self-gravitating plasma with stratified density the dispersion relation has also been investigated /14.448/. Neutral gas friction and viscosity are stabilizing , but finite resistivity is destabilizing. If the plasma is bounded by conducting walls, flute modes are absent (line tying stabilization) and also modes withfe..* 0 should be considered. Such modes grow if φ ? < \gz% \ , see chapter 16. Since feII - π/Ζ, where Ϊ is the distance between the conducting walls, we obtain the instability condition Ζ2β > π2ρα.
(14.262)
Here p is again the curvature radius of the field lines, a is the transverse distance, coil diameter or plasma radius (e.g. the distance from the axis of a mirror device to the Joffe rod, compare pp 149 - 150). So for sufficiently long plasmas instability occurs (ballooning instability). Since the curvature can change along the magnetic field lines, the growth rate γ (14.261) and also 3 become functions of the distance along the field lines, andfe,.varies too. This makes local adverse curvatures important, if particles can resonate locally in bad curvature regions. Since a bounce resonance can alter J (14.253), for stability df/dJ > 0 (14.263) has to be added to (8.142), see also p 176. If the radius of curvature p depends strongly on position, p in (14.262) must be replaced by 1/i/1, where U' = 6U/U, see p 283. This gives /9.26/, /14.288/ the maximum pressure for which the plasma can be stably confined as 3^ * π2α2/Ζ2ί/' ,
(14.264)
compare (11.43). This β is called the depth of magnetic well. The ballooning instability which is localized in a region of unfavorable curvature develops with a growth rate /9.26/ Ύ - cthl//^
(14.265)
which could be derived from (11.32). We now turn to other macroinstabilities of very
575
PLASMA INSTABILITIES
low f r e q u e n c i e s a s w e l l a s l o n g w a v e l e n g t h s ( » r^) i n high b e t a p l a s m a s . In t h e magnetohydrodynamic approxtensor imation Ckietfaj/Qj - ω/Çlj <& 1) t h e dielectric of a m a g n e t i z e d a n i s o t r o p i c plasma ( 9 . 1 9 4 ) becomes / 1 0 . 1 4 / by e x p a n s i o n of t h e B e s s e l f u n c t i o n s and f o r n = 0
s
ε
xy
Ω
= -ε
I
L*
= { ) L"L
yx
ε
= ε
Λ
ε
=
-ε
^
[ where
„
ί Ω2
*
u
2
2 ^
1
ω2
ω^
/kt
L
ω„ fe^
2
+°°
=
>
s/
k,
2 β " + ^ < >ιs σ ω 2fe Is
7
2^ 2 ^
ω π ,k,
S
(14.266)
IIM
s ss
z
,2, 4
4ω 2
= -il-g.{ ωΩ V
υ
/
^ LU 2ω
.2<
S
n
ωω
ι-, 2
S 06 /Ci · s Ω ^,, β II
2S
^
kn>
ωπ_ / ,, Λ ωΩ ωΩ \V
2 ,
2ω
7
>
2^
7
2 ^ - i
k.
>\\ J
2π
f dc||f2uc1deicn/'os(ei,c|1)
(14.267)
and /
Λ Ο
<°Ι'
dc..l27rc.dc.—
IIJ
1
β
±
ΙΙ>
ü_y.
l ( e -ω/fe )
2
(14.268)
Here Ω 8 includes the sign. When we try to redetect waves which we know from section 9.6 we find that 52fc2/u>2 = εχχ
(14.269)
MICROINSTABILITIES
576
describes the incompressional shear Alfvên wave (which becomes the firehose instability) and that 52^2/ω2 = ε
+ ε 2 /ε (14.270) yz zz yy describes an ion acoustic wave and the compressional magnetoacoustic wave (which becomes the mirror insta. Substitution of εχχ from (14.266) into bility) (14.269) gives, for a Maxwellian distribution function, for u>/k\ic.f-foj « 1, for ω2/&2
= 1 -ΐΣ(β„β
- e
i s
),
(14.271)
\\°A
where (14.272) One sees that (14.271) is equivalent to (14.85). Similarly for large ez (decoupling of the ion acoustic wave) one obtains from (14.270) the result for ions
» = -
i^'-fc)}
+ 1 + s Instability (Im ω > 0 for ^expi-iut) (nearly parallel propagation) if
-ΊΙΚ-*Ι.)
< 0
(14.273)
occurs for ^ » k . (14.274)
or for A:,. «:fej_(nearly perpendicular propagation) , if 1 +
p l s ( 1 - ß la /ß lls) < °·
(14 275
·
>
Condition (14.274) describes again the firehose instability, see (14.271) and (14.275) describes the mirror instability, see (14.88).
577
PLASMA INSTABILITIES
In addition to the comparison theorems mentioned earlier, a comparison of unstable modes in ideal magnetohydrodynamics and guiding center plasmas has been for plasma inmade /14.289/. Also an energy principle stability against low frequency electromagnetic modes for Vlasov plasmas in the magnetohydrodynamic approximation has been given too /14.290/. An energy principle for two-dimensional resistive instabilities leading to a necessary and sufficient criterion for stability has also been given /14.450/, /14.432/. A consequence of this criterion is that the current density in a plasma with arbitrary cross section should not increase to the outside. Otherwise the plasma becomes unstable against resistive instabilities (cur-
rent
density
gradient
instability,
current
gradient
instability /14.451/). A special optimal current density distribution stabilizes this resistive kink mode. The oscillations driven by current gradients have sawtooth form and have a peak at radial location rQ. Ideal MHD-theory predicts a singularity at rQ. Inclusion of resistivity and boundary layers creates a reconnecting mode /14.502/ which reconnects the magnetisic field lines at rQ and which generates magnetic lands. Kelvin-Helmholtz modes are studied in /14.489/. 14.8 TRAPPED PARTICLE INSTABILITIES
Trapped particle instabilities /13.49/ have been mentioned on several earlier occasions, compare volume 1 of this Handbook p 384 and section 14.2 in this volume. They may be classified according to the trapping mechanism as follows: 1. Trapping by electric fields /13.68/ 1.1 deviation from Maxwellian distribution /14.6/, see p 463 here, 1.2 trapping in a moving strong electrostatic wave /14.16/, see p 464 here. 2. Trapping by magnetic fields 2.1 collisionless plasma /13.40/, /14.291/, s^e volume 1, p 385 2.2 dissipative (collisional) plasma /13.40/, /13.64/, see volume 1, p 390. Let us first consider the basic electric trapping process. If we replace fix) in (3.4) by -eE or by βάΦ/άχ, where Φ = 0cos kx for an electrostatic periodic potential (or potential of a travelling electrostatic wave in the wave frame of reference) we obtain as equation of motion for a negatively charged particle
578
MICROINSTABILITIES
mx - -βΦ fesin kx - -βΦ k x (14.276) o o which has for small amplitudes kx « 1 the solution bounce frequency ωβ x = a;0sin u>et, where the electric is given by ω = fe/οΦ /m = lekE /m. (14.277) e o o For a sinusoidal travelling electrostatic wave - ut), (ω - ωρ^) in the limit of zeE(x,t) = EQsin{kx ro amplitude EQ + 0, the electron is free-streaming For EQ Φ 0 a transformation to the x(t) = x0 + u0t. u1 = u - ω/fe brings us back to wave frame a: ' = x-bst/k, (14.277) with EQ = Φ0k. For particles moving near the phase velocity ω/fe of the wave {resonant -particles) thewith Ifex1 - 2ηπI « 1, n = 0,±1,±2..., (linearized ory) , the replacement of the sine by its argument is justified. For E0 = const exact integration of (14.276) gives .,2 ^ βΦ(χ·) = J/ = const. (14.278) It is clear from (14.278) and Fig. 51 that a particle with a kinetic energy W in the interval -βΦ0 < W< +£Φ 0 is trapped by the wave. Hence we have u' = ±/4βΦ /m (14.279) max o for the maximum possible velocity of a trapped particle in the wave frame. Particles with u1 > u^ax or W > βΦ0 cannot be trapped. In phase space (14.278) defines a separatrixf see Fig. 51. The distribution function f(x\u\) has peaks in phase space wherever there is a potential trough. Trapped particles exhibit periodic motions in the wave frame. An untrapped particle decelerates and accelerates when passing over the potential, but does not reverse the direction of its motion. When E0 is slowly varying in time particle orbits may still be determined from (14.278) as long as the adiabatic condition din EQ/dt « ω β is satisfied. In the linearized theory (kx1 « 1, u' - 0, u - ω/fe) all trapped particles are resonant particles (cf pp 223 and 232). If particles are trapped by an (inhomogeneous) magnetic field, the equation of motion of the guiding center is given by (4.15) and the bounce frequency is given by (4.45). We assume that the positive ions form a stationary neutralizing background. The damping of the oscilla-
579
PLASMA INSTABILITIES
(χ')
-βΦ
u n t r a p p e d p a r t i c l e , J/>+e
-βΦ
o
\u* ( χ ' ) ί/>+£φ
ί/=+βφ
W=-e$
max Fig.
51.
Partiole
trapping
tions of negative particles is described by averaging over one ensemble of particles having velocities close to the wave velocity {linear Landau damping). Particle trapping is often said to be a nonlinear effect since strong waves act back on the particle distribution function, see p 583. Then also the damping process is nonlinear. The linear theory breaks down when the condition for validity of the linearization approximation is no longer satisfied, i.e. if τ^.ρ« τ L = = γ"1 /14.292/, /14.293/. This breakdown occurs also see p 220. Here γ is the Lanfor dfs<\/du « dfso/du, dau damping rate given by (9.169) and τ+r is the peritrapping time scale given by τβ or τ^ {bounce ods) . In such a situation ( τ ^ « τ^) trapped particles strongly influence the distribution function long before the wave damps or grows. According to (9.169) for k\DE = 10"1 and τ * ω-1, = lO"10[s] one has TL = 10^[s]. For τ^ v - τ^ or xe we have either - τ or τ ^r » τ, then the wave makes many oscillaτtr tions during one bouncing period of the trapped particles. No waves exist with ω~1 - τ » τ ,_. For feXX n77 - 1 Ό ΡΕ
tr'
DE
MICROINSTABILITIES
580
(9.169) is no longer valid and τ^ - τ, τ^ν » τ^: the wave damps out long before a particle can finish an oscillation in the potential well of the wave /14.294/. From τ * ω"^, τtT - ω~1 we obtain from (14.277), (2.19) the estimate τ 2 / τ ? * kE /4i\ne « 1 (14.280) o tr so that T^r » TL is satisfied. The inequality x^r » τ assures that the field amplitude EQ is sufficiently small to guarantee validity of the linear theory. For times > τ t r the linear theory breaks down anyway. For larger amplitudes the monochromatic electron oscillations exhibit strong distortions due to nonresonant wave-particle interaction and the adiabatic condition is no longer satisfied. Furthermore, (14.277) and (14.279) become invalid. We discuss first the integration for EQ = const. From (14.278) which is exact, we obtain dx
'
t = I
— + const.
J S2(W + βΦ cos
Defining the trapping τ|
(14.281)
kx%)/m
time by
= m/e$ok2
(14.282)
and the modulus κ of the elliptic Λ
functions
by
W + βΦ
1 _
(14.283)
O
2 " 2βΦ K O we may write (14.281) in the form
j
kx'/2
t = κτ^ tr
I
J
kx'/2
dz
rV1 - K sin—s Λ
r
^ 2
By use of the normal elliptic kind /14.296/ z d2
integral
(14.284)
O
of the
first
' (14.285) l/i 2 . 2 , o V 1 - K sin s' this may be written as t / κ τ ^ = F (kx* /2; κ) -F (kx* /2; κ) + + t0/
= f
+ t .
7T~ 2
J
581
PLASMA INSTABILITIES
which are not oscillating in the potential trough. For K 2 > 1 the particles are trapped and oscillate. To see this periodicity clearly we introduce (14.286) v = sin η = KSin(fea;f/2) into (14.284). Introducing κ' = 1/κ we obtain dV + const. (14.287) * = τ. tr\ V λ - v V 1 - κ' ι; The integral in (14.287) is given by F(n,Kf) In order to eliminate the integration constant__we integrate (14.287) over v for such a time span τ that sin η makes a quarter of a full oscillation, i.e. η from 0 to π/2 or v from 0 to 1. We then obtain the complete normal elliptic integral of the first kind
π/2 Κ(κ)
=
dz
f
J
ΙΛ
O M - K s
Κ(κ)
=
= F(TT/2;K)^
2.2 sin z 2
^d 1
4
+ ^ - + ···) dt>
f
J
+ X
ΐΛ
2 1/Ί
(14.288)
= F(7T/2;K·) .
2~~2
Then (14.287) becomes 7 = τ
Κ(κ') = F(TT/2;K')
(14.289)
since F ( 0 ; K ! ) = 0. Now the bounce period is given by ^4τ > 4i£ r . For other arguments and again #' = = x% U = 0 = 0 (14.287) reads o (14.290) * - tQ = TtrF(n(Ä") ;κ· ) which gives the solution x% (t) of (14.278). It is useful (but not necessary) to introduce the inverse function of the normal elliptic integral of the first kind, i.e. the Jacobi amplitude function η = amU/τ
;κ·) Ξ F ( ~ 1 ) .
(14.291)
Then (14.290) may be written sin am[(t-t o )A t r ;K']E sn [( t - t ) Λ ^ ; κ '] =KSin(fcc'/2)
MICROINSTABILITIES
582
where sn i s t h e Jacobi x% (t)
Then
sine.
= l a r c s i n ^ ' s n [(t - tQ) A t r > ; κ ']] .(14.292)
This satisfies the exact equation of motion (14.27 6) xQ = x% (0) is determined by tQ and for Φ 0 = const. the energy W since κ depends on W. In plasma physics the equation of motion is the characteristics equation associated with the kinetic equation, such as e.g. the Vlasov equation, see (14.249). In the wave frame x% , u* = u-w/k, ti = t the Vlasov equation associated with (14.276) reads, for electrons in an unmagnetized plasma (ions at rest) , ^dt Ξ dt ^
+ u'-^Ç— E sin fcs'|^ = 0, όχ' πΐτ-, o du
where / = f(x',u',t').
If the solution of (14.276) is
g i v e n b y x% = x* ( t 1 , x ,uQ), 1
(14.293)
E
i>' = v% (t% rXQrUQ)
where
xQ = x (0), u0 = u (0) then according to Jeans1 theorem, the solution of (14.293) is given by an arbitrary function of xQ and uQ namely f{xl,u*
l
,tl)
= f(xQ(x%
,u% ,t%),uQ(x*
,u% ,t%)
,0) (14.294)
since / is constant along particle trajectories. The function xQ (x* ,u% ,t') (and u ) have to be calculated or from e.g. (14.292). If / depends from x% (t% ,x0,uQ) is on time, the travelling wave E(x,t) = EQsin(kx1) modified according to the Poisson equation dE(x',t%) 3a:1
= -^e[jf(x%
,u% ,t')du'
- nl(14.29 5)
where nQ is the neutralizing ion background at rest. If the arbitrary function in (14.294) is a Maxwellian this leads to Landau damping, and for τ t r « τ L to a slow variation of EQ or Φ 0 with time. In order to investigate this slow variation we assume τ « τ^r « τ^ (here τ is again -ωΰΐ). τ « τ^ ρ guarantees that EQ is small and that the distortion of the monochromatic wave is slow, while τ. « τ^ assures that the rate of charge in EQ is small on the characteristic trapping time scale. Having obtained in (14.292) the exact solution of (14.276) and in (14.294) the solution of (14.293), both for EQ - const, we now determine the (slow) variation in E from the conservation of field energy plus particle energy, since the solution of (14.294) using (14.295) is practically impossible /14.297/, /13.293/. Similar results have, however,
583
PLASMA INSTABILITIES
been obtained by a perturbation analysis of the nonlinear Vlasov-Poisson equations /14.298/. For strictly monochromatic waves with spatial periodicity 2i\/k, the energy conservation f+TT/fc _d_
at
ÇL·2(x,t)/8Έ
+-γ
\u2f(x,u,t)dujdx
= 0
(14.296)
Ι-π/fe
is exact and may be derived from the Vlasov-Poisson system of equations. The calculations based on (14.296), (14.294), (14.292) are quite cumbersome /14.297/. Therefore we content ourselves with exhibiting the results. For a Landau damped electrostatic wave E(x,t)
= E {t)sln{kx
- u)t)
(14.297)
(14.296) gives for an initial condition f (x* ,u% ,0) = / (w1 ) + f 1 (uf ,0)cos kx% (14.298) and after some tedious algebra the approximate result dE ( } d tV
= 2y(f)E(t')
(14.299)
for τ « τ ^ , u'2 » eEQ/kmE = ω|/&2 (fast particles) is obtained. E is the spatially averaged electric field energy density E^(t9)/16π and y(t') is the nonlinear Landau damping rate, or better: quasilinear Landau damping rate, which is expressed by an integral over the normal elliptic integral of the first kind and over a sum of sine-terms. For tf « τ+r the untrapped electron terms dominate and y(£f) reduces to the linear Landau damping rate. In this sense it is often said that particle trapping is a nonlinear process /14.301/. For t' » τ t r the integral gives oscillations phase-mixing y(t') - 0. So for t'« x^r we have linear Landau damping, for tf > τ^ν there is an oscillation of E with a period of approximately τ^Γ and for t1 » Ttr, E tends to a level E(«>) < E(0). The trapped electrons support the wave: Landau damping disappears for waves in trapped particles. For γ > 0 a linear instability appears which later on has an oscillatory structure and saturates for t 1 » ! ^ at a level E(o°) > E(0). In both cases (y > 0, γ < 0) the time-averaged electron distribution function forms a plateau /14.299/, /14.300/ in the region where ω/k - u+r. The formation of such a plateau is charac-
MICROINSTABILITIES
584
teristic for the wave-particle interaction, see Fig. 52. The presence of collisions, which always tend to reduce the distribution function to a Maxwellian, disrupt the plateau formation and play a decisive role in maintaining the linear Landau damping for long times. For this reason not only the linear collisional Landau damping
/14.302/ but also the
dau
dam-ping
nonlinear
collisional
Lan-
ω/fc
/14.300/ has
Fig. 52. Plateau formation been investigated using a simplified one-dimensional Fokker-Planck-Chandrasekhar term (5.20). It is shown that weak collisions (3 « γ) have a negligible effect on the initial (linear) Landau damping but are effective in preventing plateau formation and in maintaining Landau damping for times close to the decay rate of the linear theory. Experiments /14.304/ with k2\2 « 1 confirm, however, the prediction of a decrease in the damping rate from the linear value. Experiments also show the growth of sidebands of the fundamental wave. The appearance of these sidebands (a trapped particle instability) cannot be explained by a monochromatic wave. It is a typical nonlinear effect /14.342/. Trapped particle dynamics is important mainly for monochromatic waves. If there are several waves (wave packet) or if the field is localized the situation is changed. For a wave packet the trapping time does not have a precise definition /14.292/, /14.299/. For the weak electron beam instability {gentle bump on tail instability) the transit time may be much shorter quasilinthan T£r. This fact forms the basis for the
ear turbulence
theory
/14.303/. The nonlinear
Lan-
dau damping involves the beating of two waves /10.13/. Two high-frequency electron waves ω-|,^ι and ω2/&2 could beat to form an amplitude envelope travelling at a speed (ω2 - ω-j ) / (^2 " &1 ) ~ άω/dk. This velocity may lie within the ion distribution function and can generate energy exchange with the resonant ions. These ions see an effective potential due to ponderomotive force, and Landau damping or growth can occur. This damping provides a mechanism for heating ions with high-frequency waves (which ordinarily do not exchange energy with ions). If the ion distribution function is double-humped, it can excite unstable electron waves (modulational instability).
585
PLASMA INSTABILITIES
Up to now we have considered trapping in electrostatic fields. We will now discuss magnetic trap-ping. This occurs in the space between two magnetic mirrors. Magnetic particle trapping may also be used to visualize the containment properties of magnetic fields /14.481/. (Then uLR - const for toroids is used; Ü)£ = eB/m = eBQ/mR) . Examples of magnetic trapping may be found in mirror machines (p 64) , in toroidal traps (p 161) or in the magnetic dipole field of the earth /14.305/. According to Liouville* s equation (5.3) the phase space density is constant along the dynamical path of the particles /14.305/. In coordinate space this has the consequence that particles can be detrapped only by collisions, by interaction with electromagnetic waves or by nonadiabatic changes of the trapping field. On the other hand /4.6/ conservation of energy requires for static fields that a particle injected with a given initial energy must return to the initial position with equal kinetic energy and therefore be capable of escaping the confining magnetic field. The motion of trapped charged particles resp. their guiding centers in an adiabatic (see p 54) toroidal magnetic field is given by (4.15) or by u Ιί + u*B)-U: 8t + »¥II ds - —(% mE du = 0.
(14.300)
Under the assumption of small aspect ratio, we may assume that ε = r/RQ is small (compare p 340). Then the magnetic field components analogous to (12.67) may be written in the form /8.48/, /8.13/, /14.291/ B.φ = Bo (1 - ecos Θ) B
= β
β
(1
£COS
(14.301)
θ)
Q ο θ "" 2 When ε «
1
^ 7 = ^ — ( 1 + s c o s 0)uM o^ dp = mc 2 2 dî 2eB R ^U + u | | ) s i n o o
(14.302) θ
(1
+ e c o s
θ
) (14.303)
MICROINSTABILITIES
586
me — {u + u N ) (qcos Θ + 2eBR-
at
° ° + ^2·Θsin Θ) (1 + ecos θ) ,
(14.304)
where r in ε in (14.302) may be considered as constant. Thus (14.302) can be solved independently of the other equations which describe the departure from the line of force. After some algebra (14.302) can be transformed into d|
at
=
±
|!d|/
2 K
2_
1
_cos
(14.305)
R q on Here κ^ = (u^-uf)2eu^. Trapped particles have a small longitudinal velocity uy = u.. A turning point occurs for the particle at 0 Q defined by 1 + cos θ 0 = 2κ 2 , which is only possible for κ < 1. Thus κ = 1 distinguishes again the trapped particles from the transiting particles. From (14.305) we obtain the magnetic bouncing period Tb = 4-2-1 \
αθ
=
_2l_
K(K)
(14 .306)
Θ V 2κ - 1 - cos Θ compare (4.45). Here K is given by (1 4. 288) . For u.. « u (14.302) - (14.304) give /8.13/ the displacement Ar
=±
m5uc
^ 2 κ 2 - 1 - cos Θ.
l
eB
(14.307)
/ε
o In its motion along the magnetic field (un > 0) an ion drifts outward and inward in its reverse motion (u|| < 0) . The minus-sign is for electrons. The radial see (12.87). displacement is of the order rLqe~^/2, The combination of (14.307) with (14.305) gives the banana orbits Θ{r), see p 159. The displacement Δξ of the trapped particles along φ during one oscillation period is given by (14.308) where J is the longitudinal invariant (4.16) given by (14.309) where Ε(κ) is the complete
elliptic
integral
of
the
587
PLASMA INSTABILITIES
second kind. Since the motion of a particle in equilibrium is now known, the kinetic equation ^1 4.300) _^may be solved by linearization / = fQ + f<\ , E = Ε0 + Ε<\, where fQ is the equilibrium distribution function which can be written as a Maxwellian in which density and temperature depend on r. The perturbation f^ can then be determined from (14.300). Long and tedious calculations demonstrate that the presence of trapped particles may lead to the appearance of flute instabilities (trapped particle instability). The trapped particles in a force tube between local magnetic mirrors (in a toroidal device) will in general execute an unfavorable magnetic drift if the magnetic field falls off to the periphery. A small perturbation thus leads to charge separation which in turn amplifies the initial perturbation. Whereas in a mirror machine all particles are trapped, the trapped particles in a toroidal trap are immersed in transiting particles which can neutralize to a certain degree the charges of the trapped particles. By use of the expressions (14.306), (14.307) etc it is possible /8. 1 3/, /1 4. 291 /, /14.306/ to derive the growth rate γ2 * ω * 2 ε 3 / 2 where
or
ι
γ 2 * /Ξω ω*/2, ι
(14.310)
m ρ
ω*
=
B*L reBn ar
(14.311)
ω
= 2lqeckOT/eBr2
(14.312)
and where m
^
B
ωn
- ω *!τΗ·
(14.313)
p din n It is shown that the instability develops mainly in the external region of the torus and that its amplitude is small for |θ| < π/2. Since there is an effect of q% = aq/dr on the instability, it may be stabilized by decreasing q% , and the stabilization criterion reads dln ? < -1 (14 3-14) din r
2
{
I*.Ji4j
(stabilization by shear). There is also a critical density gradient such that stability occurs for /8.48/
MICROINSTABILITIES
588
"4·315)
-3E-; < itfl·· r
O
The strongest trapped particle instability develops when TE = Tj /14.291/. If the plasma is not isothermal (Τ^ Φ Tj) the instability range is broadened. Other growth rates are given in /11.11/, /9.26/ by where yf is the growth γ - yfkirLI·(ntrap/nuntraç)î/2, rate of the flute instability. When particles are trapped in the magnetic field of the earth, a similar instability occurs /14.307/, /10.14/. For trapping near the equatorial region of the mirror, the magnetic field may be assumed to be BQ{z)
= £ Q (1 +bz2),
b * 9/2R^,
(14.316)
where RQ is the equatorial distance to the field line. Then the magnetic bounce frequency is given by
r
^y-^f
(14 317)
·
where μ is the magnetic moment. The Vlasov-Poisson system of equations gives after some approximations the dispersion relation (14.318) where ωρ 0 is the plasma frequency at z = 0 and (14.319) y 0 being the magnetic moment for BQ. From (14.318) it can be shown that a beam of trapped electrons causes an instability at ω ^ ω^ 0 when it is mixed with the background plasma. Recently various details of trapped particles have been investigated. Trapped particle orbits with bounce frequencies close to the frequency of electrostatic standing two-dimensional toroidal modes can amplify these. Quasi-bananas have been found /14.308/ with radial width significantly larger than that of the known bananas. Their centers are displaced from the magnetic surface on which the modes considered are on collisionless excited /14.308/. High beta effects and collisional trapped particle instability, also in non-circular cross-sectioned Tokamaks have been considered /14.309/. Finite beta effects have a stabi-
589
PLASMA INSTABILITIES
lizing influence on collisional instability. A stability condition for the trapped electron instability which is valid in both the collisional and collisionless regimes has been derived from wave-particle interaction between the perpendicular phase velocity of the waves and the gradient B drift velocity of the electrons /14.310/. As far as electrically trapped particles are concerned , *the following research has been done: evolution of the electron energy distribution due to trapping in the potential of an electron plasma wave /14.311/, the generation of trapped-particle modes by injection of electrons in a plasma wave /14.312/, instability of particles trapped by an electrostatic wave field in a constant magnetic field /14.313// electron trapping in a growing electrostatic field showing that the average field at trapping is given by ekE0/m - π 2 ω 2 /16 for growth rates γ < ω/4 /14.314/. A quasi-linear treatment of the time evolution of a large amplitude electron plasma wave showed no instability at wave amplitudes given by βΦ0 < 0.1 kBT in contrast to experimental results /14.315/. Finally an oscillation-center formulation of particle trapping in the beat potential produced by the superposition of two coheren waves has been given /14.316/. For magnetically trapped particles many details have been discussed recently: the interaction of particles trapped in a magnetic field with coherent electrostatic waves has been investigated /14.317/, trapped particle instability in axisymmetric toroidal sys-
tems /14.318/ and the electron
drift
trapped-particle
instability have been investigated /14.319/. Several papers have been concerned with the destabilization of trapped-electron modes by unfavorable curvature of the magnetic field {magnetic curvature drift resonances, electron temperature gradient drift trapped instability, trapped electron drift instability caused by bad
curvature
/14.320/). It has been shown that
the stability of trapped electron modes driven by the magnetic curvature drift does not improve appreciably, when their frequency becomes comparable with the electron magnetic drift frequency. A new trapped-electron instability has been found. It is driven by a radial gradient of the electron temperature or by the electron drift in the unfavorable curvature of the magnetic field lines for high-temperature confined plasmas in which a considerable fraction of the electrons is magnetically trapped. The mode stands along the field, is symmetric about the point of minimum field
MICROINSTABILITIES
590
and has relatively short wavelengths across the magnetic field. Trapped electron modes have also been investigated in cylindrical geometry /14.321/. Reversed density and temperature gradients may stabilize trapped particle instability /14.322/. In connection with drift trapped particle instability Coppi recently found a new mode driven by drift which he called ubiquitous instability /14.323/. These modes have frequencies in the range oo^j < ω < ω£#/ where ω£# is the electron bouncing frequency, ω-^χ = TjU>^/TE and ω| is the electron density gradient drift frequency ω* = = -kcTßkß (din n/dr) /eBQ and where ω^j < ω^-j < ω ^ < ^tE (spectrum of standing waves). For the standing electrostatic toroidal potential perturbations the expression Φ 1 = Φ (Θ ,r) exp[-i(ßt /
where q(r)
(q
= 1/ε
/ n
- q (r) )n (safety
+ in
(q (r) Q - ζ)
ip/AN
(14.320)
]F(Q) ,
factor),
qQ = mQ/nQ
and
m0,n0 mode numbers has been used. Shear in a toroidal trap reduces the growth rate of the electron trapped drift instability /14.324/. Generally, particle trapping by electromagnetic waves is not possible, however, if the phase speed of such a wave is low enough trapping becomes possible. So whistler waves may trap particles /14.325/. A second whistler wave may detrap these particles. The de trapping effect is, however, also possible in sheared oscillating magnetic fields /14.341/. Whereas for an electrostatic wave and a magnetostatic field the resonance condition reads ω = ωβ and ω = ω^ respectively, it reads ω = kzun - ηΩ for electromagnetic waves. Finite amplitude whistlers propagating in a nonlinear /14.327/ dispersive plasma give rise to self-focusing and self-trapping /14.328/. These effects are due to the dependence of the plasma dielectric properties on the amplitude of the wave fields and produce the selftrapping modulation instability (whistler modulation instability) /14.329/, which is a typical nonlinear effect. In Alfvên waves particles may also be trapped /14.326/). Trapped par(cyclotron trapped particles ticles which do not leave a restricted domain are very sensitive to collisional effects, see section 12.3. Thus the trapped particle instabilities (dissipative trapped particle instabilities /13.64/) and the plasma transport properties are influenced by collisions. Collisions may expel trapped particles into
591
PLASMA INSTABILITIES
the loss cone and the related perturbation can be The (approximative) disdamped at a frequency veff. persion relation is then (based on a Landau collision term) /8.13/, /8.48/, /14.291/
ω=
/- *
2
*2
iv r
2- + t X " ^ 7 " ~Γ'
(14.321)
The maximum growth rate is γ - /ε"ω* /8·48/. Small scale oscillations are excited most rapidly.
Very important is the dissipative
stability
dissipative
trapped
ion
in-
/8.48/, /14.330/, of fewer importance the
trapped
electron
instability
/14.331/.
There are not only linear eigenmodes of the dissipative trapped electron instability, but also nonlinear effects on the dissipative trapped ion instability, including anomalous scattering due to the development of an unstable loss cone ion distribution function. The electron mode is important for high density Tokamaks and may be stabilized by shear. It may exhibit a two-dimensional structure /14.476/. The low frequency electron modes have been identified in a simulation model to be low frequency dissipative drift instabilities. The dissipative trapped ion instability may be stabilized by mode-coupling through nonlinear saturation. It also may deplete trapped ions by anomalous
diffusion {trapped ion depletion sipative trapped electron drift
/14.469/). The disinstability has been
investigated experimentally and theoretically in cylindrical geometry (Q-machine with special mirror coils etc) /14.321/, but is not so important as the dissipative trapped ion instability. Whereas collisionless trapped particle instabilities are driven by magnetic curvature drift, dissipative trapped particle instabilities are similar to collisional drift waves in that the excitation of both depends on collisions. In addition to the usual dissipative trapped ion instability (which is basically an electron diamagnetic drift mode) there exist ion diamagnetic trapped ion inmodes of this type /14.332/ (residual
stability)
.
There exist many cross effects between trapped particle instabilities and other modes. Ion trapping may influence microwave tubes /14.340/, electrostatic modes influence trapped particle modes in toroidal plasma /14.333/, impurities may stabilize the trapped ion dissipative instability /14.334/ (but collisionless trapped ion modes are not stabilized by impurities /14.470/, if the impurity density gradients are
MICROINSTABILITIES
592
too weak), lower hybrid waves may excite or suppress dissipative trapped particle modes /14.335/ etc. On the other hand trapped particles also affect other modes. For example the thermal spread of trapped a-particles or the trapped ion mode influence the Tokamak cyclotron instability /14.336/, trapped electrons transfer energy to the ion acoustic instability, trapped particles have a strong effect on the characteristics of kinetic modes driven by an electron current along the magnetic field /14.334/, and electrically trapped particles produce new resonances due to the interaction with a whistler /14.338/. Impurities have various effects on dissipative trapped ion modes depending on whether the parallel wave phase velocity is above or below the impurity thermal velocity. In general it can be said that when the impurity density gradients are in the same direction as the electron and ion density gradients, the impurity ions are stabilizing /14.339/. On the other hand with impurities new modes may appear in the banana regime. Impurity ions can produce substantial variation of the electrostatic potential within a magnetic surface. The resulting electrostatic trapping and electrostatic drifts may alter significantly /14.477/. the result of transport theory It should also be mentioned that collisional trapped ion instabilities and non-resonant trapped particle instabilities (ω < ω^) can be stabilized by nonlinear saturation /14.340/, see chapters 16 and 19. In concluding this section we would like to mention that there exists another instability connected with the family of trapping instabilities /10.2/. This is the Hirschfeld instability, negative radiation absorption instability, nonrelativistic mode /14.343/. The energy source of this instability is a deviation from a Maxwellian distribution. Only in the case of a Maxwellian distribution does the plasma radiation temperature Tr which for non-maxwellian isotropic plasma is defined by oo
k
BTr
=
-§cr(u)f(u)u3du/
oo
^or(u)^£^u2du
(14.322)
o o equal the electron temperature TE. or is the cross section for spontaneous emission of energy Τζω by an electron of velocity u. A system in which stimulated emission exceeds stimulated absorption has a total
593
PLASMA INSTABILITIES
negative absorption and thus by Kirchhoffs law a negative radiation temperature. In this case the radiation intensity can exceed greatly the intensity from a system that has a positive temperature, because now the electromagnetic waves generated at some point within the medium are amplified in traversing it. So synchrotron radiation negative absorption is possible. It can be shown that negative radiation temperature arises at microwave frequencies when df(u)/'du > 0. A second necessary condition is obtained from integration by parts of the denominator in (14.322). When the radiation cross section ar is expressed by the absorption cross section a (u), the condition reads 3 (oa(u)u4) fèu < 0. This condition leads to a class of "trapping instabilities" by which radiation takes place in the form of bremsstrahlung and cyclotron radiation in a partially ionized gas, partly as a result of pressure broadening. The instability may be stabilized by suppressing the deviation from the Maxwellian distribution or by choosing certain stable ranges of the radiation parameters. The Hirsehfield radiation instability should not be con/13.54/, see fused with the radiation instability pp 321 and 361. There exists also a negative radia-
tion
absorption
instability,
relativistic
mode
/14.343/. This mode belongs also to the class of "trapping instabilities". In the relativistic case, the conditions for negative absorption of synchrotron radiation can also be satisfied in a fully ionized plasma, in a manner without non-relativistic analog. 14.9 INSTABILITIES IN CHARGE EXCESS PLASMAS
A charge
excess
or nonneutral
(Breit-Darwin)
plas-
ma /14.492/ is a collection of charged particles, satisfying (2.11) but not (2.7). To be in equilibrium it must rotate. Such magnetically confined nonneutral plasmas are found in electron ring accelerators /14.344/, /5.31/, in intense high current relativistic electron beams /14.345/, in nonneutral electron and/or ion clouds in toroidal and mirror magnetic fields / 14.346/. We consider first the electrostatic instability of a nonrelativistic nondiamagnetic nonneutral plasma /5.31/, /14.347/. The fluid equations of species continuity, of momentum conservation and Maxwell's equation give for a cold constant density plasma in linearized approximation the dispersion relation foi electrostatic waves. It reads, for a plasma column of radius rO,
MICROINSTABILITIES
594 K7 (k r ) I I ( f e i - J £ g g l z P k r. z'PKAk r )I7(fe P J Z a g Z 2 P 2
- 1 - Σ
- K' (k 3?„)I-(fe r ) I z P l z o - K 7 ( k 2» ) I - ( f c r ) Z z P I z o
\
ω„ (ε Ω + 2ω ) lr
S V
/
(14.323)
PJt(Tr
i l
)
v
2 g
Pg
g
g
g
(ω - fe z; 2
OS2
-Ζω ) g
r c is the radius of the confining conducting wall, Kj, Ij are the modified and Jj the usual Bessel functions,
Illkzrp ω
) =[άτί(χ)/άχ)]
s
=
ε Ω s s --
1 ±
, and zP 1/2 4πβ e n g P r o2 wΩ
χ=]
.1-21
(14.324)
g g is the rotation rate (angular velocity) of species s, is the s-component of the zero orε 0 Ξ sgn es, vosz der streaming velocity of species s, Z = 0,1,... is the order of Bessel functions and
2
2
T Z = -kZ z
Ps
1-1
s (ω - k v
where 2 v = [ω - k v g \
- Ζω )
z osz
s
- Ζω I
z osz
s)
I
- (ε Ω ss \
1-1
s
ωPs
, (14.325)
v
(14.326)
+ 2ωs )V
If the wave guide is completely filled with plasma, one has rp = r> . For a very thin plasma beam in a wave guide one has rp « rQ. For rp -* i» , (14.323) be= 0. Introducinq the effective perpencomes Ji(Tre) dicular wavenumber fef = p? /^J, where p^ m is the m-th zero of Jj(^) = 0, one may write the dispersion relain the form ^ι)Ί ω -η / (ω - Ζω -k v I 1L\ Ps/ \\ s z osz) I ίω - Ζω - k v 3/ \\ s z
Here the Cortex frequency by
) osz)
- ω
sv
h
4.327)
72
1
x
72
z
(Pötzl /14.347/) is given
+ ωsv, = - \[εs Ωs + 2ωs)) = ±ίω - ω"). (14.328) \ s s) One of the instabilities described by (14.327) is
595
PLASMA INSTABILITIES
the electron-electron two-rotating-stream instability which occurs when there are two electron components
(η#1 'nE2 w i ^ h rotation velocities ω-] = ωΕ, α)2 = ω£, defined by (14.324), see also centrifugal flute insta-
bility, rotational ponderomotive instability, p 283. ^ e For equi-density electron components (n^- = nE2) instability condition reads /5.31/, /14.384/ Αω
ΡΕ
72
ft
72
ft
, > 1 (14.329) 2 2 2 2 4 (ω ω ) Ε' Ε ftZZZ ftZ(ZZ - 4 - 1 4 6 | Z | 2 ) for I = ±λ , ±2, 6.. is the Kronecker symbol and ωρ^ = Ai\e2(nE^
+ nE2)/mE·
(14.330)
On the other hand an unneutralized rigidly rotating electron plasma is absolutely stable /14.365/ if the distribution function is a monotonically decreasand if rαω < c. EQ and Lz are ing function of EQ-uL2 the single particle energy and angular momentum around the 2-axis and ω is the angular velocity. A hot electron plasma with angular drift frequency comparable or larger than the ion cyclotron frequency can be stabilized by a small amount of cold electrons /14.475/. When (14.323) is examined in the limit of long axial wavelengths [k2 = 0) the modes may be called surface waves /14.349/. An unstable mode is the electron-ion two-rotating-stream instability, electron ion stream instability. sities are related by
η χ = fnE,
f Φ 1
If ion and electron den-
(14.331)
then the allowed values of equilibrium rotation velocity may also be calculated from (14.324) and depend on /. More details may be found in the specialized literature /5.31/. When perturbations of the electrostatic potential in a low-density, pure electron gas equilibrium with diocotron u>pE(r) « Ω2;, rij (r) = 0 are investigated the instability, also electronic instability, electron beam instability etc (p 283) is found. It is assumed that kz = 0, |ω- lu>E(r) I 2 « tiE and that the electron column is_rotating slowly with angular velocity It can then be shown /14.350/ that the ü)£(r) = b)E(r). equilibrium is stable if the radial density profile is such that
MICROINSTABILITIES
596 dUpE(r)/dr
< 0.
(14.332)
A necessary condition for instability is that db)p^(r) fèr change sign in the interval 0 < r < z» . An application of the diocotron instability is the hol-
low electron beam instability /14.351/ and the resonant diocotron instability /14.352/. Normally, the
diocotron instability has a growth rate comparable to the oscillation frequency {strong instability). It has a small growth rate (ω^ « tor) when the density gradient is weak in a low-density nonneutral plasma. The resonant diocotron instability occurs when there is a bump in the density profile. In plasma diodes large amplitude oscillations in the diode current arise at low pressures {plasma di. These oscillations occur at frequenode instability) cies ranging from a few hundred kilocycles to as low as a few hundred cycles per second and have waveforms suggesting a relaxation mechanism /14.24/. The period of the oscillation depends on the transit time of ions in the diode and the energy stems from the energy of the free particles. These free particles can be emitted from and absorbed by the electrodes. There are in addition trapped particles moving back and forth between the electrodes indefinitely. The stability properties of this system differ from those of an infinite plasma, because the latter does not contain free particles. Thus, a coupling arises between the free particles and the longitudinal plasma oscillations /10.2/. Resonant and trapped particle effects do not alter the situation as compared to that of an infinite plasma. Other authors are of the opinion that these plasma diode oscillations were initiated by high-frequency electron space charge instabilities /14.24/. The latter cause a re(Pierce instabilities) duction in the electron density within the diode but no substantial change in current. Then, on a slower time scale, the resultant positive space charge drives the positive ions out of the diode. Due to the ion inertia the process goes too far and the ion density falls below the electron density. The negative space charge inhibits the electron flow and reduces the current to a small value. On the slow time scale, the positive ion perturbation builds up again and the current swings back to the initial high level. The process repeats itself. If the growth time is short compared to the transit time of the bulk of the free particles the coupling becomes weak and the dynamics ap-
597
PLASMA INSTABILITIES
proaches that of an infinitely extended system without electrodes and the instability becomes stabilized. When a stream of charged particles is sent through an unmagnetized plasma with isotropic temperature, a bunching of the beam structure produces an induced magnetic field which gives rise to a Lorentz-force and to further bunching. As a result, a transverse unstable electromagnetic wave arises {cross-stream in-
stability
in an unmagnetized
plasma /11.56/). The in-
stability is macroscopic and absolute and resembles the Weibel instability of an anisotropic plasma.
The current
chopping
instability
{current
inter-
ruption instability) /14.25/ is related to the spacecharge and Pierce instabilities. The phenomenon is a gas discharge effect which occurs when a critical current density is exceeded and involves one or more rapid reductions in the current and sometimes complete . A decrease in plasma denextinction {arc starvation) sity or a constriction of the cross section requires an extra voltage to keep the current constant. The necessary acceleration of electrons within this region produces two double space-charge sheaths. The sheath at the cathode causes electrons to enter from this side as a beam having a mean energy equal to the sheath potential plus the mean thermal energy. The potential hump thus formed causes the ions and electrons to be accelerated in a way that lowers the density even more: the initial disturbance is enhanced /10.3/, /10.2/. It has also been suggested that a local magnetic compression of a discharge could induce current chopping. The instability can be stabilized by a thermal spread of the distribution function and by collisions. The instability of an E-layer (see p 149) is also a nonneutral instability. In a layer of mono-energetic non-relativistic electrons having no spread of guiding centers rotating in a uniform magnetic field, oscillations are unstable for an arbitrarily small average density. The growth rate is much longer than the period of rotation of an electron /14.468/. On the other hand, the sporadic equatorial E-layers in the ionosphere may be explained by the cross field in-
stability
/14.353/. The E-layer
instability
/14.354/
depends sensitively on the density of the background plasma, but is essentially independent of its kinetic structure. The real part of its frequency is determined by the gradient of the external magnetic field. Increases of the magnetic shear of the equilibrium
MICROINSTABILITIES
598
tend to stabilize the configuration at moderate wavenumbers. It has also been shown /14.366/ that charge sheets of ions parallel to magnetic field lines are unstable whereas charge sheets of electrons parallel to magnetic field lines can be stable. In connection with studies of ring currents of relativistic electrons contained in a magnetic field
(electron
ring
particle
accelerators
/14.355/) inves-
tigations have been made of the relativistic electron ring instability /14.356/. Density profiles, thermal spread and the Budker parameter n^e^-/2i\r-pm^c^ /1 5.1 93/ play a role in its theory. Also a dipole instability of a relativistic electron ring loaded with ions has been found /14.357/. Pure electron plasmas show instabilities of their
own: electron
gas instability,
instability
of
layered
electron gas /14.358/, electron plasma oscillations in the presence of Brillouin flow (see p 99) /14.359/,
electron gas losscone instability /14.360/, gas two-stream instability /14.361/, ordinary magnetic instability (plasma electromagnetic
electron electroinstabil-
ity) in electron gas /14.362/. In connection with the theory of these instabilities a thermal Larmor radius in[c^h/(ω# - ω#)2]1/2 has been defined /5.31/. Flute stability may also be found in a nonneutral plasma /14.364/. The stability of a degenerate electron plasma in a strong magnetic field (degenerate electron plasma instability) depends on the equation of state used and on the collision frequency /14.367/. In grossly nonneutral plasma an ion resonance instability has been found /14.363/. It may occur when ions of high energy are added to an electron cloud of low density (ωρ < Ω^) contained in a magnetic field. When rLI exceeds the transverse dimensions of the configuration and the ions are trapped in the strong electrostatic potential well produced by the electron cloud, the oscillatory ion motion becomes coupled to the azimuthal E*B drift of the electrons thus producing growing modes. The energy source of the ion resonance instability is the electrostatic field of the electron cloud. The instability may be stabilized by choosing the magnetic field strength between two limits determined by the electron density. Another instability related to the diocotron instability is the magnetic instability occurring in magnetrons. The properties of an inverted ion magnetron in which a confined ion cloud was produced have also been studied. Above some critical magnetic field a strong drift instability with frequency of the or-
599
PLASMA INSTABILITIES
der of 100 kHz was observed. This instability has been called inverted ion magnetron instability /14.368/. It is characterized by a spiral character, and by the electron acceleration in the electric field of the runaway regime. The instability generates intense rf emission and may be used for wave generation. It also seems to be of interest in astrophysics and planetary physics. Several other instabilities in nonneutral plasmas have been discussed in the literature. A review of glow discharge instabilities {attachment instability, recombining plasma instability, decaying plasma instability, recombination instability, thermal instability, dissociative attachment instability in partially ionized molecular plasmas etc) has been given /14.369/, /14.370/. A nonlinear rotary wave in ion plasmas /14.371/ and the transformation of space charge waves at plasma density discontinuities /14.372/ have been mentioned. Instabilities in negative ion plasmas are important for negative ion sources. Severion al papers appeared recently treating the negative plasma instability, hydrodynamic instability of negative ion plasmas /14.373/. A fast ion mode exhibiting spatial growth in the presence of a fast ion beam has been found. In these multicomponent plasmas with one negative ion species crossover frequencies (the frequencies at which the refractive index of two modes of wave propagation are equal and are of the order of ion gyrofrequencies) play an important role. External electric fields may cause an electron-ion plasma to behave like a charge-excess plasma and modify longitudinal oscillations /14.374/. Similar effects appear in rotating non-uniform plasmas /14.375/. The neutral deviation drift instability which is due to small deviations from quasineutrality has been discussed in volume 1, p 391. Also the ion space charge /11.28/, /14.376/ has been mentioned instability there (p 284). Quantum-statistical systems like an electron gas have also been investigated /5.9/. If the distribuis continuous tion function (e.g. Fermi statistics) and has a single maximum, the system is stable. A Fermi gas in its ground state (zero temperature) is able to sustain a permanent collective oscillation ω = ωρΕ for k ■> 0. For shorter wave-lengths the dispersion relation reads for neutral systems 2
ω
2
j . -37 2
2
/ c
2
= ωρΕ + 3k uF/5u>p
_ 2
7
4
+Π k
/A
2
2
/4mEupE.
MICROINSTABILITIES
600
Hereft= Α/2π, where h is Planck's constant and UF is the Fermi velocity uF = (37z32/87rm|) 1 ' 3 . This result point shows that in the ground state part of the zero energy comes from collective modes in which a large number of particles oscillate in phase. 14.10 ANOMALOUS TRANSPORT For the operation of laboratory devices the transport properties of plasma - mainly diffusion coefficients and electric conductivities - are of paramount importance. Transport properties depend not only on the force law between plasma particles (see p 120) but also on geometric factors (see p 336) /14.378/, on microinstabilities (p 348), and on turbulence, as well as on other nonlinear effects. Abnormally fast transport across a magnetic field not due to collisions is called anomalous transport. It is a phenomenon closely related to instabilities. The transport caused by microinstabilities is in general in excess of and relatively unrelated to classical transport, compare (7.85) and (12.95). Collective effects on e.g. the high-frequency conductivity (ac resistivity) of a fully ionized plasma have been investigated by Dawson and Oberman /14.377/. They found an enhancement of the classical resistivity around the plasma frequency. The effect is a nonlinear parametric effect. Other candidates for the source of anomalous transport instaphenomena are the current driven ion acoustic cyclotron bility, the current driven ion (electron) of various types and instability, drift instabilities other phenomena like magnetic effects, electric effects, fluctuations etc. It should also be mentioned, that trapped particles and untrapped particles have quite different effects on field perturbations (instabilities) . If trapping occurs, the plasma behaves like a two species plasma. Untrapped particles are free streaming with a Maxwell distribution along the field lines. Trapped particles (temperature ^/εΤχ) have a Maxwellian distribution with an additional term. (For ε see p 340). The current driven ion acoustic instability of interest here has a frequency near the ion plasma frequency. It is unstable if the current drift velocity ^ο = J/ne i s large enough that uQ > as(mI/mE)U2(TI/TE)3/2exp(-TE/2TI)
,
(14.333)
601
PLASMA INSTABILITIES
where as is given by (9.51). In many Tokamak experi< 2, so that the ments one has TE/TZ < 4 and uQ/as modes are stable. But even if the mode is stable and weakly damped, it might cause an anomalous electric resistivity. For fast ion acoustic waves ion Landau damping decreases (p 211 /9.4/). Even recently many papers have been concerned with anomalous transport due to ion acoustic instability /14.379/. On the other hand, the interaction of ion acoustic waves with a slightly inhomogeneous magnetic field generates an electric current which can produce intense and rapid changes in the magnetic field {ion/14.467/). acoustic distortion of a magnetic field Anomalous electron transport coefficients have also been derived and it has been shown not only that effective collision frequencies can be larger than in classical theory but also that there are differences in the structure of the transport equations as well (Dunn /14.379/). If it is assumed that an electron drift driven by an applied constant electric field causes the ion acoustic instability and that a steady state is formed in which ballistic clumps of plasma behave like dressed particles and collisionally scatter each other the electric conductivity is given by (Kadomtsev, Pogutse or Dupree /14.379/) σ * -[OupE/k\D.
(14.334)
The full theoretical treatment of ion acoustic resistivity must, however, comprise a nonlinear theory
(ion acoustic
turbulent
resistivity),
see later. Plas-
ma heat conductivity also differs greatly from the classical values when ion sound turbulence is present /14.379/. The current-driven ion cyclotron instability may also induce anomalous transport /14.380/. This instability may be excited in Tokamaks by finite amplitude trapped-ion modes. It is, however, found that although ion cyclotron instabilities are likely to be present, they are not likely to modify the diffusion rates originally predicted by Kadomtsev and Pogutse for anomalous transport due to trapped ion modes, see later. In recent Q-machine experiments it was found that the high frequency electrostatic (and electromagnetic) ion cyclotron and ion cyclotron drift instabilities generate anomalous electric resistivity which inhibits the destabilizing current when the plasma is stable to ion acoustic modes. Finite heat conduction effects on the ion cyclotron and drift
MICROINSTABILITIES
602
cyclotron instabilities have also been investigated and found to be destabilizing. The nonlinear development of the electron cyclotron drift instability may also generate anomalous electric resistivity. The anomalous skin effect (and the anomalous electron viscosity) may be explained by an electron plasma mode driven by a gradient in the current drift vedrift locity u0 over the plasma cross section (current velocity stability
gradient instability, current gradient in/14.381/) or by the ion acoustic instability.
Another important class of microinstabilities generating anomalous transport are the drift instabilities (mainly as trapped particle instabilities). When trapped particles are taken into account the drift waves are modified /14.393/ due to a change of the dispersion properties. In the presence of trapped particles the resonant interaction caused by particle bouncing between the mirrors is reduced by (ω/ω^)^. Then a short wave (k±rLj » 1) electron drift instability ω > u)f arises with a growth rate γ - u>i>i?/}/k±rLI, +Tj/TE). This instability is inif ω*/ω££ > SK\rLIC{ dependent of curvature and can arise in systems with an average minimum of B. Also ion drift instability may appear for (ω > (Jj) with γ - ^^jkir^j k
>1
l^LI^i'/^bI
+Tl/TE-
Since the wave number k§ in the θ-direction has a minimum value of κ (p 352) and a maximum of rj^ , the range of drift frequencies ω* (ρ 372) is given by ω* . = νηκ < ω* < vnr~\ rmn
D
= ω*
D LI
.
(14.335)
max
When some processes in the plasma generate a phase difference between the electrons and ions (having the same £x£-drift - see p 52) then the drift wave may grow due to particle-wave energy and momentum exchange. The type of drift wave generated depends on the collision frequency v, the bounce frequency ω^ (4.45), (14.306) and the curvature
drift
frequency
ωβ = kßVD = ω*, see also p 386. Drift waves will therefore be classified as follows /14.387/: 1) dissipative
drift
v * ω*, 2) dissipative
wave /14.382/
v > ω* . > ω,-,
untrapped
rmn
bE
trapped
v > ω, ρ/ v < ω*
.
particle
(14.336) mode
(14.337)
Trapped particles before they bounce and bE' collidemax
603
PLASMA INSTABILITIES
hence do not know that they are trapped. trapped electron mode, type I /14.384/ 3) dissipative ( 1 4 3 3 8 ) vE < %E' VE >
VE
K
%E'
V
E < °-'
(14
'339)
The DTEM (dissipative trapped electron mode) may also be characterized /14.391/ by ω | < νΕ/ε and ω | > νΕ/ε and u)£j < ω < ω ^ , Vj < ν^/ε. The collisionless mode mode with ω < b)fo£ is sometimes called the ubiquituous /14.378/, /14.323/. This mode does not require the ions to be in the banana regime and is unstable with and without the effects of collisions /14.415/. trapped-ion mode /14.385/ 5) dissipative V
J
K< ω
2>1'
{ V
EK
b)
bE f °
rΤ
Ε= Τ Ι ) Λ
(14.340)
The growth rate is highest (γ - /εω*) if ν^/ε^/εω*. Electron collisions increase the growth rate, while ion collisions stabilize. Landau damping by transiting ions stabilizes short wave modes. The drift wave is supported by just the trapped particles. The DTIM(dissipative trapped-ion mode) may also be characterized /14.391/ by Vj/ε < ω < ν ^ / ε , ω
< o)j>j.
6) collisionless trapped-particle flute (interchange) trapped particle mode /14.386/ (CPTM - collisionless mode) v < vDE(rR)'^/2.
(14.341)
Estimates of the diffusion coefficient due to these instabilities may be found from (7.80) by replacing τ~1 = v by γ and rL byfcj^.The growth rate γ of the dissipative drift wave is of the order of ω^ and hence (for ^m^n > ω^) (Kadomtsev formula) roughly D * yfe~2 * ^*fcj2.
(14.342)
This resembles the pseudoclassical formula (12.95) /12.86/, /12.72/, /12.97/. More refined calculations give the following results: When (14.337) is valid, one obtains /14.383/ for the electron heat conductivity
MICROINSTABILITIES
604
«E *
(14 343)
·
°Λ*ΕΑΕ.
where Cj, are numerical coefficients chosen empirically and D*
C VT 2 \E
°r
^"1/2r"1/4f
(14.344)
compare (12.95)/ p 348 {-pseudoclassical theory). Losses decrease with increasing TE (like in classical theory). When (14.338) is valid, one has /14.384/ again γ - v - ω* and ~ 3/2
E
2 *
rmn
-1
TErmn
(14.345)
El
(where κ is defined on p 352, ^^^min *-s ^ e minimum electron-tempe rature gradient drift frequency (c/eBr) (dT/dr), compare p 381. κ^ increases with TE. The diffusion coefficient is D * εκΕ.
(14.346)
Transport due to this drift regime (dissipative trapped electron instabilities) is very unfavorable since it increases with plasma temperature as τ|./14.494/. Stabilization is possible by magnetic shear. When (14.339) is valid, k is not determined by v - ω*, but by the magnetic shear /14.384/. Now the growth rate is 3/2 * -1 (14.347) γ - eJ/ ω ω ^ and the electron heat conductivity is K
E
s C 2
2*EAE>
(14.348)
where C 2 - 10 , depending on the magnetic shear strength, and D ^ V F 2 > ? J · T ^ e trapped-electron instability plays a major role in a non-isothermal plasma {TE > Tj). In the nonlinear regime of the trapped electron instability the expressions for D and κ are again modified /8.48/, /14.394/. In the nonlinear domain also, increased radial transport is possible due to magnetic island formation generated by the tearing instability /14.395/ or due to convective cells. When (14.340) is satisfied (dissipative trapped ion mode), the mode frequency is lower than the ion bounce frequency. As with the dissipative trapped electron modes, the trapped electron collisions have
605
PLASMA INSTABILITIES
a destabilizing effect (trapped ion instability /14.384/)· However, trapped ions and ion-bounce resonances have a stabilizing effect /14.385/. For collision frequencies lower than 0. 4ω^/ (mET^,/mjT^) 7/1 8 the threshold for this instability could be raised by elliptical shaping of the minor cross section or by artificial ion detrapping (ion depleting) by magnetic oscillations /14.388/. The mode is unstable and D
»
0
+V2V2)'(14-349)
where the numerator ^T 2 5~ 2 . Transport grows as T J ' 2 . In a two component plasma a new trapped ion instability may occur due to the presence of the hot ion component. The new instability leads to enhanced diffusion /14.389/. The effect of trapped ions on transport phenomena is intermixed with geometric effects. Let us recall some of these effects. In the superbanana regime (pp 162, 347) occurring in a stellarator or in a slightly corrugated Tokamak the bananas are displaced across the magnetic surfaces. This displacement is independent of the magnetic field strength so that even for very strong fields, diffusion and thermal conductivity on the bananas may be very intense in a rarefied plasma /14.392/. These effects are only observed for v < ε 3 / 2 £ £ ^ 0 (kQ is the wave number of modulation of the field BQ). The two types of field inhomogeneities in a steltarator or a corrugated Tokamak (non-uniformity of the winding and inhomogeneity due to torroidal curvature) or a bumpy torus like ELMO are described by /8.48/ (h refers to helical, t to toroidal): Bz = so M - eh 7cos(Z0 - k z) o
- ε+ t cos
θ] ,
(14.3 50)
where Θ is the (small) azimuth, z the coordinate along the torus axis and I is the number of winding depressions (e.g. I = 3 for 3 winding gaps). Nonaxisymmetric toroidal magnetic field ripple caused by discrete coils can form an additional trap for particles /14.474/). This may occur in a flux (ripple trapping conserving Tokamak /14.428/ (FCT which offers a true beta limit of 10 - 30 % instead of 3 % as previously assumed for Tokamaks). Such a toroidal field ripple can lead to anomalously large particle and energy losses /14.471/. The fact that not axially symmetric toroidal systems like stellarators or bumpy torus exhibit a substantial increase in the diffusion coeffi-
MICROINSTABILITIES
606
cient and in thermal conductivity is known since some time. So it has been shown /14.479/ that either for low collision frequency or large electric fields transport coefficients in not axially symmetric toroidal systems do not depend on the magnitude of the magnetic field. Transport of plasma across a braided magnetic field has been treated in /14.466/. Such a magnetic field occurs in a region through which magnetic field lines meander in a stochastic fashion and in which the magnetic surfaces are destroyed. This might happen due to asymmetric magnetic perturbations. Diffusion due to magnetic braiding and magnetic field reconnection (leading to disruption) are responsible for enhanced electron heat conductivity, plasma dis-
ruption
and sawtooth
oscillations
/14.497/.
For a stellarator ε+ « ε^ « 1, for a Tokamak ε^ « ε^ « 1. In a toroidal stellarator the main role
is played by localized
-particles
{blocked
particles)
trapped by the inhomogeneity due to the helical winding. This field inhomogeneity is weaker and covers a larger distance than the (stronger) inhomogeneity due to mirrors. Particles not trapped by local mirrors will be blocked (trapped) by these large scale inhomogeneities. The fraction of blocked particles is /ε^. The bounce period of blocked particles between the mirrors of the helical winding depends on the pitch angle ψ and the energy. Averaging over the energy gives ck T
^bh
" —^2 ε 7ζ^ ( ψ )
= ω
// ( Ψ } " ωΗ'
(14.351)
eBr
since /(ψ) - 1. r is the small radius. For veff - ν/ε/j > efockßT/eBr2 Ξ ω^ one may neglect ω ^ « /8.48/, and obtain for the diffusion coefficient /14.392/
v
eff
2 D
- 1 ^ 7 Z7 2 ^ ' h
For veff
v
>w
(14 352)
·
< Wfofo one has z2 ck T
D - ^ΓΤο -Â-> £.1/2 eB h
In a Tokamak
(14.353)
with slight corrugation (eh « ε,) the
607
PLASMA INSTABILITIES
main inhomogeneity is due to the toroidal curvature. The corrugation produces only a small fraction / ε ^ of blocked particles. The diffusion coefficient is now given by /8.48/
Dα
1ieh\3/2
ι(τι) 0
Ό
V
1
-vDo> D
= c
(14 354)
·
r
R
a
eau
where v^ = ^ ^ ^ / ^ ' o th î^^ o ^V^ ^ diffusion) . Transport coefficients are modified by trapped instabilities /14.392/. The trapped-particle effects are strongest in highly rarefied plasma, when the electron collision frequency is smaller than the drift frequency. At low collision frequencies but V E > ok-βΊIeBv^ the main role is played by the dissipative trapped-ion instability. From (14.342), (14.349) and then D - T^E^DQ/TEZV
(14.355)
is obtained /8.48/, where Tj- < TE, ε = r/R, v ^ = v-j diffusion for c-f-fo = ctfo-g. We see that at Tj - TE the at v^ = v ^ exceeds the classical value DQ coefficient by a factor ε"^ and decreases as vE increases, see also p 345. When (14.341) is valid, i.e. at very low collision
frequencies v^ < ck^T/eBr*, the dissipative trappedion instability is modified into the trapped-particle
flute instability, and D and κ are no longer dependent on v#. For ω < ^hE' oscillations in the trapped electrons with ω > ω| may be excited. The trapped particles are then a source of oscillations. For ω < b)foj the main instability source is the resonant interaction between trapped ions and the mode ω > u)j /14.393/. Such flute-like oscillations (quasi-flute modes /14.378/) with frequencies ω « OJ^J have been considered /14.396/ taking into account temperature anisotropy and other effects. The instability is called flute-like because it resembles the ordinary MHD flute instability in a mirror machine, in that it is driven by the combination of the radial pressure gradient and the unfavorable drifts of the trapped particles, and in that it has nearly constant perturbed potentials along the magnetic field lines. The flute mode takes place with a frequency /14.384/ ω 2 - ε 1 / 2 ω*ω π ,
γ 2 = -ε 1 / 2 ω*ω η ,
(14.356)
MICROINSTABILITIES
608
where ω η = kv„ = k ckOT/eBR. u u y D
(14.357)
This mode can give rise to transport similar in scaling to DBohm (7.85) /14.387/. Stabilization of the collisionless trapped particle mode is possible by finite beta and cross section shaping /14.400/. For magnetic wells radial plasma transport due to low frequency instability (ω « üj) is described for dB/dr > 0 by the diffusion coefficient /14.399/ D]L - rLIckßTI/eBR(TE/TI
+ m^m^.
(14.358)
Odd modes /14.378/ for which the perturbed potential changes sign somewhere along a field line have been found in collisionless and collisional regimes /14.398/. Doubts have however been raised /14.455/ if these odd modes exist in reality. Coppi showed /14.397/, /14.378/ that a periodic dependence of magnetic drift on particle motion along a field line gives rise to flute-like perturbations. It is characteristic for this instability that the phase speed changes sign according to the sign of (1 - TE/Tj). For TE > Tj the mode propagates in the direction of the Larmor drift of electrons, whereas for Tj? < Tj the mode propagates in the opposite direction. Shear is effective in reducing the anomalous particle diffusion due to the collisionless drift wave instability. There are some other instabilities which may influence plasma transport properties. An instability for which the toroidal configuration is insignificant temperature instability. It yields a is the drift thermal conductivity K * 1θ"2ε""2κ ,
(14.359)
where κ 0 is the classical ionic thermal conductivity /8.48Λ Historically, the screw
instability
in a
partially
ionized plasma (see p 292) seems to be the first instability connected with anomalous transport. It has also been investigated recently /14.401/. The onset region instability causes an increase of the cathode of the wall ion flux in the plasma column. Also the two-stream
instability
(two-stream
ion
cyclotron instability) has been investigated as a candidate producing anomalous resistivity by ion
609
PLASMA INSTABILITIES
bunching /14.402/. When the electron drift speed is larger than the electron thermal speed, the electronion two-stream instability occurs. Then the electronion collision frequency v#j is said to be increased instability, to the growth rate of e.g. the Buneman see p 472, such that V
eff
* V(V
W
) 1 / 3 i
I
(14.360)
This increases momentarily the effective electric resistivity, but since the beam loses its energy, the ion acoustic instability will appear. Sagdeev reviews electric conduc/7.7/, /1.34/, /1.20/ the anomalous tivity
produced by dc current
driven
instabilities.
The following table summarizes the situation: instability threshold frequency growth conductivity Buneman / 1 4 . 4 0 3 / ( c o l d e l e c t r o n - u n > c,l7P . , v D thE ion two-stream) von
sound
,
^ V^ >a
Drummond-Rosen-
bluth /14.405/ ,. ' (two-stream . v Ί , ion cyclotron) fied Buneman)
Bernstevn
/14.407/
<ω
^
σ
1/3
-1 1/3 ù} PFÎmT\ p 4 7 2 —^—(—) * 2 xm^l E 2 o)-Tvn ne a0TT PI D SI
Q
„ v^> a D
electron sound / 1 4 . 4 0 6 / (modi- vn < e . ,
imw\ ω^ί— PE\mT) I
n
ü-r
T
I
« Ω_,
Q
*Λΐ
**£
ID cthE ,7_
g £>
c ft£,
The de resistivity values described roughly by these growth rates are, however, based on the nonlinear stage of the respective instability. Anomalous transport processes associated with the lower hybrid instability have also been discussed /14.408/. There appears an enhancement of the electron transport of heat and momentum due to electron turbulent motion in the field of thermally excited lower hybrid waves. Nonlinear and parametric effects are included. Anomalous heat transport may also be due to tearing instability /14.409/. The appearance of trapped particles increases the resistivity as a consequence of the fact that only the circulating particles can carry a current and that in
610
MICROINSTABILITIES
the presence of a dc electric field parallel to B the trapped particles lose momentum to the confining magnetic field and therefore provide a frictional momentum sink for the circulating electrons /14.410/. Also a localized collisionless instability that occurs in a nonuniform or skin current flowing along an extervisnal magnetic field is shown to produce anomalous cosity giving cross field diffusion of the current. This instability occurs under conditions where the ion sound wave is stable /14.416/. Trapped particle instability theory /14.411/ is also important for describing the effects of impurity ions. The trapped ion modes may stabilize or destabilize impurity /14.412/. Impurity ions, also α-particles, may excite special drift modes. Since in laboratory plasma °th-imp « Gthl <
611
PLASMA INSTABILITIES
ω - = cE./reB el
(14.361)
1
appears /8.48/. For u>ei > ω^ (for ω^ see p 606) the drift trajectories are strongly distorted. Then the bananas in non-symmetric systems will be untrapped. This leads to a diffusion coefficient D - e?v1/2(feDT)25/u) ,e2EBr t
for v < ε^ωβ£.
B
F o rv > ε
ΗωβΙ
eL
o n e
(14.362)
°btains
D - elel/2c2klT2/ve2B2r2. t h B
(14.363)
Particles trapped in strong electrostatic waves may also undergo trapped particle instabilities /14.417/. Nonlinear waves with electrically trapped particles can display new physical properties. An electric field externally applied along a magnetic neutral line results in a redistribution of plasma and excites instabilities /14.418/. Other efrefects which have been investigated are anomalous
sistive
instability
/14.433/, conductivity
sum
rules
island
for-
in two component plasmas /14.419/, transport coeffi/14.420/ and effects cients in relativistic plasmas magof strong magnetic fields /14.421/. Very strong netic fields not only alter the kinetic equations /5.9/, /14.421/ but also the transport coefficients directly. Another magnetic source of anomalous transport is the break-up of magnetic flux surfaces by
kink
and tearing
instabilities
{magnetic
mation) . Across magnetic islands transport is enhanced. If there are adjacent island structures with different periodicities magnetic braiding occurs /14.422/ (see p 606). This means that large radial excursions of field lines and even contact of the plasma with the vessel wall occurs. Break up of flux surfaces modifies trapped particle instabilities. Besides instabilities there are other processes which might be responsible for anomalous transport. Due to long range polarization fields arising from scattering convective instabilities a quasiclassical process /14.423/ may occur. This process can be regarded as being the transition between a stable collisional plasma and a fully turbulent plasma. When a microinstability grows, the fluctuating fields produced by the instability scatter plasma particles and in velocity space /10.14/. As we will cause diffusion see later, this (nonlinear) process can be described
MICROINSTABILITIES
612
as induced scattering of a particle with momentum p conby a wave with momentum %Kt where ft is Planck's stant. (Real quantum effects are however negligible.)
The trapped
ion scattering
instability
is also de-
scribed in a review /14.378/, which classifies nearly all other collective modes in confined toroidal plasmas as follows: slow modes (ω < CÜ^J « ω ^ ) trapped ion scattering instability impurity sound instability impurity instability interchange trapped particle instability dissipative trapped ion instability quasitrapped ion instability intermediate modes (ω < ω ^ ) circulating ion instability electron drift instability dissipative electron drift instability ubiquitous trapped electron instability trapped electron scattering instability fast modes: ω > u t r a n s i t I circulating electron instability current electron instability. Another process leading to enhanced transport are fluctuations. The spectrum of fluctuations is quite different in an unmagnetized three-dimensional plasma from that in a two-dimensional (i.e. magnetized) plasma /14.424/. In the limit of extremely high level of fluctuations Böhm diffusion ^λ/Β appears. Also the fluctuation spectrum of a low density (u)pj < üj) twodimensional plasma can produce Böhm diffusion
("vor-
tex diffusion", "two-dimensional diffusion"). Studies of plasma diffusion across a magnetic field in two and two and one half dimensional models /14.429/ show that at large magnetic fields collective transport by convective motion can dominate over diffusion due to particle interaction. Plasma transport properties transverse to a dc magnetic field can be dominated by fluctuations in the electrostatic field whose time dependences are slower than the ion gyro-frequency. In twodimensional turbulent fluids (interaction of line vortices) and in two-dimensional guiding center plasma the negative temperature instability may appear/14.425/. Fluctuations and turbulence are basically nonlinear phenomena and will be discussed later (chapter 19). Fluctuations of an enhanced level can be thermal turbulence. Periodic fluctuations average out and give no contributions to anomalous transport.
613
PLASMA INSTABILITIES
kB.
/
jo
Fig,
53.
Geometry
in Stringer's
theory
A general theory of fluctuation driven anomalous transport has been given by Stringer /14.426/. The geometry used is depicted in Fig. 53. Low frequency potential perturbations are described by plane waves Φ ( ί , £ ) = $(x)exp[i(k
z - ut) ] .
y+k y
(14.364)
%
The assumptions ω « üj, krLI « 1 and 1 « kzX « « (βΦ/kßT)"3/2f where λ is the mean free path, are made. If collisions are ignored, the kinetic equation in guiding center variables reads df au df d u f s J J s z , s 1 = 0. + («·ν)/ + Λ dt du dt 2 dt z J
(14.365)
du
Expansion of the species distribution function / in s powers of e^/k-nTs gives B
f
=
** s
f (0) + f J
s
(1) + f ( 2 ) + *s
•'s
(14.366)
Using = 2knT /m , ω- = ω - k u 'ths B s' s 1 z so as well as the drift velocities
(14.367)
MICROINSTABILITIES
614
T an dT 1 V = Dns vin~-d' DTs 7~ß ~d# s s s a n d u s i n g γ « ω-j^, ω<\ = ω ^ + ί γ , S t r i n g e r / 1 5 . 5 6 / t h e f u n c t i o n -2Z1 = 1 + # Z U ) V
=
— I —
/FJ
— CO
'
expands
X
dz; = 1 - 2*e * l e * du + ifuxe
VX
(14.368)
(14.369)
J
for real arguments x and then for small x, i.e. for th<< ^/k-z' which holds for most instabilities. With the definition of heat current (6.17), the heat flux in ^-direction may be defined as c
1/3ΦΓ1 2.(1),3 \ „n. (ΛΑ d U /· (14.370) «ax " S \ ^ J 2 ^ ?s This gives the heat flux and the thermal conduotivity for electrons and ions as a function of the observable quantity Φ =
H
/π y 2 c , k
E
E l eB\
-¥υυ0ΤΕ)\ΐψ-Ε\ ^J
' ,y J / 5 ω eß U
"tftia ω
7
r T
z oE
7
y DnE (14 371)
-
E
3. I 4 y\vDnI
βΦ
DTI
' T. r
Vi'
The particle flux and the electric current ^";s " σΙΙ^ο^ Z-^tf c a n a ^ s o ke found. The transport coefficients satisfy the Onsager relations of irreversible thermodynamics (except in turbulent state and for diffusion anomalous processes). For drift waves the coefficient
is
n
β
25yk
v _(fe v _ + 2 a l k 2 / ω l) . 2 y nDnE y nDnE S ζ' r \ βΦ \ 2
Ύ 25,
^Φ
fc2 Τ ' ^ Τ ;
Since experimentally γ 1 ^ * £ 1 40
2
'fr Φ
k 2(/
,2 u 1 #Where U
e$/kßT
Vs
7
S
r VM
- 0.03,
one
I
-
(14.372)
has (14.373)
615
PLASMA INSTABILITIES
which should be compared with Kadomtsev's formula (14.342). The results for drift waves may always be compared with the results for ion acoustic waves, since from Fig. 39, p 3 54 (volume 1) it can be seen that the drift mode becomes an ion acoustic wave for large ^2 >:> ky For the diffusion driven current {bootstrap current) , which has the definition
(14 374)
·
h = Kv*> y
where Yz is the particle flux in the ^-direction, the result is
h - *o-r^r·
* ^i$-J
·
(,4 375)
·
All these values of transport coeffficients will be enhanced by toroidal trapping. It should also be remarked that in experiments, drift waves have been detected which did not give rise to anomalous transport. During the growth phase of a finite amplitude wave Φ the particle flux Tj = -< θΦ/3ι/) nf>/B is outward, during the decay phase of the damped wave the flux is inward. The outward flux may be responsible for losses of plasma occurring in a stellarator {pump-out). The pump-out rate is defined by 1 = /din n\ _ /din n\ \ an J no instability* τ \ an iinstab. pre sent (14.376) Pump-out is possibly connected with enhanced loss due to Ohmic heating current in the collisional and intermediate regimes /14.430/. Pump-out should not be confused with burn-out, the rapid decrease in the density of neutral particles in a plasma due to the fact that the mechanisms of neutral particle formation are slower than the ionization, charge exchange etc. In concluding we would like to mention that anomalous transport processes have also been investigated recently using numerical codes /14.431/. In this confor a viscous, nection a newly found energy principle resistive, heat conducting, finite Larmor radius electron plasma is of interest /14.432/. Runaway transport in Tokamak has been discussed in /14.472/. Increased electric resistivity may also be due to
MICROINSTABILITIES
616 ouvrent
density
gradient
instability
/14.473/. Cross-
field energy transport by plasma and ion waves has also been found, even in thermal equilibrium, because wave propagation is uninhibited by the magnetic field. Fusion reactions also modify transport /14.478/. 14.11 SCALING LAWS AND DISCHARGE REGIMES
Plasma transport properties decisively control the scaling laws for the construction of larger and larger devices and dictate also their operational regimes. Authors mainly use either the electron heat conductivity /14.435/ or the diffusion coefficient /14.436/ to define various regimes. For given density, E = const, and with a constant toroidal magnetic field Mercier writes for the electron heat conductivity as a function of r, T, BQ the general equation (for K Q see p 608) K
E
=
K
(/(r)Ta(r/Vß
(14.377)
and then specializes it for the following regimes classical (collisional, p 124, Sp = Spitzer) K
Sp
s
pseudoclassical
T5/2
'
(empirical law - see p 348)
K
ps * n2/Bl
or -nßZeffI~2Tß^/2, (Artsvmovich
plateau
scaling
V^E'
(14.379)
where I is the total current /14.443/).
(p 346) (empirical) K
pl
banana
(14.378)
nT3/2r/BQ/
*
(neoclassical, K
p 346)
2 2 2 3/2,7,1/2 - n r / DB~r ' T ' , Θ
nc
(14..380)
(14..381)
Böhm (p126) (assumption not proven) κΒ trapped
(14.,382)
T,
«
particles Ktp
*
nT3/2r/BQ.
Recent Tokamak experiments have shown κ7 •z1£2n-1T-3/2 /14.442/. *
(14..383) 1-3/2
617
PLASMA INSTABILITIES
as function of Based on the diffusion coefficient the collision frequency one may classify seven microinstability regimes /14.436/. They are Böhm Dn = cTj\$eB, (14.384) D
b
see (7.85), p 126. 22 Neoclassical diffusion is - q rLE\)gj see (12.89), (for q see p 164). Kadomtsev-Pogutse (intermediate collisional) 1/3 2 1/3-4/3 4/3 EI E I I \ D K= 2/3 1/3 1/3 4/3 4/3 \1 + ~ / ' (14.385) (r Θ) ' « / n;/ e ' B ' E n
i
t
where r^ = din n/dr, Θ = (rr /B)/d(B§/r)/dr. Έseudoclassical (collisional drift)
V
-e " » e * V 8 î , 1 + W '
(14.386)
where the const is 10 or 5, see (12.95), to be determined experimentally. For decreasing v one has DB > DK > Dps. If v decreases further, four "collisionless" regimes appear. They are: collisionless drift regime (minimum of D and K«) (14.387) trapped
electron,
high collision
regime
(14.388) ^ _ \ Q -2 3/2-3/2-2^2 -2 D -2 // c T^e B I/ \v^+10v , or: 3^n r ' R E E o) where v Q = trapped
rT]/201/2/Rrnml/2,
electron,
T E o see (14.395))
low collision
regime
/14.437/ (for
3/2 (14.389) £> 0 = ε ' 21- v^/5 or = const. This formula is valid when the bounce frequency exceeds the effective collision frequency of trapped electrons. Finally, we consider for very small v the trapped ion collision-less drift instability regime
MICROINSTABILITIES
618 0
D
+r
2ε
a
5/2~2„2 oT 9 9 o
!
5·
(14.390)
Furthermore, other regimes like the Tokamak empirical diffusion regime /14.438/ and the steZZarator empirical diffusion regime or the convective cell regime have been defined. Actually, the stable operating regime of Tokamaks is merely restricted by instabilities: the disruptive instability sets in both at high and low currents and at high density. So we have a low-current operating limit, a high-current ing limit and a high density limit of about
operat-
10^2 one has the run-away
for
10^4 [particles cm~3]. For densities lower than about limit
(runaway
regime
E > Ecritt where Ecvit ~ n/TEt s e e P 1 2 1 ) /14.490/. Sometimes the various plasma regimes in a Tokamak experiment have been characterized by 3 dimensionless
parameters. These are the streaming ξ
=
u
/a
U
D\\E thE'
D\\E
=
parameter
V
n
e
(seep 387) (14.391)
and
γ^ = V ^ o " Φ | | / η 5 '
(14.392)
where λ„ is the electron collision mean free path, and « τ\'2/ητΕ,
Δ = rE^/TE
(14.393)
where T ^ J is the characteristic time for thermal energy exchange between electrons and ions (when Tj«TE)
and is given approximately by T^/^A/n-rln aspect ratio and τ ^ is the energy
Λ. A is the
replacement
time
(the time required for a plasma to lose a quantity of energy equal to its average kinetic energy). Other authors /14.440/ prefer observable parameters like the effective electron
by
energy
EE
3 o go 4 E||
or the energy
replacement
T
τ X
E
=1^EolhoL 4
E..<3,.>
replacement
time given
( 1 4 3 9 4 )
IH.JS4J
time
'
(14395)
U4.J»bJ
where o indicates the peak temperature (at the center
619
PLASMA INSTABILITIES
of the radial profile) and where En is the induced longitudinal electric field. I S the profile averaged current density. Other observable parameters are the ratio of the longitudinal resistivity to the classical resistivity, the ratio of the applied electric field to the classical runaway field, the streaming parameter and electron poloidal beta defined by
*PB-
2
™oVßp·
(14 396)
'
These parameters may be expressed in terms of TEo, n Q and the total current I: these parameters may be used regime, the to define regimes like the low density slide away regime /14.488/ (5x1012 < nE < 1.5*1θ13, - 0.25^, the transitional 0.7 < n0/I < 2, i>icrit (2 < n0/I < 4) or the quasilow collisional regime (nQ/I > 4, 1 0 ^ < nE < 5x10^4, resistive regime T%0 - lV2#1/2) i n which the heating of ions is collisional. Here^nQ is the density on the axis measured in 10^ 3 cm"37 τΕο is the dimensionless peak electron temperature, and I is the dimensionless total current. I is measured in 100 kA. Other authors /14.441/ define regimes by equations of state: banana regime —Ό 9ft
plateau
(14.397)
nT u '^° = const, regime
ηΤλ · 4 6 = const, Pfirsch-Schlüter regime
(14.398)
nT0'58 = const. (14.399) A parameter expressing the influence of the collicollisionality, sion frequency on Tokamak regimes is defined by C(r) = Ra(r)R3r/2/r3/2\rT ° ° C(r) = vEIA/ubE,
t2
or by (14.400)
where A ^ RQ/r is the aspect ratio and ω£# the electron bounce frequency. Small influence of collisions is defined by low collisionality (C < 1 ) . (p 4) is important for Since the Lawson criterion the construction of fusion power reactors, it is of
MICROINSTABILITIES
620
interest to calculate ητ for various regimes. For a total current I in [MA], temperatures in [keV] and a containment time calculated according to p 126, B in 50 [kG], one obtains for a given $pE the results pseudoclassical 14 2 1/2 ητ = 2χ1θ' I τΙ/Δ/Ζ _. (14.401)
E
eff
Here Zeffe is the effective charge, compare (2.73) trapped electron I 3 / 2
I 4
■ ^VEW ητ = 6x10'"l"£*e;:^Z ^ J / V ? /V ^ , trapped
electron
II
ητ = 10 1 3 Ι 2 τ1 / 2 Λ 1 / 2 /Ζ trapped
(14.402)
t.
ion
_,
(14.403)
ejj
ητ = 3 x 1 0 1 6 J 4 5 2 ß 2 E Z ß / / ^ 5 / 2 ( 1 + Τ Ε / Τ 7 ) 2 / 4 Τ ^ 1 / 2 . (14.404) Here A is the aspect ratio and the temperatures reactor are profile averaged values. A Tokamak fusion is expected to operate in the trapped ion regime and will require a current of about 10 MA or more for ignition (A = 3, B = 50 [kG], a = 360 [cm], q (a) = 3 ) . For a D-T reactor with 0.1 % molybdenum impurities one expects ignition at η#τ = (2-5)χ1014, Τ = 8χ10 7ο Κ (ignition device). In February 1977 at MIT ητ =2x10^3 at Tj- = 1θ7°κ has been reached and on PLT (Princeton Large Tokamak) 1 0 1 3 and 2x10 7o K are expected soon. As a general tendency the performance improves with increasing ε and decreasing A. Energy confinement time increases with plasma density and radius /14.482/. The plasma current I is ^eaB^/q. It is now of interest to understand how ητ and T scale when device parameters change (scaling laws). One important scaling law describes the dependence of the electron energy confinement time on the minor radius of toroidal machines. One obtains from experiments /14.440/, /14.442/ 3/2 τ
*~
3ό/Δ « * 7 V 2 r 3 / 2
eff
A l s o τΕ = 3 x 1 0 " 1 5 n ^ 1 j / 2 a 2 / j
1 0
·
(14
[ s ] h a s been g i v e n .
'405>
621
PLASMA INSTABILITIES
A maximum of ητ# as a function of nB% appearing at 10 3 e 11 /7( a # t ) 3/7,4-2/7^-12/7 has been found. For the electron temperature experiments yielded <ΤΊΡ> = Ζλ/2Ιλ/2/λΟα t
or * Iq^2,
(14.406)
ejj
(or TE s jV2#1/2 i n dimensionless units). More sophisticated formulae may be found in /14.384/ - MATT 1136. Βτ scaling laws are discussed in /14.486/. The scaling of plasma beta in a Tokamak has also been investigated /14.444/. The scaling law for Tj in the neoclassical plateau regime is given for hydrogen by /14.387/ (plateau regime) in the form T [eV] = 6x10~ 7 (I£,R 2) 1/3 -L
(14.407)
~U
and ^/2^1/2a-1/2 ±n the banana regime. A very wide range of present experiments seem to satisfy the scaling T „ [ S ] = 2χ1θ" 3 (<η>α) λ / 2 Βα,
(14.408)
or (a)1/2J. More generally τΕ = Cnaa^I^A^B£, where the powers α.,.ε have to be determined from the re[s] satspective experiment. Also τ - 3x10"1 ^a2nqV2 isfies experiments. The Tokamak plasma Ohmic heating current may be evaluated from I - q*'2 or j = 5x1ο""3αΒ / (Aq) (1 + e 2 )/2,
(14.409)
where I in [MA], B. in [kG], A = R/a (aspect ratio), ellipticity (e = b/a). Then the current density is given by j = I/i\a2e
(14.410)
= 5x10 17 3 I2/ea2.
(14.411)
and
In many experiments (see Tokyo Conference /11.57/) plasma density varies linearly with the plasma current. When instabilities with growth rate γ are involved, from D - y/k2 and p 126, formulae of the type e.g. (collisionless current driven drift mode) t
k2Bn Ξ τ * kT ^
U
y E
(14.412)
MICROINSTABILITIES
622
have been found. Stellarators obey scaling laws similar to those for Tokamaks. The Kurchatov τ
ΕΕ
scaling
law
( 1 4
~- **™~**\ol'
·
4 1 3 )
where Q = \η{ττ
+ ΤΛ V
(14.414)
is the energy content and V is the volume, seems to be satisfied by high current stellarators. The relationship T2?£ - na?- seems to be very well established for all devices. The particle
containment
Tp - 3.5τ# is proportional to the energy time
time
containment
τ^-
When writing scaling laws, some authors use the
Artsimovich
parameter
ζ = (l p B t/? 2 ) 1 / 3 ^" 1 / 2
(14.415)
to formulate dimensionless equations. An accurate calculation of the neoclassical parallel conductivity σμ is essential for obtaining an estimate of the effective Ί Ä
Ση8Ζ1/ηΕ
charge
(14.416)
s*E
and hence the impurity content of a Tokamak /14.483/. Since the neoclassical conductivity relates the parallel current density j \ \ to the electric field averaged over a magnetic surface j'
= o../
(14.417)
the impurity content can be estimated from an approximate analytic conductivity formula /14.484/. In all collisionality regimes, neoclassical transport theory predicts a toroidal flow driven by radial gradients of temperature, density and the electrostatic potential /14.499/. Scaling laws must be invariant under transformations leaving the plasma equaignitions invariant /14.500/. Scaling laws for the tion parameter Τητ have also been given /14.501/.
stability
or the mirror
instability
are of this type.
All instabilities may however be classified by the properties of the plasma, see section 10.1. One may use the classification scheme of waves (see chapter 9) or a slightly modified Mikhailovskii scheme: 1. Homogeneous collisionless plasma 1.1 unmagnetized (isotropic and anisotropic distribution function) see section 14.2 1.1.1 electrostatic instabilities 1.1.2 electromagnetic instabilities see section 14.3 1.2 magnetized, ^IIS0 1.2.1 electrostatic 459
460
MICROINSTABILITIES
1.2.2 electromagnetic 1.3 magnetized, klBQ 1.3.1 isotropic distribution function see section 14.3 1.3.2 anisotropic distribution function see section 14.4 1.4 magnetized, oblique wave propagation see sections 14.3 and 14.4 2. Inhomogeneous collisionless plasma, see chapter 13 and section 14.5 2.1 unmagnetized 2.2 magnetized, straight field lines 2.3 magnetized, curved field lines 3. Kinetic theory of macroinstabilities and the influence of collisions on microinstabilities see sections 14.7 and 14.6 4. Special effects 4.1 trapping instabilities see section 14.8 4.2 charge excess plasmas see section 14.9 4.3 anomalous transport see sections 12.3 and 14.10 4.4 beam plasma systems see chapter 15 4.5 parametric effects see chapter 17 4.6 nonlinear effects see chapter 19 4.7 laser plasma systems see chapter 20 Another scheme for an inhomogeneous two-dimensional low beta plasma would be that by Coppi /14.7/: 1. MHD modes, independent of I in the sense that for the limit l/a -> °°, where a is given by the characteristic inhomogeneity scale and I represents the order of magnitude distance between a minimum and the successive maximum of the B field along the w ^ere same line of force, ω > ^/^/^ > cthl^' 0} ω bs ^ cths/^' £> ^-s ^Cïe k ° u n c e frequency. 2. Fast kinetic modes with o-f-^/l > ω > o-^-^j/l. c < 3. Slow kinetic modes with ω < Gthl^ thE/^· These three types of modes are drift modes or curvature driven modes. Another scheme given by Coppi for the low frequency, low beta limit is: a.1 hydromagnetic modes b.1 density driven modes a.2 fast kinetic modes b.2 curvature driven modes a.3 slow kinetic modes c.1 wiggly (ballooning) modes d.1 resonant modes c.2 flute-like modes d.2 non-resonant (fluidlike) modes. Coppi gives general dispersion relations for a.2 and a.3. The ordering procedure mentioned on p 386 (volume 1 of this Handbook) may also be used to classify microinstabilities. Another scheme runs as follows /11.82/
461
PLASMA INSTABILITIES
MHD model
ω « ωLI
guiding cenU)«Ü) L J ter plasma Vlasov fluid ωLI model hybrid kinetic model
includes X»r^j gross stab. Λ ^ X»rLI
\>r LI
excludes microinstabilit.
gross stab., finite rL effects
kineeffects
finite r effects finite r^ effects finite r^ effects
microeffects very high ω
full Vlasov model The general theory of microinstabilities is based on the dispersion relations for waves which we discussed in chapter 9. These dispersion relations can be derived from the Vlasov-Maxwell equations (9.151). This system has a manifold of equilibrium solutions some of which we have discussed in section 5.3. Collective phenomena among plasma particles allow these equilibrium distribution functions to relax to more stable solutions (of lower energy states). These are stable states of non-thermodynamic equilibrium. (5.25) (i.e. 8/ o /3u 2 <0) is the sufficient but not necessary condition for any isotropic and monotonically decreasing distribution function of kinetic energy fQ(u2) to be stable. On the other hand, Sf/du > 0 is not sufficient for a plasma to become unstable, see section 14.2. It should also be recalled that the Vlasov theory is valid only for times short compared with the collision time. If collisions become important, the situation may change, see section 14.6. On the other hand, an unstable plasma in which instabilities grow slowly compared with the time between two collisions may be considered Vlasov-stable. The (sufficient but not necessary) stability condition (5.25) is called Bernstein-Newcomb-Gardner-Rosenbluth theorem /14.2/, /9.26/. It predicts stability of an unmagnetized homogeneous plasma with an isotropic equilibrium distribution function decreasing monotonically with velocity. It may also be shown that any isotropic function / 0 , decreasing or not, is stable, if / is a function of (u2 + ufj + u|) . Also all one-dimensional distributions f(u) decreasing monotonically. away from its maximum are stable. The class of functions stable according to Gardner's theorem are all locally isotropic in some frame of reference. Similar results may be obtained from the thermodynam-
462
MICROINSTABILITIES
ic point of view for monotonically decreasing distributions as a consequence of the conservation of energy and the continuity equation in y-phase space, i.e. the Vlasov equation. Gardner's thermodynamical proof is also valid in the presence of self-consistent electric and magnetic fields and is nonlinear. Gardner1s general proof is however of little practical importance since it does not permit proofs of instability. No universal rule is known for finding the constraint which must exist if the system is stable, see section 10.4 and /10.9/, /14.3/. We will use a Lyapunov function of type (3.34) to prove the stability theorem /14.4/. The Lyapunov functional is quadratic in small quantities and is a constant of the motion. When the functional is the sum of positive terms, it follows that the equilibrium must be stable, since the functional, being bounded by its initial value, cannot grow indefinitely. Being a constant of motion, the functional L is independent of time. This can be verified by taking the time derivative of L and using (9.151). As in (3.34) L is the total energy and is given in cgs units by
L = ί { # 2 / 8 π + (B
-BJ2/8T\ o
2
<14·1>
v,
if' and / are equilibrium values, the summation m u {f f )/2 s s s~ o
+
G(fs)]dujdT.
Here BQ 0 over s extends over all species and G is a function of / to be determined. We make a series expansion in powers of the amplitudes / = f0 + f-\ + f2> E = ° + E-\ + E2r B = 9 B O + 5 1 + B 2 , G(f) = G(fQ) + G'(/0)/-, + G'(f0)f2 + + /fG"(/0)/2 + ... and substitute into (14.1). Then, correct to second order, we may write L = ΓίΕ^/δττ + Β^/8-π +
G'(/0))(/1s+/2s)
+ l\[(mu2/2
+
+
f2uG-lf0)/2]dZ}dx
(14.2)
Now, if we choose G such that for each species mu2/2
+ G 1 (f Q )
= 0,
(14.3)
L will be a quadratic functional. For this to occur fQ may depend only on u . If (14.3) is satisfied, the r.h.s of (14.2) is equal to a sum of nonnegative terms provided G"(fQ) > 0. On the other hand, from ~1 , (14.3) we obtain for G" the result - (m/2)(dfQ/du2)
PLASMA INSTABILITIES
463
which is clearly nonnegative as long as d/Q/du^ < 0. Hence all plasmas for which the several f0„ satisfy (5.25) are stable. It also can be shown /14.4/ that even for the non-thermal equilibrium, S = -/£(/)dudx can be interpreted as a generalized entropy and L as an associated free energy, see (10.23). The Lyapunov function approach has also been used for an inhomogeneous two-stream plasma /14.29/. 14.2 MICROINSTABILITIES OF A HOMOGENEOUS UNMAGNETIZED PLASMA When considering what we have said in section 9.7 on Landau damping and the damping rate γ ^ df/dcXr we might conclude that the plasma wave is unstable when there is a "bump" in the distribution function so that inverse Landau 0//Β£)ω/£. > 0 {Landau instability, damping /14.6/). Although this criterion can be understood physically in terms of resonant coupling between the phase velocity of various waves and the speed of particles, it suffers a drawback because it is not invariant under Galilean transformation /14.37/, /4.11/. Particles which go faster than the phase velocity of the wave in a bump region will outnumber those which go slower, and this produces a bump
instability, instability,
trapped trapped
particle instability, trapping ion instability, trapping insta-
bility in moving waves /14.16/, which may be stabilized by dissipation. The particles faster than the wave will be retarded and the wave accelerates those which lag slightly behind it. With a positive slope of the velocity distribution there is a net energy transfer from the plasma to the wave. The necessary condition for instability df/dc > 0 is however not sufficient. For a Maxwellian distribution displaced by w, a longitudinal plasma wave with phase speed c ^ = cQ + w (c0 is the most probable speed of the undi-splaced Maxwellian) is expected to be damped (df/dc < 0 ) , but a wave with > 0) , °ph = °o "" w mi<3ht b e expected to grow (df/dc since in the moving frame there would appear to be more particles faster than the wave than slower. However we will show that single-humped distributions like a displaced one-dimensional Maxwellian distribution are stable. This paradox is resolved by realizing that the slow wave c , = c0 - w has a negative energy in the laboratory frame of reference. There is more energy in the quiescent, drifting Maxwellian
MICROINSTABILITIES
464
distribution than in the presence of an oscillation. Thus the wave gains energy from the resonant particles but loses negative energy and hence damps out/14.58/. The introduction of dissipation (Krook equation, see p 82 of volume 1) may however cause the wave to grow if particles of positive energy are present /14.59/. In strong electrostatic waves a significant number of charged particles can be trapped near the bottom of the moving wave troughs. These particles act coherently as a bunched beam and produce growing waves by a similar mechanism (trapped particle instability
in a strong electrostatic wave, trapped particle stability of large amplitude, bounce resonant
ininsta-
bility) . Other authors /11.21/ define the trapping instability as an electrostatic macroinstability occurring in a toroidal configuration as a result of flute instability among particles trapped between two magnetic mirrors, see section 14.8. Stochastic insta-
bility
/14.26/ (instability
in stochastic
heating
fields) is a term used by Soviet scientists for a special trapping instability in electric fields. The trapped particles have an energy close to the "edges" of the well (produced by an electrostatic wave or by magnetic mirrors). Thus they may oscillate across the wells, up the edges. By stochastic disturbances of the electric field, a random acceleration of these particles then occurs which on the average leads to an increase of the particle energy with time and the ejection of particles from the wells. The motion of the electrostatic waves is the energy source of the instability which might be stabilized by sufficiently small amplitudes of the electric field fluctuations. There are in general no trapping instabilities for transverse waves, because there are no particles going with the phase velocity of these waves, which exceeds the velocity of light. The Nyquist plot discussed in section 3.4 is however an excellent means for finding electrostatic stability conditions in an unmagnetized plasma, since the number of times the plot Re D versus Im D encircles the origin as ω varies from -°° to +<» (conformai mapping of the real ω-axis) gives the number of unstable modes. The dispersion relation (9.153) may be specialized to a collisionless plasma (vs = 0 ) . According to (3.45) it can then be written in the form
465
PLASMA INSTABILITIES
n
1
7f
1 - 1
^Ps(dfso
do
_
SK
= D
+ iD . = 0,
(14.4)
where for real ω, according to (9.161), 2
-L.0O
n - i _ yÜP£ pf 8 ^ ° 2 ωτ
s k
\
/ c=u/k
de
(14 5)
(14.6)
where P indicates again the principal value. On the real ω-axis (which has to be mapped onto the D-plane) , we have D (ω = -°°) = 1, Dr(u = +°°) = . In order that we have an instability/ = exp(2i\in) according to (3.9) and (3.11), we must have Im ω < 0 for modes ^exp(iu)t). According to Table 8 on p 263 (volume 1 of this Handbook), k is real for an absolute instability so that ω/fe is complex. To have an instability it is necessary and sufficient /14.8/ that the plot C of the real ω-axis onto the £-plane encircle the origin of the Z^-plane, see p 47. In order to be able to do so, the plot must cross the real axis in the D-plane, or Dj, = 0 for a particular cm = = ω /k so that from (14.6) it follows
(*L
c
= 0.
(14.7)
m
Since the imaginary part D^ changes its sign when going from D^ < 0 to D^ > 0 we must have dD^/du < 0 and hence from (14.6) > 0
(14.8)
m
must have a minimum at o . Due to (14.7) we or f8 may write the integral in (14.5) for real ω in the form
466
MICROINSTABILITIES
λ P ί ^ Γ 2 —^-7T =P 1 —ΤΤ TT-lflc) J 3 Ö c - ω/fe J σ - ω/k d c L J
'
- f (ω/fc)] dc = J
J
(14.9)
+00
Çf{o) - f(u/k)
=
i
(*-ω/Λ)
+ 00
since
i
^
2
+00
^'
(g)
da = f — ^ ( g ) *
77?
(c
- c
0 )
d g and because the
equilibrium distribution function does not depend on time and on space. Then D - = 0 at % = kcm and
^'r ~= 1 - Ι-^| — ~ ' ^ 2
,
^-^
/7 x 2
This can be regarded as a Kramers-Kronig
dö. (14.10) dispersion
relation /5.9/, /9.5/, /9.55/. Equation (14.10) is then used to obtain a plot of Ζ?(ω) for real ω. The requirement that / s o should have a minimum implies that it has two or more maxima at e-i/^ etc, since /(+00) = /(-00) = 0. The Nyquist plot D (ω) for real ω is then drawn by calculating Dr at ω = kcm, kc^ and kö2· If / has its absolute maximum at c* , f(c^) > f(c) for any c and according to (14.5), (14.4) we have > 0. Dr at kom and ko2 can be positive or negDr(kc\) ative. But since / ( ^ ) > f^°m^ i1: follows from < £>ρ(ω = ko2) · So it is a (neces(14.10) that Dr(um) condition for the appearance of sary and sufficient) an electrostatic absolute instability that the equilibrium distribution function / Q have a minimum at °m
(/' (cm)
=
+
°°f
i
°^
anc
* that (Penrose
(G) - /
/14.8/)
(σ ) 7y
(c-c
criterion
r
de > 0.
(14.11)
m If a distribution satisfies (14.11), it will be unstable only in the range of k in which £ ρ (ω= kc ) <0. This range can be deduced from the Nyquist plot, see Fig. 42. It should be mentioned that stability criteria could also be derived by investigating ensemsee e.g. p 279 in /5.13/. bles of Vlasov plasmas, For v Φ 0 (use of the Krook equation (5.18) instead of the Vlasov equation) the criterion is anal-
467
PLASMA INSTABILITIES
ogous; according to (9.153) a term ^v appears in the denominator and the plot C is shifted slightly. The Penrose criterion will in general have to be applied for each direction of the wave normal, when the distribution function is three-dimensional. Two-humped and multipeaked distribution functions, see Fig. 4 2 , which are unstable according to (14.11), can result from a plasma containing at least two identifiable streams or beams. For this reason the resulting instability is called two-stream instabilibeam plasma instability, twoty , beam instability, humped instability, double stream instability, bunching instability or also longitudinal instability /14.9/. On the other hand a distribution can b e t w o humped and still be stable, see Fig. 4 2 . This indicates that Gardner's theorem (stability of a singlehumped function) is sufficient for stability, but not necessary. T w o beams with distribution functions constitute a good example / 9 . 5 / . If / 1 ( u ) and f2^u) the thermal velocities the directional v e Ί,2 and >:> e values of locities v<\ 2 satisfy \u-v^ ' °*\ I » ΰΛ ?ot hthe values of 2 the distribution functions 'f^ (u) , fntu) will be close o-~ -ri— . ^ c. . c?2 _ the Λ.Λ-- minimum value of / I.v.^ - . v.2 I. » to zero. For and the system of two will also be close to zero and beams will be unstable. If however \v^ -t>2 I « c-\ +
unstable Fig.
42.
Nyquist
marginal plots
for
Z(u/k)
stable from
(14.4)
468
MICROINSTABILITIES
ion velocity distributions however do not necessarily need beams to occur. They may also be produced by the application of short pulses on dc biased grids /14.60/. Examples of stable and unstable distribution functions are also /9.26/ 1. any one-dimensional distribution function with a single maximum like fM according to (5.22) is stable /14.32/, 2. any spherically symmetric distribution function f(u2+^2+ u 2) (isotropic distribution) is stable /14.31?, * 3. an anisotropic distribution function that decreassee (5.25). es monotonically with u 2 is stable, 4. a two-dimensional distribution function f(ux+uy*uz) stable, with one maximum is 5. the Jackson
distribution
function
/14.33/
M ( u - u 0 ) 2 + a 2 + ( u + u 0 ) 2 + a 2 ] is unstable, because it has two sufficiently separated maxima. It should be mentioned that the Penrose criterion systems : If the is valid also for quantum-statistical distribution function is continuous and has a single maximum, the system is stable, see p 325 in /5.9/. If the distribution function has an interval Ac in which no particles are present, f(àc) = 0, ("hole" in phase space), the plasma is always unstable, since - f(kc) > 0 and may exhibit periodic behavior f(c) /14.42/. This instability is weak and limited to a narrow interval ω^/fc (electrostatic plasma with discontinuous velocity
instability of a distribution
/14.30/). Electric fields externally applied are shielded by a plasma. If for any reason there is no such shielding the plasma is micro-unstable. In particular it is unstable when the electric field strength has a maximum inside the plasma /14.31/. If a distribution function is unstable, to reach stability it is necessary to add particles and to fill nearly the entire well in velocity space. On the other hand more ions than electrons are needed at a given velocity to generate a hump and an adjacent minimum, since / - foE + mEfQl/mj /2.12/. Distribution functions
f
so
=
of
m s
the
l\2W^Ts\
displaced
exp
m (u u
s ' os) 2knT B s
type 2
+ exp
describe counterstreaming plasmas
(^ os
m (u+u ) s os 2kOT B s (14.12) = streaming
PLASMA INSTABILITIES
469
velocity). They generate the count er streaming instability , double humped instability, electron instabilbeam instability, electron-proton two ity , electron stream instability, electron-electron instability, double distribution instability, two-eleor on-stream instability, electron-ion instabilities and ion-ion instabilities, electron-ion streaming instability, double electron ion instability, electron-ion two/14.10/. stream instability (= Buneman instability) Electron instability, electron wave instability, or electron plasma wave instability are the names for the electrostatic two-stream instability in which ions do not participate. In the Buneman instability, the ion-ion instability and the electron-proton twostream instability ions do however participate. Otherwise the electromagnetic electron-ion instability is excited by magnetic fields. Shifted distribution functions of type (14.12) may be investigated with the help of an expansion with respect to hermite polynomials /14.17/. Two counterstreaming electron beams with a temperature TE perpendicular to the beam direction may be described /2.12/ by the distribution function foE = (mE/4i\kBTE) exp (~mE (c^ + c\) /2kBTE) · • [& (cy-u0) + δ {cy + uQ)] whereas for the ions fQj = = 6(cl, This distribution is unstable if the electric field E of a wave is Ik. From a Nyquist analysis the instability condition is fc232 + ωρ^Ο - mEu£/kBTE) < 0. Such current driven instabilities are also called pinching instabilities because the pinching of a perturbed current increases the perturbation current. A special case of the counterstreaming electron-electron instability is the finite length instability(finite plasma instability) /14.28/. When two counterstreaming electron beams of finite length are considered, electrostatic growing oscillations with frequencies connected with the transit time of the particles appear. Usually the two beams are considered in a magnetic field. The boundary condition effects (finite length effects) vanish, when the transit time approaches infinity. The mutual velocity of the counterstreaming beams is the energy source of the instability which may be stabilized by increasing the length of the device, changing density and velocity length appropriately. In plasmas also other finite may be found /14.28/. So e.g. electron (plaseffects ma) and ion waves at lower frequency than the plasma frequency have been detected experimentally /14.38/. It seems, however, that this effect may not be due only to the transverse size of the plasma, but also
MICROINSTABILITIES
470
to nonlinear effects and pseudowaves (see also (9,42) may be diagnosed by the exin volume 1 ; pseudowaves perimental fact that their velocity depends on the voltage producing them, whereas the velocity of a (linear) wave is amplitude-independent /5.13/), /14.61/. Another possible explanation is a slight deviation from a Maxwellian distribution function (suprathermal particles) /14.41/· It is of interest that the basic features of the two-stream instabilities can be explained also in orbit theory and in fluid theory /3.1/, /9.26/, /2.10/, /2.3/, /2.12/. If the ions are at rest and two electron beams designated by I = 1,2 move in opposite direaction, then for ui ^ exp(ikz - iut) the linearized continuity equation (n^ « nQ^r u^ « uQ
(14.13)
The equation of motion gives four real roots
-iu>ûii + iku^ -eE/mg. ol nûiI (14.14) The Poisson equation reads ikE
three real roots
= -4πβ(η1 + 9i2) . (14.15)
Elimination of both and & both ui gives the dispersion relation of the two/9.1/, stream instability /9.55/ 2
2
ω ρ1 /(ω - kuQ^) + (14.16) 2 2 1. + ω ρ2 /(ω - kuo2) two real roots The solution of (14.16) P2 -- vt°' i.e ω = ku0+ for oop2 + ωρι is called the fast space charge wave and ω = = kuQ - ωρ* is the slow space charge wave (which ω/u >/i o2 o1 is a negative energy wave, see p 256). Fig. 43. The two-stream When the dispersion reinstability. Plot of l.h.s, lation (14.16) has four of (14.16) versus k real roots, the plasma is
\J
471
PLASMA INSTABILITIES
stable. For two real and two complex roots the plasma is unstable. For equal density and equal speed (U 0 1 = ~^o2 = uo) o f the two electron beams the plasma is unstable for kuQ < /2ωΡΕ. The maximal growth rate γ is obtained when k-m ~~ 0)p/3/2uo and is given by /14.9/ (14.17) ] max
P
When a single low density beam (nQ<\ « n02) °f speed u0 passes through a plasma of density nQ2 then (14.16) gives by ωρ-j = /αωρ 2 ' α = n ol/ n o2 K< ^ 2 2 /,..«. .. ,,.. ,.. ,«. (14.18) 2 ^. ....2 Λ ωρ5,/ω + αωρ£,/(ω - feuQ) = 1 and the growth rate r-
,, 2
γ = /αωρΕ/
/7 2
(b)pE/k
2
1X
1/2
(14.19)
uQ - 1 )
All (electrostatic) bunching instabilities discussed in this section have γ < ω?£' or γ < ω ρ ρ if the electrons are stable (mj ·> °° gives stability) . Oscillations described by (14.18) have a group velocity ^u Q and are therefore called beam oscillations. The real and imaginary parts of the solution of (14.18) have been plotted in Fig. 44 for a «c 1 . /ku„
k mu ο' /ω.^ PE Fig.
44.
kuQ/uFE
Beam instability according to (14.18), (14.19)
MICROINSTABILITIES
472
Dispersion relations of the type (14.16), (14.18) may also be obtained from (14.4) by inserting the two beam distribution function /3.1/, /4.11/ f(c)
= [&(c-uQ)
+ δ(σ+« 0 )]/2.
(14.20)
When the electron component of the plasma moves relative to the ion component with speed uQ we have from (14.16) /15.43/ ü>p£/(ü>-fcu0)2 + ω ^ / ω 2 = 1.
(14.21)
This equation is of fourth degree in ω and has four roots. Two lie close to ±copj and two lie close to kuQ. Longwave oscillations (kuQ < ωρρ) are unstable {Buneman instability /14.11/) with a maximum growth rate γ = ωρχ/3^/3/24/3^1_/3 /9.26/,/15.43/ near the resonance condition ωρ# = ku0 and ω = ωρβ (mjr/mj) V 3 e The alternating current instability /14.35/ is an instability which is produced by a two-stream instability of large amplitude. When the electron beam of density nQ and speed u0 > ctJijr is emitted perpendicular to a cathode and is absorbed at the anode that is separated from the cathode by a distance I, the Pierce instability {space charge instability, ion space charge instabili^ ty /14.12/) is generated. The linearized equations for perturbations K\\UQ read then /9.26/ -toon + un' o -iuu* -Φ" =
+ uQu%
+ nu* o
= 0
= -{e/m)§*
(14.22)
4-nen.
h, û are the perturbed quantities, Φ is the electrostatic potential and ' indicates derivation with respect to z. Since the cathode and anode potentials are fixed and speed and density at the cathode are given constants, the boundary conditions are Φ(0) = = Φ(Ζ) = 0, u% (0) = n(0) = 0. The solution of the system (14.22) yields the dispersion relation of the /9.26/, /14.12/, /10.14/ and a Pierce instability purely imaginary frequency: Re ω = 0, γ i\nl/AuQ. In all beam and two-stream instabilities the energy source of the instability lies in the kinetic energy of monoenergetic beams. Stabilization is possible by a thermal spread comparable to the relative beam velocity since this destroys the coherence.
473
PLASMA INSTABILITIES
The dispersion relations derived so far are valid in the cold plasma approximation, i.e. •th < ω/k or u When the thermal spread of the distrioE > kBTE/mE.
bution function is such that the particle distribution overlaps the region of unstable waves beam temperature effects, i.e. the effect of resonant particles (moving with u = ω/fe) must be considered. Such a distribution function of warm electron beam passing through a Maxwellian electron cloud with a is shown in Fig. 45. Inspeed u\ > ο\^Ε = 2kBTE/mE
unstable region
Fig.
45.
Bump on tail function
distribution
elusion of a pressure term Vp = ykßTVn, γ = Cp/cv, - see (9.49) - into the equation of motion and a Döppler shift by uQ in (9.56) yields the dispersion relation /14.34/ \i = 1,2)
ωP2
"ΡΪ
(ω - ku
„ ) - a- Zc
/
7
\
2
2
7
2
= 1
(14.23)
(ω - ku 2' " ao Here ωρ^ are the stream plasma frequencies or the electron resp. ion plasma frequencies, a| = ykßT^/m^ are the sonic speeds for the two beams or for electrons and ions, see (9.49), (9.50). For nearly cold plasmas (α^ « uQ^) we come back to (14.16). For large thermal speeds an inhibition of the instability is found if α-j + a 2 > uQ^ - u o 2 /2.10/. For a-j + a 2 < uQi - uQ2r the two-stream instability is maintained. Similar results are obtained in kinetic theory
MICROINSTABILITIES
474
from the insertion into (14.4) of an appropriate bump in three dimensions on tail distribution function (14.24) E2
S(c
- > * m {o
E z~Uo)2" ΔΚ
Β
Ε2
J-J
+eXP
x
)6(o ) y
""'"'"
t
L
2
V*2Ε2 J /
"^Β
and / 0 j = 6 (cx) 6 (Ο^) δ (e^) for the ions. Here nE = nE<\ + + nE2 » nE2, TE2 << TE<\ , u0 » 2k-ßT^/m-^. For nearly real ω expansions, using u)£j « o)L, n#i » n^2 anc^ (9.166), give /2.12/, /14.5/, /9.17/ for the gentle bump instability
"i * ωΡΕ(1 + 3 k 2 x L l } and for nE-\kßTE-\ ω
;
s
» nE2mEuo' 71
W
PE1
2
(14 25)
'
^ΌΕλ
-r8X3-exp fe
λ
£Ε1
<<:
"*
(""ii" 1 ) Ν ^
"Z?E1
26)
Since ω^ Φ 0 the bump instability as well as the beam (oscillating) instabilities. instability are periodic When temperature effects are taken into account the beam instability may be called hydro dynamic twostream instability. Various dispersion relations and growth rates have been derived /9.26/, /14.9/, /2.12/ for e.g. 1) a cold beam in a cold plasma, giving (14.18), 2) a slightly heated low density beam in a slightly heated dense plasma, 3) a beam with large tail thermal spread in a dense cold plasma (bump on instability, electron sound instability, electron instability /14.15/, /14.36/), resonant bump on tail 4) a beam with small thermal spread in a dense cold plasma, 5) a cold beam in a dense hot plasma, plasma /14.9/, /9.26/, 6) a relative motemperature effect tion of electrons and ions (Buneman instability in a hot plasma and others as in current carrying plasmas /14.295/), 7) and for the excitation of ion-electron oscillations of a hot plasma by ion beams etc, see also chapter 15. It should be mentioned that in multi-
PLASMA INSTABILITIES
475
invariance of component plasmas the lack of Galileian formulae for the growth rate of longitudinal electrostatic waves has been claimed /14.37/. Furthermore, a beam of energetic electrons (or ions) passing (e.g. parallel to BQ) through a cold plasma can excite ion waves which grow rapidly at the expense of the kinetic energy of the electrons (ionwave instability, ion plasma wave instability, ionwave instability in a magnetized plasma, current driven mode), electron plasma instability /14.13/, driven by the small electric current due to the relative motion between the ions and electrons). The phase velocity of the excited wave is at an angle to the magnetic field, but independent electrostatic oscillations are set up on each flux tube. A large thermal spread smears out and damps the instability, which taps its energy from the electron current along the magnetic field. Experiments on standing ion waves in thermal equilibrium show /14.44/ that two types of unstable ion waves exist: one has a phase velocity corresponding to the minimum of the distribution function which is almost independent of the voltage applied, the second has a phase velocity corresponding to the discontinuity of the distribution function and varies linearly with the square root of the voltage. The two waves may be denoted by two-stream instability and bump instability, respectively. Current instabilities (current driven instabilities) are of various types: on pp 364 and 374 of volume 1 we mentioned a current driven drift insta, on p 374 one finds also the inertia currentbility on pp 374 and 335 one finds convective instability, the nonlinear current driven ion sound wave instability /12.63/, /14.55/, on p 375 the ion wave current /13.34/, and on p 374 the skin current instability driven wave instability. Recently, in addition to the current driven long-wave ion-acoustic instability, a new current driven short-wave ion-acoustic instability at velocities lower than those needed for the excitation of the long-wave ion acoustic wave has been on an ion acoustic detected /14.64/. The modulation wave is shown to be unstable in a direction oblique to that of the wave phase velocity /14.65/. A survey of computer simulation of current-driven microinstabilities in homogeneous collisionless plasmas has been given by Chodura /14.66/. Finally a special current instability (current sheet instability) i.e. instability of the electric current itself is mentioned on p 318, see also /2.12/ p 505. There exists also
MICROINSTABILITIES
476
a current
driven
surface
wave instability
/14.51/,
collisional instabilities excited by currents may be
found in /14·53/, a current
induced
two-stream
insta-
bility has been discussed /14.54/ and nonlinear effects on current instabilities have been discussed in /14.52/. Current^ driven
standing
ion waves have been
used to excite travelling ion waves /14.56/. Standing electron plasma waves have also been reported /14.57/. Current instabilities are generated by a wave propacurgating parallel to the current. Electromagnetic rent instabilities are generated by linear electromagnetic waves in a current-carrying Vlasov plasma; they have lower thresholds than the electrostatic ion acoustic instability /14.62/. In chemically active molecular plasmas circularly polarized electromagnetic waves may cause - even in the absence of external electric fields - a new type of instability, the in-
stability
in active
molecular
plasma /14.63/.
When the plasma is warm there is a heat transfer from the hot electrons to the ions by Coulomb collisions. When T-p > YJTJ with yj being the specific heat ratio of the ion gas 7 this leads to heating of the ions in places where the ion density perturbations have their maxima, thus giving rise to growing ion sound waves (ion
instability,
sound wave instability,
ion sound wave instability
ion
in fully
acoustic
ion-
ized plasma /14.14/). This instability is often termed macroscopic, see section 12.2. The thermal energy of the electron gas is the source of the instability which may be stabilized by viscosity and heat conduction. It resembles the macroscopic acoustic wave in-
stability
in a partially
ionized
plasma,
see section
12.2 and chapter 19. Ion acoustic instabilities may also be excited by strong electromagnetic waves /14.40/, /14.62/. At low density the mean ion velocity is greater than the ion wave velocity in the plasma frame. Therefore it is possible to observe, instead of forward and backward ion waves, fast and slow waves in the laboratory frame (Buzzi /14.14/). Inserting distribution functions of the type (14.12) into (14.4) yields (9.167) and (9.168), if u,QQ = 0. According to (9.167), (9.168) ion waves propagate and are only weakly damped for TE» Tj, Cph= as-isoth>
°thl"j'
but are heavily damped for T% < Tj. When the electrons move relative to the ions with a speed u Q , the Penrose criterion together with (14.12) gives /12.2/, /14.10/ stability for uo
< 1.3(kBTE/mE)U2,
if TE = T ,
(14.27)
477
PLASMA INSTABILITIES and for
U
1 /?
o
<
m
(fe B V I
)
' ±f TEy>
Τ
Γ
(14.28)
If u is large enough (u0 > αβ), this stability criterion is no longer satisfied and the ion acoustic wave grows. The growth rate of the ion acoustic instability is then given in one dimension by /9.69/ ku_ - ωω
/™7Γ if O1/ ^^ o o P.. /3/2 2.2 y Q8/w'
T
w E»
τ
τ- I
(14.29)
The dispersion relation for ion acoustic waves based on a linearized kinetic theory for Maxwellian electrons and ions is /9.69/ 1 + ω ^ [ 1 +i/^zEW(zE)]/k2v*
+
+ ωρχ[1 + i/nraJf/(aJ) ] / f c 2 ^ = 0 , va = (Τα/ιηα) 1 / 2 , α = £ , I . F o r w h e r e za = iß/kvav/2, zE « 1 , ζχ » 1 , W{zE) * 1, W(zj) * ί[λ+λ/2ζ} + j + e x p ( - s ? ) . An e x t e n d e d N y q u i s t t y p e a n a l y s i s / 9 . 1 7 / g i v e s f o r t h e ion wave instability f o r | R e a ) | » | I m ω| and |Re ω/fel » c?* t h e d i s p e r s i o n r e lation ω - ω ρ (1 + τηΕ/2πι].) - iS-nrrij./ (S8mEk +
3,3
λ
+ 3fc 2 fc ß ^[1 + T^/T^] )
expi-rrij./
(mE2k
/2mE
+
2 2 λ )) +
(,/^/^|)exp(-3/2 - Λ"2)
Unless Tj « Tp the waves are heavily damped. hydrodynamic The assumption TE » Tj leads to the approximation. In (14.29) the ion Landau damping term has been neglected, compare with (9.168). In the three-dimensional case one has to replace k(u0~u>r/k) by the instability cone ^ w Q - ω ρ . It is possible /4.11/ to derive from γ = 0 the curve of marginal stability uQ{kc) , where the critical wave number indicates the onset of the instability. A study of ion /14.43/ reveals acoustic waves using Green's function that for TE < 3Tj collective interaction is unimportant. Then the wave properties are determined by phase-mixing of freely streaming ions. Even for fl = 0, for u = Cpfo the waves are damped. Relaxing the usual assumption of constant electron temperature gives the
478
MICROINSTABILITIES
modified ion-aooustic resonant instability /14.45/ in which electron density and temperature are coherently oscillating. The ion background is cold, at rest, and monoenergetic in space and time, but also variable ion-electron temperature ratios have been discussed /14.46/. Using Θ = Tj/TE, y = kaE/u>pE, k = wave number, μ = mE/mT and aE (9.49), ax (9.50), p 188 volume 1, γ^ = γj = 1, from continuity, momentum and Poisson equations in linearized form for a collisionacoustic less plasma the dispersion relation for ion waves 1/0 2ω 2 = ω^[(1 +Ch/2)y + (1 + z/2)]· 1± ,
4y,y [θ+Π+θι/ )] 2 2 2 μ+μθ# +(1+2/ ) is obtained. The range of validity is limited by Landau damping. There exist however undamped ion-acoustic oscillations even in kinetic theory. Using a selfsimilar distribution function Pitayevskii found /14.47/ a new branch of undamped ion acoustic oscillation. He has also investigated the breaking of ion acoustic waves. Ion acoustic waves may also be excit(ion bursts) (Cherenkov ion acoused by pseudowaves tic instability due to ion pseudowaves /14.48/).In turbulent plasmas a new quasi-acoustic oscillation (second sound wave) has been found /14.49/. The ion acoustic instability should not be confused with the acoustic instability in a partially ionized plasma and the acoustic instability nor with the magnetoacoustic instability {magnetic acoustic instability), see pp 318, 335 of volume 1 and chapter 20. Ion acoustic instabilities have been used as a source for travelling ion waves. When a grid is inserted between the hot plate and the opposite cold plate of a Q-machine, a positive dc bias applied to the grid with respect to the hot plate ( 2 - 3 5 volts) drives an electron dc current in the region between the hot plate and the grid. If the current exceeds some critical value, a low frequency ( 3 - 3 0 kHz) current driven standing ion acoustic instability (λ = 2L) appears between hot plate and grid. By changing the distance L between the hot plate and the grid the frequency of the instability is modified. The frequency is inversely proportional to L. Since the ratio of the frequencies for sodium and potassium plasmas is approximately equal to the ratio of the square root of the ion masses ( % a / % ) 1 / 2 at a fixed value of L one may conclude that the oscillation is a standing ion wave. By applying additional external
479
PLASMA INSTABILITIES
signals to the grid, traveling ion waves are excited for various frequencies at a fixed L in the region between grid and cold plate /14.50/. This allows measurement of the dispersion relation outside the instability region. The instabilities described so far have been electrostatic (ΒΛ = 0, curl £<| = 0) and due to charge bunching. All of them had anisotropic distribution functions since isotropic distribution functions are always stable. Particle streaming (beams) represents an anisotropic departure (e.g. Tj_ » T\\) from the (always isotropic) thermodynamic equilibrium. Anisotropy is also a feature of magnetized plasmas, see section 14.4. Besides the electrostatic instabilities in thermally anisotropic field free (B0 = 0) plasmas there exist also electromagnetic modes in isotropic and in anisotropic unmagnetized plasmas. Before specializing to an isotropic unmagnetized plasma we woul
!
= —
τπι
s
\κ· c - ω)
—
-
(14.30)
ÔC
By inserting the perturbed current density into Maxwell1 s equations (9.151) and assuming vanishing space charge and current at equilibrium, one may find the dispersion relation (9.153) (for v = 0) or (14.4) for electrostatic modes, and the dispersion relation (9.175) for electromagnetic modes. Equation (9.175) may also be written in the form /14.18/, /2.12/ c k
V_
kc2
- L>ω„ - L)ωηPs -5 -r—^— de, (14.31) Ps de kc -u) s s J x x where k = kx, ky = k = 0 has been assumed. Equation (14.31) is also valid, when we replace c^ -> e| in the numerator. (14.31) is never satisfied by isotropic fQ with ω = ω ρ + ίω^ and ω^ > 0: These modes can be driven unstable by temperature anisotropies, by beams etc. However, first of all we would like to investigate the electromagnetic stability of an unmagnetized plasma with the isotropic distributions / = fOQ {c£ + + 0* + c^) . in full analogy to the PenroseScriterion = ω
MICROINSTABILITIES
480
(14.11) it can be shown /14.18/, /2.12/, /14.19/, /14.20/, /2.9/, /3.1/, that isotropic distributions and monotonically decreasing distributions, as well fo(-c) as functions symmetric about the origin fQ(c) are also stable to electromagnetic (transverse) perIn full analogy to the turbations (Kahn criterion). method by which the Penrose criterion has been found, the Kahn criterion has been derived from (14.31). To use this technique the unstable roots of (14.31) have to be found. However, the crossing of the real axis in the ω/fe plane at (ω/fe)^ = cm demands now that the distribution function have a relative maximum at cm /14.18/, /14.20/. For instability it is therefore no longer necessary that the distribution function be two-humped. Be cm a relative maximum (^f0s/^ox)cm =0, then Kahn*s instability criterion reads +00 k
a
m-^PB~k s
G
- He J ka - ka s ·* x m
*»„< 0.(14.32)
—oo
The necessary and sufficient criterion for instability of a single-humped anisotropic plasma reads
k2c2 + 2 ω ^ | ^ ( ^ - -^\\ί„τ.+ζτί~τ\ά°<0.
(14.32a)
It also can be shown /14.18/, /14.21/ that in an unmagnetized plasma transverse waves are more unstable than the longitudinal modes. Thus a single-humped anisotropic plasma or a distribution f(oLc^ + acf· + c|) r a < 1 , may be unstable for electromagnetic (and electrostatic) modes. Another example is the bi-maxwelli= const an plasma defined by (7.81). We assume n(x,y) and normalize / such that n = 1. This distribution is stable to electrostatic perturbations, but according to (14.32) it is unstable. This may be found by a Nyquist plot /2.12/. For instability we then have
T
IE/T\\E - 1 > φ 2 / < 4 ·
(14 33)
·
Waves with ?* || 2-axis propagating in the colder direction are unstable. It can be shown /14.18/, that under certain conditions even a very small anisotropy having T± > Tn can produce instability. Mainly the very lonq waves are unstable and the maximum growth rate can be found to be /2.10/, /14.20/, /14.2/
PLASMA INSTABILITIES
481
ym = (32fe5T2/27ïïm52)1/2(1 - Τ 1 / Τ υ ) 3 / 2 ω ρ .
(14.34)
Quite generally, according to Furth /14.18/ it can be stated that the growth rates of the electromagnetic if instabilities are less than uQu>pi?/c or u0upj/c electrons alone are stable. In the general dispersion relation, anisotropic (bi-maxwellian) distribution functions couple longitudinal and transverse waves. The coupling terms have destabilizing effects /14.18/. For a bi-maxwellian plasma {anisotropic velocity distribution function*') (13.41) the dispersion relation for electromagnetic waves (14.31) may be written in the form /2.12/, /14.18/ for waves parallel to the re-axis k2χc2
2 = ω 2 + LΙωΙ JT s\\, - 1) +LΤωPs ζ Μ , Ps (Tsi s Ζ(ςs )Τsi'JTs\\ ' s s (14.35) where ζ^ = u/kxS2kBTs\\/ms and Ζ{ζ8) is given by (9.170). In the limit ζ8 « 1, ω 2 < k$c2, mE « mz one obtains
ω = ik /k
x B2WmE^TEU{TEl/TE\f
-1 * φ 2 / ω Μ >
/Τ
ΕΙ·
(14.36) This mode exists only in an anisotropic plasma. It grows at the expense of the excess electron energy k B^TEi-Tm). When a two beam distribution function of the type (14.20) is used for a first and for a second electron beam or ion beam, electromagnetic counterstreaming instabilities may be found /14.19/. Instead of using kinetic theory one may use^two-fluid theory. For a ^ave f = kx, ky = kz = 0, E = Eyt Ex = Ez = 0, B = κχΕ/ω, crossing two streams of electrons with density n/2 and unperturbed velocity Uy = ±u with stationary ions, equations of motion and of continuity give /2.10/, /3.1/ 2~2 2 22 2 2 k c + o>p(1 +k u /a) ) = GO (14.37) (Weibel instability , transverse wave instability r transverse instability, anisotropic temperature instability) /14.19/. Transverse instability is a rather general name which is used for some particular Some authors call a bi-maxwellian distribution anisotropic /4.11/ p 510 or /13.49/ and /5.14/,whereas others call it isotropic /10.14/ p 38 or /14.17/.
MICROINSTABILITIES
482
electromagnetic modes such as e.g. the Weibel instability. The terminology of these instabilities is not very strict. Lehnert /10.2/ gives the following defi-
nitions: the temperature
anisotropy
transverse
wave
instability /14.39/ is due to the temperature anisotropy T|| - TJL and concerns electromagnetic waves propagating across a magnetic field in a plasma with anisotropic temperature. The transverse wave instability /14.19/ or anisotropio velocity (Weibel instability is due to an anisotropy of the electron instability) velocity distribution in an unmagnetized (or magnetized) plasma. For u ■> 0 we obtain (9.46) from (14.37). The Eccles condition, see section 9.2, is still satisfied for u Φ 0. The dispersion relation (14.37) is shown in Fig. 46. K2
-u
Fig.
2 2,-2 ίύρ/α
46.
Dispersion Weibel
curve of instability
the
The growth rate is given by 2 2 2-1/2 γ = uu^k^ for u « c,
2
mEu
+k c )
= kBTE
w Δ
» u?,mE
,
(14.38)
(strong "temperature"
anisotropy). In kinetic theory /14.22/ one obtains instead of (14.36) the dispersion relation fc252 + co|(1 + + u2Zi2c^}l) = ω 2 , where Z is given by (9.170). Another formula is /3.1/ - compare (14.31) c2k2
+ü>p(l + ^c^tCèf/dc)
(2Î-c - 2ü))"1dS)= ω 2 ,
483
PLASMA INSTABILITIES
where the appearance of c± indicates that the velocity distribution f(~c) is anisotropic even in an unmagnetized plasma. (Then c\ is the component perpendicular to the direction of anisotropy.) Besides the purely transverse (Weibel) instability a coupled transverse-longitudinal mode separating into a quasi-transverse unstable mode and a quasi-longitudinal ion acoustic mode has been found. The ion mode is driven unstable by its interaction with the quasi-transverse mode. The Weibel instability is a microscopic electro-
magnetic anisotropic
temperature
instability
involv-
ing only the electrons. It can be described as the spontaneous formation of local pinches with currents parallel and antiparallel to the axis of maximum thermal velocity. The Weibel instability may be stabilized by imposing a magnetic field perpendicular to the low-temperature direction. Other electromagnetic beam instabilities have also been investigated. An anisotropic ion "beam" with thermal spread 1/2
foi
= (m I / 2 i r Vll , i w I / 2 i r Vlll ) 2
• exp (-mTcIxl2^B^m
(14.39)
2
) expi-mj-Cj.
l/2kBTJi)
in a background of hot electrons
fQE = ^mE/2^BTE)3/2^{-mEa2E/2kBTE)
, 2 ^ (14.40)
is unstable /2.12/, /14.18/ to electromagnetic waves \\x if
T
IL/TI\\ - 1>
Φ2/4Ι
(14 41)
'
with a growth rate ω^ - (^ρχθ^/ο. If the electron background is cold, the same distribution functions are valid, but TE « Tj. Then the electromagnetic wave propagating in the ^-direction is unstable for a certain range of kx. Quite generally, we see that currents in a plasma generate instabilities for waves propagating parallel to the direction of the current. Another practical example of a highly anisotropic electron distribution in a neutralizing ion background is the electrostatic confinement instability occurring in an electrostatic confinement system proposed by Farnsworth /14.23/.
484
MICROINSTABILITIES
14.3 MICROINSTABILITIES OF A HOMOGENEOUS MAGNETIZED PLASMA WITH ISOTROPIC DISTRIBUTION FUNCTION If a homogeneous Vlasov plasma is magnetized it becomes anisotropic but may retain its isotropic distribution function, see p 229. It is a matter of taste whether a distribution function /(ej + cfi ,cz ) symmetric around the direction of the magnetic field, of which a bi-maxwellian (7.81), (13.41) is an example, is called isotropic or anisotropic, see p 481. In this section we intend to discuss instabilities of a homogeneous magnetized plasma with a truly isotropic distribution function. In this situation no new instability appears - only instabilities known from section 14.2 are modified. However, new instabilities appear when an anisotropic distribution function is assumed, see section 14.4. In a magnetized plasma there are longitudinal (electrostatic) waves which can be treated in electrostatic approximation c ■> «> and other waves which are essentially transverse, see section 9.8. The electrostatic instabilities grow more rapidly, especially for low beta /14.67/. These electrostatic (longitudinal) modes are however not influenced by the magnetic field Bo, if they propagate in the direction parallel to ~ÈQ, see p 196. If they do not propagate exactly in the direction of B0, one has a coupling between the longitudinal and transverse waves. This coupling is destabilizing for the longitudinal and stabilizing for the transverse modes /14.17/. For the electrostatic approximation (2 + «>) the coupling goes to zero, but the electrostatic modes are still modified, see p 197 (volume 1 ) . However, for a magnetized plasma there does not seem to exist such a general stability criterion as that of Penrose (14.11) or that of Kahn for electromagnetic modes, see p 480. But for stationary ions a stability condition for the electron distribution function of a magnetized plasma with respect to longitudinal (electrostatic) modes propagating at an angle Θ between 1c and BQ may be derived /9.5/. In the electrostatic approximation c ■> oo, n -> oo according to (9.18), and A -* 0, see (9.35), longitudinal modes are described by A = 0. From (9.33), (9.20) we then have ea,a:sin2e + + e22cos20 = 0. The elements of the dielectric tensor are given by (9.185) or (9.194), pp 231 and 234. For long wavelengths kcpfo « ΩΕ or rL < λ (small Larmor radius, see p 233), -ehe argument kxcx/Q,E of the Bessel function is small. For exact parallel propagation
485
PLASMA INSTABILITIES
(θ = 0/ kx = kI = 0 ) , the argument is zero. So we have for long wavelengths ^ethE <<: ω^ o r f o r P a r a l " lei propagation J n (0) = 0 for n > 0, see p 234. Since this is equivalent to putting n = 0 we obtain from (9.185) the dispersion relation for longitudinal electron
B0
modes
propagating at an angle Θ with respect to
/9.5/, /14.68/, /9.4/ in the form
4cos 2 e + f° roic )
j
+ 00
·ί(
4 g sin 2 e
feilen - ω
•^o(ö||>
fe|.ö|. - ω + Ω„
II
2Ω E (14.42)
Λ> ( °||>
k,.c ..- ω - Ω„>
de
For 5Ό0 = 0 or θ = 0 one comes back t o ( 9 . 1 5 3 ) z I n t r o d u c i n g Ψ1 =2 / ' {c |() c o s,2c 6 + fe|| [ f ç ( ö || + QEk \0 " - /"0 (e11 - Ω^/^||) ] 3ΐη θ/2Ω 5 ϊ one can r e w r i t e (14.4^) i n t h e form +CO
f
*'(c.i)
?
d
° II" ω / ^ il
fen
^ n = - T2 - · II ,x
(14.43)
Comparing this with (14.9) it is easy to see that the necessary and sufficient stability condition for /0(£||) is that Ψ (OH) have a minimum. Thus the elecis stable for all tron distribution function fQ(c\\) ω/k || = c\\ for which Ψ (c ^) has a minimum and the condition + 00
|—-IL-dc
i
II
\\m "
< 0
(14.44)
is satisfied. This is the case for even distribution functions with one maximum. For Θ = 0 (parallel proand pagation) Ψ'(Ο||) -* fo(c\\) (14.44) becomes identical with (14.11) and longitudinal modes propagating by the magnetic field. A along B0 are not influenced magnetic field reduces the manifold of stable distributions: it "destabilizes" longitudinal modes. Stability criteria for fixed k, Θ or üE have also been found /14.69/. Transverse (electromagnetic) modes propagating along BQ (kx = 0, kz * 0, if ky = 0) satisfy the dispersion relation (9.195). Defining the reduced dis-
MICROINSTABILITIES
486 tribution
/ 1 4 . 1 8 / , / 1 4 . 7 0 / , /14.74/
functions
i 00
00
9
v-»
^so(cl'Cll)
3
f
° 21, , foil ° 2TKo«Ld«L
(14.45)
00
Fsi(cu)
=
2^(cl/2)fso(ci,cn)dci o
one obtains the dispersion relation (k = k ) ω2 = a V - ωΐω2
U
all
s i II
( 1 4
.46)
When this is compared to (14.43) one can derive the following necessary and sufficient criterion for instability. For om defined by =
Hs[usFs\\
0
( 1 4
S
"47)
and for +00
2t2
~2v2iT
2 Ρ"ΊιΓ°Ι|",<Ι||'°Ι|'-'"<>ΐ"Ί|' U c — OO
c
ir m
) +Ω
β
>
0
(14.48)
i n s t a b i l i t y e x i s t s , if ZF „ (o - Ω /fe) S " * 2 ÖC m
82F . (c - Ω A ) 1 «1 % s 1> 0 . ( 1 4 . 4 9 ) r. 2. do m
For vanishing magnetic field, (14.48) can be trans(14.32). An isotropic formed into Kahn's criterion distribution function fso(°fi+°^) gives /14.18/ from (14.45), (14.46), (14.47) the single solution cm = 0 and then (14.48) gives the contradiction - 5 2 > 0. We conclude that electromagnetic
BQ are stable
for isotropic
For propagation perpendicular
modes propagating
along
to ^ Q (kz=0r
kx Φ 0 )
distribution functions.
electromagnetic waves with^E||B0 and mixed transverselongitudinal waves with ElBQ are possible, see p 237. When there is no current parallel to the magnetic vanishes, so that field, that is if ffSQc..dc..
487
PLASMA INSTABILITIES + 00
a
cndcn
2
=0
(14.50)
these two types of waves do not couple. This is so because according to (9.190) Όχζ =Ό ζχ = D z = D =0. For isotropic distributions transverse waves are always stable but the longitudinal waves may become unstable for non-monotonic isotropic functions (beam instability, Bernstein instability). For monotonie anisotropic distribution functions the longitudinal waves are stable. The nearly longitudinal waves are the Bernstein waves, the transverse waves are the cyclotron waves. Both of them will be discussed in section 14.4, since electromagnetic modes are heavily influenced by a magnetic field and since electromagnetic instabilities can appear only for anisotropic distribution functions we will postpone the discussion of electromagnetic instabilities to section 14.4 (anisotropic distribution functions). Criteria distinguishing between convective and absolute instabilities have also been found for magnetized plasma /14.92/. A general discussion of modes propagating in an arbitrary direction with respect to the direction of the magnetic field is not possible. For low frequencies, i.e. ω « kc, we may neglect the coupling and disregard the transverse waves. When we neglect the electromagnetic modes we have quasi-electrostatic instabilities satisfying (9.15) and (9.11). For ky = 0 the dispersion relation for quasi-electrostatic waves becomes /9.4/ k2e
+ k 2e
X XX
+ k k (ε + ε
Z ZZ
X Z
XZ
) = 0.
(14.51)
ZX
When the distribution function is isotropic we have to use (9.185) whereas for an anisotropic distribution function the elements of the dielectric tensor have to be taken from (9.194). If one neglects in (9.185), (9.194) the high frequency electron cyclotron resonance terms ηΩΕ - but keeps the ion terms - a further simplification is possible. In many plasmas one has ωρι » Ωτ and since ω < kc was assumed one has for ion waves with ion cyclotron frequency ω - Ω Γ (λ - rLI) 1 ω
ΡΙΚ<
PI
°/rLI
J
LI
(14.52)
MICROINSTABILITIES
488
or with (2.21), (2.40), (2.25) with cth (TE - TT).
ß « 1,
* u^
and (8.3) (14.53)
Thus we conclude that our approximation is valid /14.71/, /14.18/, /11.40/. only for low beta plasmas At low beta, electromagnetic effects increase the magnetic energy so that only electrostatic modes can become unstable and transverse modes may be neglected. Assuming a simple Maxwellian distribution for the ions, and for the electrons a Maxwellian distribution displaced by the relative electron-ion velocity u0 of the form ^exp[(c? + {cu - u0)^)/2kßT] one obtains the dispersion relation of the hydro dynamic twofield. Since stream instability in a uniform magnetic we are concerned with a wave nearly at the resonance with the ion-gyrofrequency, we may neglect all the ion terms except n = -1. The integrations over on and c^ respectively give W-functions. The argument or the ^-function (9.196) is small for electrons, since u f l °thE ^ 1 anc^ large for the ion W-function because otherwise the mode is Landau damped. (In the longwavelength limit correlation damping dominates Landau damping /14.81/). So the ^-functions may be expanded. The real part of the dispersion relation then becomes /14.18/, /11.40/ 2
2
PE
PI
-e
1 1(XT)—£—
= O,
(14.54)
χ Λ 2 c Γ ΐ'ω-Ω. °thE °thl thl 7,2 2 where xj -= k^r^j =_ k^c^^j/Q,^is the argument of the has a modified Bessel function. Since I<| (xj) exp (-xj) maximum of 0.22 at kfrlj - 1.5 one has - for TE^Tj the result that the mode has the frequency
ω = üj
+ ω0.22ΤΕ/ΤΙ
* Ωχ.
(14.55)
An instability ω - Slj/2 has been found also /14.80/. Substitution of ω into the imaginary part of the dispersion relation gives for TF - Τγ the rough instability criterion u
.. ^ 13c., T (14.56) cr%t tnl for the hydrodynamic two-stream instability in a magnetic field. When TE/Tj increases {TE > T£) , the critical relative velocity ucr£t decreases, if the waves propagate exactly parallel to the direction of the magnetic field, the instability threshold is increas-
489
PLASMA INSTABILITIES
ed to ucvit - cthE - 40ethI, but the growth rate is decreased to -Ω^- whereas the growth rate for oblique propagation under angle Θ is larger, i.e. γ ^ oip^sin Θ (uip^cos θ) 1 / 2 /2/Ω^ or γ - ωΡΕ /14.18/, /14.73/. An abundant literature exists on the twostream instability of a magnetized Vlasov plasma /14.72/ since the two-stream instability is probably the most extensively investigated microinstability. Because ω - üj- according to (14.55)fsome authors call this instability, which is due to the intermixing of a weak stream of electrons or ions with a background plasma and the coupling of the slower electron (ion) cyclotron mode with the ion (electron) cyclotron mode
of the plasma, an electromagnetic
cyclotron
wave
in-
stability /10.14/. The other low frequency (ω - 0) wave instability which we mode /14.82/ is an Alfvên discuss also in section 14.4. For laboratory experiments the theory of the instability in finite systems is of importance /14.75/.
The finite
plasma instability
{finite
/14.28/f /9.6/
length instability), produced at a frequency connected with the particle transit time by two oppositely directed electron beams of finite length in a strong magnetic field is a beam instability. Finite plasma effects have also been discussed for cold electrons and hot ions in a magnetized plasma /9.6/, /14.76/. The Vlasov-Poisson system of equations gives for a plasma slab of thickness a in the ^-direction (BQ parallel to z) a transcendental equation for ζ Ξ (ωρ^/ω^ttL - l)-1/2 fe-αζ fe.aC 1 + ccot - ^ — = 0 and 1 - ctan - 4 j — = 0. (14. 57) If we denote the solutions of (14.57) by ζη, for the frequency ω of the instability ω
η
=
ω
Ρ^η
/ (
^
+ 1)
one has (14
·58>
which should be compared with ω 2 = ω^(^ | ( /^ 1 ) 2 /(^^/^ 2 + 1)
(14.59)
for a wave in an infinite plasma. We see that in a finite plasma the wave numbers kn, k\ are discrete. The growth rate of the instability is nearly the same as for an infinite plasma. For a half-infinite plasma contained by a high-frequency electromagnetic field in a distance I from a conducting wall the stability
MICROINSTABILITIES
490
condition cot(j/o)2/52 - k2l) > 0 (14.60) has been given /11.100/. The two-stream instability of a two-component plasma confined in a strong axial uniform magnetic field /10.2/, /14.10/, is also called Buneman instability /13.67/, /14.77/. The Farley instability /14.91/ is a
variation of the ion sound two-stream
instability
in-
duced in a uniform weakly ionized plasma by an electron drift through ions in a direction perpendicular to the magnetic field. The electron drift is often produced by crossed electric and magnetic fields. In the Buneman instability any transverse motion of the charged particles is neglected. Electrons and ions have different drift velocities/ so that uQ Φ 0, and different temperatures; the velocity distribution is Maxwellian. Due to the relative motion uQ a current appears and the Buneman instability is a current driven instability. Bounded and unbouded plasmas have been discussed. Also the computer simulation of current-driven microinstabilities has been given in a review article /14.66/. For bounded plasmas the thresholds given by (14.56) must be replaced by other expressions. As the temperature ratio TE/Tj increases the number of unstable modes in a bounded plasma increases: TE/TI = 1, n = 0,1; TE = 2TJf n = 0,1,2; TE = 6Tj, n = 0,1/2,3. Because of the stronger Landau damping higher modes are difficult to observe. All results are sensitively dependent on both geometry and density profiles, if there are any. Mikhailovskii defines /9.26/ the hydro dynamic Buneman instability '/3 by Tg * Tj as well as by frequency ω= 2^/^pE(m^/mj) /m )1/3f s e e p 492. and growth rate γ = ωρΕ2~4/3/3(m The electron-ion instability is defined as due to the relative motion between ions and electrons in an unmagnetized plasma, see p 469, with growth rates proportional to the derivative of the electron (ion) dis-
tribution. Electron-ion
streaming
instabilities
(elec-
trostatic and electromagnetic mode) are the names applied in a magnetized plasma. The streaming takes place across the magnetic field. This terminology is however not used strictly /14.9/, /14.78/. The electrostatic electron-ion streaming instability
/14.78/ is found for ωρΕ
» ÜE,
TE > Tj, Uç >
ο^Ε.
It is due to the resonant coupling of a Doppler-shifted ion mode and an electron cyclotron mode. Its growth rate is
491
PLASMA INSTABILITIES
γ - (mE/mI)UAüE.
(14.61)
The energy source of the instability is the electron current across the magnetic field. The instability involves the whole electron distribution. It occurs only in sufficiently dense plasmas and is stabilized by the damping of the electron cyclotron and Doppler shifted ion modes. The electromagnetic electron-ion streaming instability
has a monoenergetic cold ion beam directed
across a magnetic field as its energy source. It arises in a large amplitude whistler (collisionless or collision-dominated). The transverse growing electromagnetic oscillations are produced by a coupling between the ion beam and the whistler wave (anisotropic distribution function!). The growth rate of the instability is limited by damping effects and transverse and wave numbers /14.78/, /10.2/. Electron-electron ion-ion instabilities are (electrostatic) two-stream instabilities/ either in an unmagnetized or in a magnetized plasma /14.79/ and arise from electron (ion) distribution anisotropy. There exist no electromagnetic instabilities due to two streams in one species. Current driven instabilities are either field aligned current instabilities (current parallel to BQ as in the Buneman instability or in the cross stream instability, p 300, in which the waves propagate acurrent cross the field) / or they are cross' field driven instabilities (current perpendicular to BQ) . A current parallel to the magnetic field can cause a variety of instabilities /9.26/. For parallel propainstabilgation of the mode (k\ = 0 ) , an ion acoustic For k ι * 0 an ion cycity is excited for w* < o^E. lotron instability (anisotropic distribution, see section 14.4) is excited for u0 < c^^g whereas for u o > °thE the electron-ion two-stream (Buneman) instability may appear. Current driven ion waves (ion wave instability in a magnetized plasma, current driven mode /14.13/) are driven by an electron current along the magnetic field in a plasma with zero ion temperature. For zero temperature the distribution functions are δ-functions. For equal masses of counterstreaming particles moving with ±uQ, ωΡΙ = ωΡΕ, Qj = ΩΕ the dispersion relation for electromagnetic modes in a magnetic plasma is very simple /2.6/ 0 0
0
~Z- 2
c k
o/0)
- ω
ω - ku
+ /cu
2 , 2 /
+ ω Ώ —— Ί
\
O ,
T;
Ρ\ω + kuQ - Ω
θ
+
7
τηι
ω - kuQ + Ω /
\
Λ
= 0.
492
MICROINSTABILITIES
This expression has singularities at ω = ±(ku0 - Ω ) . When 5 2 & 2 /ω£ > 2ku0/ (Ω - kuQ) not all roots for ω are real. Therefore this is the condition for instability. kuQ > Ω is always stable. degenerBesides the normal ion-wave instability a has been reported which is a ate ion-wave instability non-dispersive low-frequency mode (ω « Qj) for T E\\ > TI\\ f ° r Anisotropie distribution function /14.83/. The influence of conducting walls on the ion wave instability in a cold current plasma has also been investigated /14.85/.
If Tj Φ 0, we speak of ion sound instability,
ion
acoustic instability (fully ionized plasma: collisional /14.13/, in a collisionless shock wave /12.64/, p 318, nonlinear current driven mode /12.63/, p 374, in a weakly ionized plasma /14.84/). Ion wave instabilities and ion sound wave instabilities may also grow wavelength due to an ion beam (chapter 15). Long ion-acoustic waves in a magnetized plasma in a gravitational field have been discussed in /4.86/. Mikhailovskii /9.26/ gives the following survey for instabilities driven by a longitudinal current: 1 . u0 > ci.jtE (high current velocity) , weak magnetic (electrostatic field, Qj? < ωρ^- Buneman instability , kz = LùpE/uQ, electron-ion streaming instability) ω - γ - (mE/mj-) '' ωρ^, see also (14.61). 2. u0 > c-f-hEr strong magnetic field, Ω^ » ω-ρΕ elecJ ω - Ω^, k~=flE/u0, tron cyclotron instability 2 4 3 1 3 / JT . There is γ = (/?ωΡ£./2 / ) ((mELupEs±n Q/mz^E) also a second branch described by the dispersion relation 1 = ω ^ / ω 2 + O U ^ C O S ^ / (ω - kzuQ) . 3. u0 < c-f-fo-g, u0 > c-f-foj, TE » Tj. Then one finds high ion acoustic modes. For kj_ = 0, the magfrequency netic field does not affect the instability. These instabilities have a maximum growth rate as in the case BQ = 0. They may γ - (mE/mj) Î/2ωρι, be excited also by runaway electrons. 4. uQ < c^E, u>/kz < ctfoE, uQ > c-f-foi* In contradiction to the ion acoustic modes it is not essential that TE » Tj. Now electrostatic ion cyclotron instabilare found /14.104/. They grow in a plasma ities with TE - Tj if u Q is not much smaller than ο^^Ε. For TE - Tj no ion acoustic instability appears. ion-acoustic in5. Theoretically, also low frequency stabilities could be excited. Their growth rate γ^ is small and proportional to ^foE/^^z· If Ίη be the growth rate of the high frequency ion acoustic mode, one has Yi/yn - Ωχ/ωρρ. The low frequency mode plays a more important role in a collisional
493
PLASMA INSTABILITIES
plasma. Additionally /14.93/, for counter-streaming ion currents along a uniform magnetic field a purely growing instability exists with a growth rate as high as 16 times the ion gyrofrequency. When the streaming ions are only 1 percent of the stationary ions, the growth rate is still 4Qj, but the real part is near the lower hybrid frequency. Finite beta effects (3 > 0.0001) increase the growth rate. For ion beta a 1 three instabilities appear /14.103/: a long wavelength kink-like mode and two instabilities with k > *Vj/ the electromagnetic ion acoustic instability
and the whistler
current
instability.
These instabil-
ities have substantially lower thresholds than the electrostatic ion-acoustic instability, and are favored by increasing ion beta. The whistler current instability heats the ions. Instabilities for ion beta > 1 and kr^j > 1 have also been investigated /14.94/. Instabilities with low phase velocities offer a possibility for ion heating /14.102/. Cross field current driven instabilities form an important family. Theories published may be classified as follows: 1. isotropic distribution function, homogeneous plasma /14.87/ 2. anisotropic distribution function, see section 14.5 3. inhomogeneous plasmas /14.88/, and also non-linear theories. 2
A
It*o
"y
gradient drifts Vrc
Fig.
47.
Cross field instabilities
current
driven
MICROINSTABILITIES
494
We discuss here the isotropic theory. According to Hellberg /14.89/ we assume the geometry configuration as shown in Fig. 47. The basic physical assumptions are 1) high frequencies ω » Ω^, 2) electrons see the ExB drift, ions don't, they are at rest or have straight line orbits and have a Maxwellian distribution, 3) electrons drift on self-consistent magnetized orbits and have a Maxwellian distribution, 4) low beta homogeneous plasma in a homogeneous magfor gradients, if netic field, local approximation any (compare p 366), 5) linearize and integrate along unperturbed orbits (characteristics), compare section 13.4, p 366 6) quasi-electrostatic approximation. When uQ is again the relative motion between electrons and ions the dispersion relation for^electrostatic modes propagating perpendicular to BQ is given /14.89/, /14.87/, /9.26/ by 9
9
1'p /
\
ω- k u
I
_±?>
(3^ 7π η= ι -οο
V2k
IAA r~>\
Here Z
I
=
*/kGthIA'
nE =
Z
( ω
k U
y o'nÇlE)/^kz°thE
"
-v2y? K r y LE = e * I
(14.63) 22 Γ (fer*) n n y LE' to (9.170)/ p 227 is have been used and Z according
the Fried - Conte plasma dispersion
function.
netized
instability
The fol-
lowing regimes have been considered: 1. Hot electrons, cold ions, TE » Tj. For y„ > m^w 0 also the electrons have to be assumed to be hot. 1.1 Low frequencies, n = 0, Zj- » 1 , %E « 1 ΐοη acows, tic instability 1.1.1 kyrLE » 1/ magnetic field plays no role unmagion acoustic
(ion wave)
ω = feajd + fe2x2)-V2f s e e (9.76), Landau damping as T j increases / 1.1.2 kyrrq « 1, long wavelength, magnetized acoustic
\%on wave)
instability
ion
/14.66/, frequency
given by (9.76), dispersion relation
(14.64)
495
PLASMA INSTABILITIES
where ω = ω - kyU . Oscillations are excited at ω - topj. The growth rate is enhanced over the unmagnetized case for this regime, but smaller than for 1.1.1. Growth rate /9.26/ γ = u)/iïs(u - ω/fe) (ω/^α χ ) 2 Γ ο
for s £ 1,
(14.65)
where s = km^2/kz/2mj and kz « fe . As ωρΕ/Ώ,Ε decreases this mode vanishes for ωρΕ = Ω^. 1, 1.2 Cold electrons, cold ions, T« à Tj, zp» 2p » 1f n - 0, fluid mode only, kr^g « 1. This regime gives the (linearized) modified two-stream instability /14.66/. In this instability the Doppler-shifted electron plasma oscillation is nearly perpendicular to B . A negative energy mode (ω * kyUQ - kzu>PE/ky) couples with the lower hybrid mode. The dispersion relation for the linearized modified two-stream instability reads 72
2 1
* ~2
- 7T(
2
2
1
Γ2 + —
(14.66)
k (ω - k u) ω y y ° for kz/ky < (mE/mj)^ / 2 . γ, ω - ω^#. The maximum - Vm^/m-r. As kz/ky ingrowth rate is seen for kz/ky creases, the wave-wave coupling decreases and the modified two-stream instability goes over into the ω
magnetized
ion acoustic
instability.
1.3 kz/ky -*- 0, ω a iïE, kyrrE » 1. Then only a single (resonant) value of n is of interest ω = k uQ - | w I Ω .
(14.67)
This dispersion relation describes the electron
lotron
drift
instability
(beam cyclotron
drift
cyc-
insta-
bility) /14.66/. Here a negative energy Dopplershifted slow electron Bernstein mode couples to ion acoustic modes. 2. Electrons and ions at equal temperatures, TE = Tj. In this regime no unmagnetized ion acoustic instability nor conventional electron cyclotron drift instability appears. > 1 no instability can appear. 2.1 For kyrEE 2.2 By Landau damping the negative energy slow Bernstein mode will be driven unstable and the Bernstein 0.3ct^E. instability appears for uQ > 3. Hot ions, cold electrons, Tτ » TE. 3.1 Now the Bernstein instabil%ty, still with narrow angle appears, but with reduced growth rate γ^Ο.ΟΟδΩ^,
MICROINSTABILITIES
496
depending on the thermal spread of the hot ion stream /9.26/. 3.2 Further, the modified
two-stream
instability
is
present. 3.3 As k2/ky increases, the modified two-stream instability goes over into the electron acoustic insta-
bility
(electron
sound instability)
/14.90/
ω = k u - k a cr (1 + X^fe 2 )" 1/2 , y o
here a SI
z bl
D
(14.68)
'
YkBTT/mE.
(14.69)
The growth rate is large ^ {QjQE) V 2 or in another approximation
^ = (f) 1 \ ^ i w ( 1 + À D f e 2 ) " 3 / 2 / f e 2 c L r
(14 70)
·
The electron acoustic mode has negative energy and is unstable due to ion Landau damping. Mikhailovskii states /9.26/ that the thermal spread of the ions has its greatest effect on the oscillations with maximal / 3 no incos Θ. For cos Θ - 1 and c^j > uQ(mE/mj)^ stability is possible. The largest growth rate occurs for maximal cos Θ - (rn^/rnj) ^/2 for given UQ/C^^J m
^olcthI^'2
a n di s
ωρχ(1 +ω^/Ω|)~1/2. (u0/cthI)
^/2.
Electron acoustic oscillations are excited if u o K cthl* With increasing u0 the electron acoustic instability goes over into a hydrodynamic instability in which the energy spread of the electrons and ions is unimportant. If however uQ » c-f-hj a high frequency kinetic instability is discovered /14.101/. Its growth rate is greater than the maximal growth rate of the hydrodynamic instability. For unmagnetized ions (Im ω » tij, kc+^j » üj) but strongly magnetized electrons (ω «: ÜE, κ ^ ^ « Ω Ρ ' M a x w e H i a n distribution functions and |ωf ^>kc+^I cos ΘΙ the dispersion relation of the new instability reads (in a frame in which the electrons are at rest) ^ 2 /n2 2 1+t*pE/nE-upEcos Λ
-z2 Z 2 Q / 2 _,_ . r E 2 /7 2 2 ^ θ/ω +v/^zEe ω^/k cthE +
^FIU+i/i\zI^(zI)]/k
cthI
= 0,
where z^ = u//2kc l E
« plcos ΘΙ, 2 . = (ω - κ·ΐι)
/}/2kc£}lj,
497
PLASMA INSTABILITIES
W(z)
= e~z
11 + 4= U * dt) ,
see p 372.
In concluding this section we would like to mention some special instabilities like the ion-ion /14.95/ which is of interest cross field instability for theta pinch implosion. The stability of the (bumpy) theta pinch has also been investigated in the framework of the guiding center plasma equations /14.96/. It should also be mentioned that thermodynamic-kinetic methods have been applied to investigate the stability of current-carrying collisionless plasmas /14.105/. There are also some interesting instabilities in relativistic plasmas. The relativistic
strong
electromagnetic
wave instability
/14.97/ is a
nonlinear effect and is generated by a strong electromagnetic wave. Both electrostatic and electromagnetic ) may be excited dependmodes (both for £ll^0 and ing on the strength parameter v = eEQ/mgCU0, such that 1 « v « m-j-fm-j?. The strong electromagnetic wave could be either a wave in the plasma or an external pump wave. Longitudinal modes exist in a frequency band near the plasma frequency and mainly propagate at small angles with respect to the external magnetic field, their phase velocities being greater than that of light. Thus longitudinal wave generation is not favored. Also circularly polarized electromagnetic waves have been studied. The unstable modes pinch the azimuthal plasma current. This new instability seems to have interesting applications in laser-plasma physics and for rotating neutron stars (particle acceleration near -pulsars) . Another microinstability in homogeneous magnetized relativistic plasma is the
resonant
diffusion
instability
in a relativistic
col-
lisionless plasma /14.98/. It is generated when acollisionless relativistic plasma is immersed in a homogeneous magnetic field, in the presence of a cold non-relativistic plasma. Then unstable electrostatic flute modes become coupled with selected particles. A resonance occurs with the associated waves, since the particle mass is no longer the relativistic rest mass. There is then a resonant diffusion of relativistic particles and an energy transfer from the particles to the waves. The kinetic drift motion of the relativistic particles is then the energy source of the instability, which might be stabilized by decreasing the number of relativistic particles compared to that
MICROINSTABILITIES
498
of the background plasma. Other instabilities are the Breit-Darwin plasma /14.99/ or the linear sideband modes eigenmodes /14.100/ which appear at nearby frequencies in a collisionless plasma when a small amplitude monochromatic electron plasma wave is launched. The modes are called linear since the amplitude of these excited sidebands depends linearly on the amplitude of the launched wave. By nearby frequencies one means that the ratio Δν/ν < 0.1, where Δν is the frequency separation from the frequency of the launched wave v Q . The amplitude of these modes exhibits large spatial oscillations while the amplitude of the exciting wave damps monotonically according to the linear Landau theory. These sidebands may also be called -plasma echoes. 14.4 MICROINSTABILITIES OF A HOMOGENEOUS MAGNETIZED PLASMA WITH AN ANISOTROPIC DISTRIBUTION FUNCTION Anisotropie velocity distribution functions may be either symmetric around the direction of the magnetic field, bi-maxwellian (7.81), (13.41) or generalized non-maxwellian distribution functions generated by particle beams, particle injection or particle losses. In several methods used for plasma production, energy is transferred to the electrons by accelerating them in a particular direction. Since in some plasmas the relaxation mechanisms are slow the electron distribution functions will be anisotropic. Thus the plasma acquires an anisotropic temperature even in the absence of a magnetic field. An anisotropic or bi-maxwellian distribution function of electrons provides a coupling mechanism between longitudinal and transverse modes. Such anisotropic distribution functions not only modify known instabilities, see also section 14.3, but they also excite new types of instabilities. To see this we discuss a simple example /2.6/. First we write the dispersion relation (9.195) for electromagnetic waves propagating along BQ for a one species plasma in the form
-oo o
|| II (14.71)
499
PLASMA INSTABILITIES
(see also (14.71a) on p 501). The factor 2π comes from integration around the s-axis and the factor 2 cancels because del,2 _= 2c\dc]_. When we insert into (T4.7V) the beam-like anisotropic distribution function (T,, -> 0) ;f0(öj/S||> = δ ( σ ||)/ 0 (ο 1 ) '
(14.72)
where / Q is an arbitrary equilibrium distribution/ we obtain 2 2 2 ω (ω ± Ω) - ωρω(ω ± Ω) 9 (14.73) S z (oo ± Ω ) ζ
+ ü>p</2
or for a two species plasma /10.14/
-, + 4E
,2
+
(ω + Ω^) 2 2 2 ω
1 -
4l
ωPE ω (ω + Ω„)
ω
2
c
]-
(14.73a)
Ί
ΡΙ 1 ω(ω - Ω ) J '
where
= c )c d (14.74)
ϊ> //V i'*ll î *A
tains for ions an overstable bility , see (3.9))
wave (oscillating
insta-
^/e/2, ± ωρι<ο-2 > 1/2.,-/^ (14.75) I For isotropic distributions the imaginary part vanishes. Let us now derive a stability criterion for transverse waves propagating along j§ which is simpler than (14.47) - (14.49). For propagation along £ Q longitudinal and transverse waves are decoupled. For one species (14.46) may be written in the form (for transverse waves) ω = Ω
2 ω
~2 7 2
= o k
F (ω/fe. ,C||) ko..
- ω ± Ω
(14.76)
■do,
where
FWk,on)
= [<ω - onKn)FsU
- F;JU|
.(14.77)
MICROINSTABILITIES
500
If damping and growth are present for a given distribution function, ω has to change its sign at some k. Then there exists one value of k where ω is real (curve of marginal or neutral stability). According to (9.161), the integral in (14.76) may be written at such points as
(14.78) So the curve of marginal stability is given by (14.79) It is, however, very difficult to derive general stability criteria /14.164/. It is therefore useful to consider electron resonances (ion motion negligible) and ion resonances separately. For ion resonances in a two-component plasma with distribution functions ( 1 4 8 0 ) Όΐ=^οΙ(*Ϊ+νΦ · foE=^oE^\+aE°\)' the criterion (14.47) - (14.49) for waves propagating
along
B
gives
o
= Ωτ(1 - α~ 1 )Α.
= [SlT/k] m
I
m
I
(14.81)
I
Neglecting the electron terms we find that this satis- cj/k) > 0. Subfies (14.49), since UpjOLjF-r ( [üj/k]m stitution of (14.81) into (14.48) gives the result /1 4. 17/ that under the assumptions Çlj/ajk » cthlt ct^j - cthE, k^c^aj. > Ω 2 |Ωχ(1 - *[/az)/k- tiE/k\ » cthE, the waves are stable if °A
<
^Τ\\
(T
||/T1 "
1)
/Ti'
(14.82)
where cÄ is given by (4.65), p - njmj. This result can also be obtained /2.6/ from (14.79) by inserting (14.80) into (14.71). So for T\\ > Tj_ and small values of B0 we have stability, but the plasma becomes unstable if the magnetic field increases. Instability arises only if Ω* > ω|τ^/ [Τμ (Τ|| /Τ ι - 1 ) ] and then only for fc2 < fc2. p o r rji > rp^ w e have always a transverse instability for k2 < fc2 = {Ω2(1 -T||/Tj_)2 + {electromagnetic + ω|τχ (1 - T||/T_L)/T||}/32, see (14.92) wave instability, transverse electron wave instability , Weibel
instability
/14.116/). In an unmagnetized
501
PLASMA INSTABILITIES
plasma according to (14.33), instability could be generated for a small anisotropy by reducing k: modes with long wavelength are easier to destabilize. In a magnetized plasma k2r^j > 1 is required so that a field-free motion of wavelength 2i\/k persists. 1 is equivalent to k2c2/u^I = kïr^j/ftj > Vßj# k2r2LI> So the transverse mode propawhere ßj = 4-nnkßTj/B^. gating along B0 is unstable if Ω2(1 - Τ||/Τ1)2/ω^χ + Τ1/Τ|| - 1 > g"1.
(14.83)
The circularly polarized mode, propagating parallel is stable for T\\/T± - 1 > β"1 and unto ί , E^ik\\B0 stable for T|| > T\ and is called fire-hose instabili-
ty, whistler instability, ion Alfvên wave instability
Alfvèn wave , garden-hose
instability, instability,
see (9.136), p 214 in volume 1 of this Handbook. From (14.83) we see that low beta or λ > rL stabilizes this electromagnetic instability. The results obtained /10.14/ may also be derived by using (14.71) for many species in the form integrated over e^. This reads n - 7
2
-
D - k c
2
2
-
ω
'2
J.
+ b)pF
[("-fc0||)/og(0||)-fc<0^>/2.a/og(0||)/a0||^ ]
ω - kc.. T Ω Γ
J
l l +
ω
E
+
°\\ (14.71a)
2 ( ( ω - ^ | | > ^ ο ΐ ^ | | > ^ ^ I J ^ ' ^ o J ^ H ^ 3Ö„ Ρΐ] ω - /cou + Ωχ
where -Ω# is for the right and +Ω# for the left hand circularly polarized wave. The singularity ω±Ω-& 0 in the form (14.93), so that even for ω -> O a small anisotropy T. > T results in instability. ^ " The results obtained so far may be specified as follows. When instead of (14.72) we insert / 0 s = 5 ( c ) and expand about ω - 0, we into (14.71) (cold plasma) obtain /9.6/ (9.94), i.e. an Alfvên wave (Alfvèn wave . Without the expansion in ω we obtain instability)
MICROINSTABILITIES
502
(9.90)/ see p 197. In the frequency range iïj < ω « Ω^ we have (9.101)f which means a whistler or helicon
(whistler
instability,
helicon
instability).
To take
thermal corrections into account /14.156/ we find for ω « Ω,
2
'
k
2
=
°A
f—2
(P\ "Pll 2
p . = L\m If -^άο _L s r os 2 s
+ 1
\
P
*
J
l "Pll n
2
+
(14.84)
(14.85)
P A 1+^/δ2\ P^ / see p 214. We see that a hydromagnetic instability
(fire-hose instability, garden-hose hose-instability) occurs, if P H - ρλ > QCA.
instability,
(14.86)
Let us say some words regarding this family of instabilities. The Alfvên-wave instability /14.106/ is defined as an electromagnetic microinstability generated in a cold magnetized plasma when the particle energy is larger in the direction parallel to the magnetic field than in the direction perpendicular to it. The instability is due to the centrifugal force, which acts on the plasma flowing along a curved field line, see Fig. 20, p 213. The curvature of the field lines is produced by a first perturbation and is enhanced by the instability. The whole field pattern oscillates back and forth. There are two modes: the
fast
Alfvên-wave
instability
is due to a large aniso-
tropy involving the whole plasma and the slow Alfvênwave instability which exists at small anisotropies. The slow mode is driven by the tail of the velocity distribution which is in resonance with Qj. The energy source of the instabilities lies in the T.. - T. temperature anisotropy. By making the magnetic field strong compared to the plasma pressure anisotropy the instability (anisotropic pressure instability) may be stabilized. There exists a higher order CGL theory and also a guiding center Vlasov theory of the Alfvênwave instability. The Alfvên-wave instability may be stabilized by finite Larmor radius effects. Resonance effects between helicons and Alfvên waves are known (low density pinch instability, growth rate
PLASMA INSTABILITIES
503
γ <υ c2/3feuy3 /14.112/). The whistler instability /14.107/ (helicon instability) is also an electromagnetic microinstability similar to the Alfvên-wave instability, but it occurs near fij, where adiabatic invariance is destroyed. The whistler instability is produced by electrons exchanging energy with a whistler wave (9.TOO). Since the transverse phase velocity of the whistler mode can be much less than the speed of light, there exist particles which can travel with the wave. This whistler mode is also called pressure driven whistler instability . It is stabilized by relativistic effects and by removing the anisotropy. In the frequency range üj « ω « QE there exists another whistler related mode, the current driven relativistic whistler instability /14.108/. It occurs in a plasma with low-energy thermal particles and with a group of hot relativistic electrons. The latter move in flat helices. The hot electrons further produce an electric current by moving along the magnetic field. The flat helices make possible a resonance between the hot electrons and a whistler wave, which grows. By decreasing the plasma density the frequency range for instability is reduced and stabilization may become possible. A current driven whistler instability (whistler current instability) appears however also in a non-relativistic plasma /14.449/. It can give rise to an anomalous and heats the ions. A current driven Alfresistivity vën instability driven by the resonant Landau and ^ transit time terms for propagation not parallel to B has also been investigated /14.449/. There also exists a surface-wave helicon instability /14.113/. Whistler instability may be destabilized (enhancement of the maximum growth rate) by the presence of cold electromagplasma /14.150/. The same is true for the netic ion cyclotron instability in a mixed warm-cold plasma (warm and cold ions) as well as for the ion whistler instability /14.162/. The fire-hose instability (hose-instability , garden-hose instability, also called transverse wave instability, high-frequency electromagnetic wave instability) /14.109/ may be considered as a whistler instability in a warm plasma. It may be termed macroscopic. For sufficient transverse anisotropy T^/T^ * 1 not only the bounce model pinch instability, a form anisotropy of the fire-hose instability, but also flute instabilities appear e.g. in a cylindrical pinch /14.110/. The fire-hose instability may also be excited by external gravitational forces combined
MICROINSTABILITIES
504
with a magnetic field in a stratified collisionless
plasma (internal
gravitational
instability
/14.111/).
The necessary and sufficient condition for stability against the (electromagnetic) firehose instability (see p 214) is P
l " Pll +
ß2/y
o
> 0/
(14.87)
against the (electromagnetic) mirror later and p 214) is 2 P i + B2/MO
instability
- pj/3p„ > 0,
(see (14.88)
and against the (convective) internal gravitational instability (in an external gravitational potential Φ) is ρ'Φ' + ( Ρ ι + β 2 / 2 μ ο ) · 2 [ ΐ / 3 Ρ | | + (1 - p 1 / 3 p | | ) 2 ?
2
•(2pi+BVy0-Pi/3P||)
-11
'J < o.
(14.89)
The primes i n d i c a t e d e r i v a t i v e s . The growth r a t e s a r e (9.136), i . e . Ύ = [^ 2 (P|| * Pi " S 2 / y o ) / p ] V 2 for t h e firehose
instability
γ = fc[6P|| - p2i/(pi
for the mirror
instability.
(14.90)
and + B2/2yo)]1/2/p1/2(14.91)
Propagation oblique to ~BQ may be described by the superposition of two plane waves exp [i (tût - kzz - kxx)] giving a wave travelling in and exp[i(ut - kzz + kxx)] the direction of 2 , but not a plane wave described by e*v[i(u>t - kzz)] cos kxx /2.6/. A magnetic field subjected to this wave looks like the one shown in Fig. 20, p 213 (mirror instability). The magnetic resonance between such a wave and particles travelling with the phase velocity ω/fe^ along the field lines excites this instability. No instability exists unless there is a minimum in the distribution function (integrated over o±) at some value of cu. If the particles have greater energy E^ than E\\ tney become concentrated in the region between magnetic mirrors thus increasing the particle pressure which expands the magnetic field etc. Let us now turn to electron resonances. In this case either a relativistic theory has to be used or
505
PLASMA INSTABILITIES
based on (14.80) instability in the frequency range ω * QE may occur for αΕ > *\ , e + hirL > cthE\\' in0)2 » 4keQg, b)pF » ^ühn^/m-r (electron cyclotron /14.17/, /14.114/, see stability, Weioel instaoil%ty later). For high frequencies (ω £ Ω# ω > ωρ^) the cold treatment is adequate and shoulâ be carried out relativistically. For low frequencies (ω « üj) we have the hydromagnetic limit, compare the firehose instability. The instability of the two circularly polarized transverse waves propagating along B has also been investigated using the Nyquist technique /14.115/ assuming infinitely massive ions. For isotropic distributions in the plane perpendicular to the magnetic field, growing waves with k < kQ are obtained for TjL > T|| , where kQ is given by
32
* o - 4 s ( ^ - 1)+ (1 -¥-LÎ4-
(14 92)
·
The frequency of instability at threshold k = kQ is ω = ±(1 - Tu/Ti)QE.
(14.93)
When finite ion mass is taken into account /14.67/ for ω ~ Qj the firehose instability appears. Even if there is insufficient anisotropy for this instability, Landau there are other instabilities due to inverse damping of ions satisfying ω - kc\\ - Ωχ. In this case of slight anisotropy the growth is small and given by Y = Ω
χ θ χ ρ[4(ί:" 1 ) ]·
(14 94)
·
All these electromagnetic instabilities (Alfvên instability, firehose instability) are unimportant in low beta plasmas and are of interest mainly for cosmic space plasmas. For plasma devices the electrostatic (longitudinal) Harris instabilities (cyclotron instabilities) are more dangerous. This is however not true in general. Under certain conditions /14.146/ the electromagnetic instability in contrastreaming collisionless anisotropic plasmas may have larger growth rates than does the electrostatic instability. Up to now we discussed electromagnetic waves propagating along BQ. We now turn to electromagnetic £ Q . Electromagnetic waves waves propagating across with λ « rL will be unaffected by the magnetic field. Later on we will discuss longitudinal modes also. When (14.50) is valid, electromagnetic transverse
MICROINSTABILITIES
506
waves {cyclotron waves) with E^BQ and mixed transwaves) verse-longitudinal waves with EIB0 (Bernstein propagate in an uncoupled manner perpendicular to ÎQ /14.117/. For isotropic distributions transverse waves are : always stable, but anisotropic distributions may become unstable. Transverse waves propagating across BQ (fe|| = 0) are described by (9.31), (9.190) which may be written in the form +oo oo
2 „2 ; ω„ I I cuJ (k c./iï ω - c k.1 - 2ÏÏO) L* PsJ J II n x Γ S , n
s
)
-oo o
(14.95) J
OS
dc.. °l "
J
S I
OS
ω I dc.. cl ηΐί
s
_
- ω
J
c
Qs\
\\ dc. // J 1
II
1
compare (9.203). Dispersion relations for wave propagation across the magnetic field in hot plasmas may
be simplified by introducing the magnetoplasma persion function
dis-
π
f {z)
u
=
sin* πωί°°3 ( ω φ ) o
ΘΧρ [
= ΎΖ exp(-2) I (z) n =-oo
n
"a ( 1
+ COS
φ ) ] α φ
=
ω w
n
see /14.165/. It is connected with the Gordeyev integral, see p 513. Deviations from a Maxwellian fQ do not modify the imaginary part of ω determined by (14.95). If all fQQ are Maxwellian, the modes are stable, see (5.25) and section 14.1. Instabilities near ω 2 belonging to n = 0 may appear only for anisotropic distribution. For distribution functions of the type (14.80) the necessary instability condition Υω 2 (1 - a"1) < 0 or Ψ > Tx (14.96) s I I 1 £ Ps may be derived /14.17/. If one assumes λ « rL, kcj_/ü « 1 , and stationary ions the Vlasov equation gives the dispersion relation /14.117/ 1 =
[ α Λ(ω
2
-5 2 ^ 2 )]+ (ω^ 2 <α, 2 >)/[(ω 2 -2 2 ^ 2 )ίΩ 2 -ω 2 )]. (14.97)
507
PLASMA INSTABILITIES
The c o n d i t i o n f o r s t a b i l i t y i s J2 2 2,^2 Ω >
[ωρ<ο„>/ο
- Ω ]
(14.98)
(14.99)
If ion motion is taken into account the dispersion relation is modified and the stability condition is P|| < #2/μ 0 . This criterion agrees with the criterion for non-existence of the firehose instability in the case of P\\ » Pi. Another analysis (Hamasaki /14.117/) shows that no instability of electromagnetic modes propagating perpendicular to BQ occurs if either ßn < 1 h growth rate is given P r i f athEl - cthE\V T The ° mΛVmaximum ■,',1 by i1/2 'thEl 'thE\i J Y = (\ !3"X - μ ' PE 2 2 (14.100) Ö'thEl | C^thEW j1n Also another growth rate has been given /14.117/ namely for λ -*■ 0
fV -
y2~- n|(1 - 1/0 ||E ) r è]}E=^nTUE/B2o.
(14.101)
Comparison with the firehose instability (propagating along BQ) shows that larger values of 3|]# - $]_E are required to excite the new instability. According to (14.101) it grows at a rate less than Ω^ whereas the firehose instability growth rate is less than Ω/. An electromagnetic microinstability of this type
is the anisotropic temperature instability, wave instability, Sharer-Trivelpiece temperature anisotropic transverse wave high frequency velocity space instability
transverse instability, instability, /13.31/,
/14.18/, /14.39/, /14.118/ (see also p 481). It arises when electromagnetic waves propagate across a magnetic field in a plasma with anisotropic temperature. This instability is different from the firehose instability for which the waves propagate along the magnetic field. The present instability has a higher threshold of electron pressure (and needs a greater T\\) to occur than the hose-instability, but leads to larger growth rates (14.100), (14.101). By decreasing T\\/Tior choosing other plasma parameters in a stable range, the instability may be stabilized. This type of instability may arise also in an unmagnetized plas-
MICROINSTABILITIES
508
ma and is then sometimes called transverse
ty /14.118/, transverse
wave instability,
instabili-
Neibel
in-
space
insta-
stability (which occurs however most easily for a wave propagating parallel to the magnetic field). Not only electromagnetic but also electrostatic modes may appear in a magnetoplasma. Perpendicular or oblique to the magnetic field electrostatic electron modes (due to a larger Τμ) may become unstable but also electrostatic ion-ion instabilities exist /14.118/· The mirror instability (see p 213 in volume 1 and p 504 is also a microscopic electromagnetic instability propagating across an inhomogeneous magnetic field and will be discussed in the next section. (The term mirror instability is however also used to designate the instability of mirror devices.) Electrostatic waves propagating oblique to the magnetic field in an anisotropic plasma with ^μ + Ο were first investigated by Harris (Harris instability,
velocity
anisotropy
instability,
velocity
bility, /14.119/). Later his results were generalized for T|| Φ 0. For isotropic electrons instabilities appear near ω - Qj for Tj\ > 8Tj||, whereas electron oscillations become unstable at ω - ηΩ^ for an anisotropic electron distribution if TE± > TE\\. If the electrons are assumed to be cold/ the ion cyclotron instability appears for ωΡΕ > Qj/2 and Tji > 2 ^ j | | /14.17/. Therefore electron heating is stabilizing since larger ^ I J / ^ J N values are needed for instability. The basic (nearly perpendicular) Harris instability has its energy source in the thermal energy of the particle gyration and is an electrostatic microinstability in a homogeneous magnetized anisotropic plasma. Since particle velocities in the direction parallel to the magnetic field are assumed to vanish (Tu ■> 0) and velocities of the transverse ion gyratory motion are assumed to be sharply peaked (e.g. around c*), a strongly anisotropic distribution for electrons fQ(e)
= const*
δ (c\\) 6(c,
- c*)
(14.102)
is assumed. Such distribution functions may be generated by beams or currents which may excite cyclotron instabilities /14.128/, but also strong electromagnetic waves may excite cyclotron instabilities /14.133/. This function has a hump, from which energy can be fed into a growing electrostatic wave at ω - ni]. The mechanism works also in the case of finite ion and electron temperature provided T^/T\\ is large enough. (Stabilization may be obtained by reducing this an-
509
PLASMA INSTABILITIES
experiisotropy) . T^ » Τμ is realized in injection ments. When the distribution (14.102) is inserted into (9.194), a dispersion relation for the Harris instability is obtained /2.10/ in the form 2 -h» O
(khk2)j2n{k
2
n=-oo
2
c*/«z
(ω/Ω^ - n) 2
2
(k l/k )n(J n)
ΏΕ
(ω/Ω^ - n) k^e*
(14.103)
= 1
(for ions at rest). With the Nyquist technique it can be shown that whenfcj_e*/ft> 1.84 there are growing threshold for the waves ω ^ Ω^ if ωρ?? > Ω„ \density Harris instability!. In plasmas of finite length the bouncing of the ions (or electrons) in a magnetic well shifts the frequencies of the unstable modes /14. 124/. The Harris instability has also been observed in an ion cyclotron resonance heated stellarator /14.125/. Of more interest than the Harris instability, for the Soper-Harris distribution function (14.102), is the instability, Timofeev isntability, which takes into account ion and electron temperature effects /14.120/. When in a fully ionized plasma confined in a homogenous magnetic field the perpendicular ion and electron temperatures exceed the longitudinal ones, one
speaks of a cyclotron
instability,
synchrotron
insta-
bility /10.2/, /14.121/. In this case the cyclotron motion can couple to a number of longitudinal or transverse waves such as Langmuir oscillations, Bernstein modes and electromagnetic waves. One may speak of electron cyclotron instability, electron cyclotron resonance instability (hot electrons, cold ions) and of ion cyclotron instability, ion cyclotron resonance instability (hot ions, cold electrons). The ion cycllotron instability may be convective as well as absolute /14.127/. Instabilities may however also be caused by cold electrons (whose density determines the upper hybrid frequency) and by hot electrons (loss-cone distribution, see later). This (convective) instability is called cold upper hybrid instability /14.155/. Also a low frequency flute cyclotron instability is known /14.452/. There are two types of cyclotron instabilities: either there is considerable particle motion across the magnetic field and a humped particle distribution
MICROINSTABILITIES
510
peaked at other than zero energy or there is no hump, in which case the particle motion along the field is essential /11.21/ (two-stream cyclotron instability). In order to investigate instabilities near harmonics of the cyclotron frequencies we start /9.6/, /2.10/, /4.11/, /14.119/ with the dispersion relation for quasi-electrostatic modes (14.51)/ (9.11). Using (9.194) in the form 2
e
kls ,πΩ I s \ ö. 1
= (1
- 4s/("2)6kl
dfJ
os 3c. ^L
«
3Jf
*
v
os li W 3 ο „ / ηr Ω " " H ' ^s
l^të
nω T n ski + kl· ' \\°\\
(14.104) ω
"
'
where Τη8^ι is again given by (9.186) one obtains the Harris dispersion relation X^4s(
2
?rrSfe
J
σ2(&.σ./Ω ) « I l s
ηΩ
+ K.G..
- ω
df
- Λ u
9e
i
df
os
ll9
i
c
*
(14.105)
= 0.
This ε is also called the dielectric response function, compare (9.162) for an unmagnetized plasma. Using the Plemelj formula (9.161) we may decompose ε into its real and imaginary parts J2(k.cjQ
-H
dcJ
n
(fe.c./Ω i l s
LA
)
§ηΩ
s
(14.106)
) δ (ω - fr.,0.. - «Ω ) || Il s
(14.107) II 3c | ( To have i n s t a b i l i t y one can c h o o s e df0/dc\\ or df0/dc\ p o s i t i v e . 3 / 0 / 3 c n > 0 means a two-stream p a r a l l e l t o field, b u t 3 / 0 / 3 ^ i > 0 or k | | 3 / 0 / 3 e | | + the magnetic
511
PLASMA INSTABILITIES
0 give a new situation. + 2niïsdf/dc2\> We may now investigate six special cases of Harris type instabilities. 1. cold electrons, hot ions {T^/m^« Tj/mj). In this case (hydrodynamic approximation /8.13/) we have k\ c-rp ι/Ωο - 0. Since the small electron mass is in the denominator of er (in the plasma frequency) the electrons contribute more to er than do the ions, whose contribution may be neglected. Integration by parts gives zr = 1 - ω 2 kj?/o)2fc2 a n d Zr = o results in = ü)p£>cos θ. Θ is again the ω = ß(fe) = Ω^ = upEKz/k angle between the wave vector and the magnetic field. For small growth rates we may use (9.166). This gives /9.6/ with ε^ from (14.107) the result
^kY^Psc s,n £ 6 (Ω. - fc..£.
2}dcJn(kici/Qs)
-
nü
,ηΩ
) i
(14.108)
so + k *f so] II dc{ dct
df
Ί ^1 A t w o - t e m p e r a t u r e d i s t r i b u t i o n f u n c t i o n of t h e Jf_
so = n s exp[ ^
gives Ύ?
"°Ϊ / α Ϊβ " cN/aNs
]/π
3/2
form
a,,Ils a,I s
/9.26/
= la. V 7 Τ Ί Η Χ ? t- fe î^e /2] xn ( f e Ks / 2 ) 5^2β,η k
(
Ζ'.
•lu* Ω ^—- ηruù Ω fe a
ll lls
a..
\
(14.109) 2ηΩ Il Ils
Ίΐβ.
ais
:[
k a
\\ \\s
}
-
This γ is of the order of (mE/mj-)^ '^ftj. Here a\\E is very small (cold electrons). Another result neglecting Landau damping for η 2 Ω 2 » k2T, k->/w„ may be found b "** in /4.11/. In the derivation of this result (14.109) the formula oo
I
2 2 T-2, v -v _ 1 -s / 2 T , 2 / o x KTn(s2*)e dr = ^e Ι^(β /2),
compare p 23 2 of volume 1, has been used.
(14.110)
MICROINSTABILITIES
512
Ζ^{χ) = /ï\e~x is the imaginary part of the FriedConte-function (9.170) both for small and large argument, but in the latter case only for Im x = 0. A graphical representation of γ^ as a function of tik = ß(fc) /9.6/, /14.119/, /14.120/ shows that near multiples of Ωρ there can be maxima of y, if ωρΕ > ntij. This is an approximate necessary condition for the appearance of the ion cyclotron instability. This instability is generated in the vicinity of Ω η = ηΩ by the coupling of the electron plasma oscillation and the ion Bernstein mode. This can take place only when ωρ# > nüj.This determines a minimum electron density nE > n2£2/4ïïS2mj for the instability. For large n the stabilizing terms 'V/n (aj?j/a?j) in (14.109) are predominant. This gives the condition that for Q(k) - ntijmodes will be unstable only if /14.120/
Γ
||Λΐ B all/all
K
A'
(14 111)
-
This means T\ j > T \\j or that the ion cyclotron motion contains excessive kinetic energy which may be fed into the electron plasma oscillation at Ω^ £ nüj. Finite temperature T,, is stabilizing. It might be worthwhile to mention also that ion cyclotron instabilities at half the ion cyclotron frequency have been observcyclotron ed /14.134/. Furthermore, there exists a thermonuclear instability /14.145/, which is due to a group of fast particles formed in thermonuclear reactions. The growth of this instability is proportional to the square root of the particle density. In a Tokamak this instability is suppressed by the magnetic field corrugation. Another instability in a cold electron hot ion plasma is the cross field current driven ion cyclotron instability /14.147/ in an E*B rotating plasma. It is generated by the coupling of an electron acoustic mode and a Doppler shifted ion cyclotron mode. 2. Hot electrons , hot ions. If the electron temperature increases artÉ if ωρ# < nQj-, an increasing negative bump near Q(k) = Ω^ = 0 grows further and can stabilize modes near the first harmonics of Ωρ. If the electrons become however too hot (TE/mjr » Tj/mj) , acoustic the modes ω - ^TEkßk\\/mj; become unstable (ion instability, magnetoacoustic cyclotron instability /14.458/) at ω < ωρχ. Since the energy of the ionacoustic wave is positive (9 Re ωε/dax > 0) they become unstable if the cyclotron emission of the ions is larger than the Cherenkov absorption of the electrons
513
PLASMA INSTABILITIES
and ions. For hot electrons the density criterion for instaThen the ion acousbility is then n-j- > n2B2/Ai\mj-c2. tic wave is the basic excitation and will couple with the n-th ion Bernstein wave. Soper and Harris /14.120/ have shown that the dividing line between cold and hot electrons is defined by 5 V (14.112) V < (Soper-Harris instability) and that T\\z < TjjVU + γ) / is necessary for instability. where y2 = mpT \\E/mET\\j (for Tj_ = T||/14.122/) Based on the Gordeyev equation and the Gordeyev integral, Timofeev /14.120/ had givinstability) reen a region of instability {Timofeev presented by 1
(14.113) T H WE Hence low T^j and high T^E promote the instability but 2Ti|j < TJ_J is also required for instability. A sufficient condition for stability is therefore T. T < 2T||j. Another scheme is given by Mikhailovskii T
/9.26/ for a plasma with 3.f0j/3öj_ > 0, athil ' 2cth\\Im If k^T^/m^c2 - < ω 2 / Ω 2 < 1 one has the hydro dynamic cyD
clotron m E^mI
ti
1
±1
Jrl
instability, k.. = ^,ηΩ^/ω , γ - ωρι for <ω < an ρχ/^χ 1 d the kinetic cyclotron instabil2
2
9
i t j / fe |j = nQj/Cf-fav, γ = kßTgüj/mjC . j f o r mE/mj < üdpI/Qj < < (mE/mj)^/2. F o r kBTE/mzc2 > 1 , ωρχ/Ω2 < 1 one h a s a 9
2
cyclotron instability, for copj/fij > 1 one has high frequency instability. Of interest also are electroin a plasma static electron cyclotron instabilities consisting of several components having different electron temperatures /14.149/. Under certain conditions a cold component may significantly change the structure of the dispersion relation even if the relative concentration of the cold component is small. The maximum growth rate may be much larger than that in the absence of the cold component. The presence of a hot component may decrease the growth rate. For isotropic electron distribution characterized by TE and an anisotropic bi-maxwellian ion distribution Mikhailovskii /14.459/ gives the following formulae: unmagnetized plasma: for ω « kc^^g, kc+^j, instability occurs if Tmax/Tmin > 1 + c2k2/upi, where
MICROINSTABILITIES
514 T
max
= ™**^LI>
Τ\\Σ)'
=
?min
m i n
^ll'
Τ\\ϊ> ' 9 r o w t h
rate γ - (mE/mj) ' ^thl^PI^' ^ ~ ωρρ/2· F o r a magnetized -plasma several cases have to be considered: (mj/mE) a) Tyj > Tj_j, kz = 0. For ω « QE and g « instability is possible for very large temperature anisotropy, i.e. if TJJJ > IJ_J {mj/m^) 3/2β-1 /2. b) T i r > q , J f fel = 0. Reasonably large magnetic fields affect" ônly^the electron motion. For β * ml/mE and 3 - 1 one has the ion cylcotron instability, γ - Re ω - fij. For kz < ®>E/cthE a n d 1 < ß < mr/m^, perturbations with k■ = 0 have ω - Ωχ. Then the interaction of resonant ions with waves can lead to instability. The growth rate is given by .2 7 2 r2 C k y Yi\m Qj. /2\j Ω z - 1|exp{ -2 2 Ύ = m k Q ιη Ez thE III J K PI z°thE
Αττ~
/ττΩ' k
z°th\\I
c-
H
2 2
Il
z
ωPI f ,v k\ c) T > T fe^ z ' Φ ~0, l * 0. The low-freUif ^ = a0,finite quency U instability"in pressure plasma is the ill ^ e ^re" mirror instability. For 11 j » Ij|j/ ω » ^zcth quency ω is purely imaginary. Tnis is the hydrodynamic mirror instability {probkotron instability) for $11 > T\\l/Tll* F o r Tll > T\\I o n e h a s instability for ω « k„e + i, when <-zcth 1 + k2z\\ + (f ^ ( ΐ 1 τ / ι , I - 1) |χ)/2]Α2 li mirror instability. is satisfied. This is the kinetic 3. #
ΡΕ
>
η
®Έ
or
*-he condition n% >
n^B^/A-nm^c^.
From a
bi-maxwellian electron distribution function for instabilities near ηΩ# (electron-cyclotron instability, electron-electron instability) the necessary condition for instability II E T
ÎE
2n
(14.114)
has been obtained /14.123/. It might be of interest to mention that the electron cyclotron instabilities suppress runaway and thus runaway instability (p 453 in /1.35/). High frequency instabilities /14.126/
PLASMA INSTABILITIES
515
arising from electron anisotropy are called electronelectron instabilities (if there is self-resonance for the electrons)· Ion cyclotron instabilities due to a coupling between the gyratory motion of hot ions and hot electrons (moving along BQ) are called ionelectron instabilities. One speaks also of cyclotron double distribution instabilities, mainly if there exist cold and hot particles of the same species. The excitation of electron plasma waves (electron acoustic waves) in a plasma with anisotropic ion distribution has been treated in /14.140/. In conducting media also acoustic-electric (electroacoustic) waves are known /14.456/. 4. Cold electrons, cold ions. In this case both the ion and electron distribution functions are represented by ό-functions of the type (14.102). Instead of (14.103) a dispersion relation for hydrodynamic modes in an anisotropic plasma may be derived from fluid theory. It reads /14.114/, /9.26/ for cos Θ = 1 (£lB0)
A similar result may be derived from the general dispersion relation by assuming ot^^E » c-^hWE' n e 9 l e c t ~ ing the ion contribution to the permittivity, assuming Ιω - nQ,E\ » k c^||F (neglect of resonant particles) , and by assuming K.rr^ < 1 (long wavelength approximation) . For Tu/T^ < f<-]_r^E one has nonresonant instability /9.26/. For short wavelengths also ^irLE > 1) a dispersion relation and an instability has been derived/9.26/. condition T\\/T^ < (kj_rLE^2:n)'''] and ions. The 5. Relative drift between electrons instability occurring in this situation is the twostream instability in a magnetized plasma, two-stream cyclotron instability, two-stream ion-cyclotron instability, two-streaming cyclotron instability /14.129/. In many cases authors assume a simple isotropic (Maxwellian) distribution for the ions and a displaced Maxwellian electron distribution, see p 488, /9.6/, /14.129/, /2.10/. One then obtains the hyd.rodynamic in a uniform magnetic field two-stream instability ion cyclotron instabil(electrostatic current driven ity)
and the formulae
(14.54) - (14.55). For Re ω > ηΩΣ
growing modes exist. For waves travelling at a large angle to the magnetic field the instability can be excited by an electron drift uQ along the field. When
MICROINSTABILITIES
516
the relative velocity uQ > ω/k^ the energy of the drifting electrons is fed into the Bernstein mode and a growing Bernstein mode appears. Its growth rate must overcompensate ion Landau damping. Since the mode propagates almost across the magnetic field, u must be large enough to exceed the phase velocity ω/feji along the field. If the relative motion of electrons and ions is due to different angular velocities in a rotating plasma, one speaks of the cyclotron resonance instability (in rotating plasma) /14.131/. A new almost electrostatic temperature anisotropy instability
has been found for a bi-maxwellian plasma in a
uniform magnetic field. Instability exists for 3M = > 0.591 when Γμ = 0 and ions are neglect= Ai\nkßTn/B2 ed /14.160/. The growth rate is -Ω^ and the polarization is almost electrostatic for almost perpendicular propagation. The instability is obtainable only from the complete dispersion relation. A drift of Maxwellian electrons (streaming along the magnetic field relative to Maxwellian ions) generates in a current carrying magnetized Vlasov plasma a whistler
current
instability
/14.449/ (ion beta >1,
^rLI > 1)· T his electromagnetic current instability has lower threshold than the electrostatic ion acoustic instability /14.161/. 6. Loss
cone
distribution.
In a plasma confined
between magnetic mirrors only those particles located in a certain domain of phase space, i.e. outside the loss cone, see p 56 of volume 1, can be trapped. The resulting loss cone distribution function is clearly not Maxwellian and is partly anisotropic. It vanishes within the loss cone Θ < θ 0 or for c^_ < I o\\ I cot θ , 4 / (
,e.=const)
O
m
I
Loss-cone distribution Fig,
48.
X
x
Double humped distribution Loss-cone
instability
517
PLASMA INSTABILITIES
but is isotropic for c outside the cone. The loss cone angle dc is defined by cos Qc = (B {s) /B (s0) ) " 1 /2. This anisotropic distribution function gives raise to the (electrostatic) loss cone instability, Dory-Guest instability, Dory-Guest-Harris instability, instability
of
mirror
traps.
Since the distribution function
f0 of cx integrated over Cy and cz is double-humped having a central minimum due to the lost particles, see Fig. 48, the appearance of this instability is understandable. For low plasma densities it is electrostatic and has frequencies which are harmonics of the ion cyclotron frequency. For finite beta an electromagnetic loss cone instability appears, see later. When we integrate an equilibrium distribution ,cz) such that function fQ(cx,c +00
f
•Ό
(Q
) = fff (c JJ
#
J o
— 00
,c
,c
x' y' z ^
)dc
dc..
y II u
and calculate its derivative at c~ = c we obtain ' X o /4.11/ + 0°
dfo (c x ) J ns v' * àc ^
/· /· ' o df Γ Γ "1 , 2 o2, 1/2 3c. O P"1J
C
hl· J
_
J œ
dc. ( c. - c )
J
1
;r c o ^ο ~ 1 o (14.115) may become positive for where cj = c~, + Cy. Now f'Q(e0) cQ < cm and we expect instability for ω/fej^ < cm. To investigate the loss cone instability, distribution functions of the Dory-Guest-Harris type fso{cl'cz)
^ ^exp[-cJ/aJ-^/a^]
(14.116)
have been investigated (I a positive integer). In the simplest case one might assume k^ = 0 (flute model) and one obtains from (14.105) the dispersion relation /9.6/ 2
Σ k~ s,n -^f ωρ
2
where
c
C
nti
VUJ SH
/ n i = °r '"V (ω - ηΩ ) '
. (JL £ » j Ä j a ;
(14.117)
(14.118) le. de. n\ Ω / y 1 1 s (14.117) has unstable roots for distributions of the type (14.116). Since it would be too tedious to investigate all sn
MICROINSTABILITIES
518
possible solutions of (14.105) for fen * 0, we make several approximations /2.12/, /4.11/. First we assume that the electrons are isotropic and cold, or ω » k\\ctfoE. (There exist, however, also investigations of the electrostatic electron cyclotron loss cone in/14.163/). Second, wavelengths longer than stability the electron gyration radius will be considered (kr^g«
1 or Ω# » ^l°lE'
small
Larmor
radius
limit).
Due to this assumption only the n = 0 term remains for electrons. Due to the first assumption we may expand 1
- fe, Then t h e d i s p e r s i o n 2 ,2 ΡΕΚ\\ 2 ω
ω
_
v2
K
k 1 \l°\\ = — + —V 1 - + .. . ω
relation
- Ιω ρζ 2 dc
(14.105)
+ 00
4 iί
n
(14.119)
fnti
de
reads 3/
oJ
de,
nj)oi
■ fc,iMvvy -o. Il 3c,
(14.120)
ηΩ Γ + k.| c.. - ω
w h e r e de = 2i\c]_dc±dc\\ , s e e ( 9 . 1 8 0 ) h a s b e e n u s e d . S i n c e ωρ^ » o)pj s e e ( 2 . 2 1 ) , t h e u s e o f Ω^ = ωρ^/cii/ki s e e p 511 a n d o f ( 9 . 1 6 1 ) g i v e / 2 . 1 2 / II ^ = ω^—p + k
ω
(Ω^ -
^ oJ
ηΩχ)·
3ÖM
2 2 . v 2ÏÏ ω t i
0
3/,
oJ + ηΩ X . 2 I j
ω
ΡΕ
fT2/*l°l\ J
* Ίΐο
• 2
(14.121)
e.|= (Ω, -nüj)
/k.
To be able to find instabilities Ω^ ηΩ should I! wecondition have o)p£ > ηΩρ which results in the density k\\
y =
k
2
2n2
ω ρ π nη Ω uT Ç , . 3 / 2 J n\ k\\(kBT £ I/mI)
2(Κ1°ΐ\Γοΐ\ dc±. Ω Γ Λ de. I 0 . 1 = 0 I I
(14.122)
519
PLASMA INSTABILITIES
The sign of this integral depends on c^ and its location relative to those c^, for which Jj^kj^i/Œj) is small, γ is positive and we have (the fastest growing) instability, when kxc > nüT (14.123)
Im
I
(because J^ has its maximum nearfcj_£j_s ηΩχ and when > k\_c \/Ωτ this maximum is within the positive kj_cm/iïj domain of (df/dcj)e., =0)· Consequently the rc-th harmonic may be viewed as having the same effect as a beam moving along the magnetic field with the velocity nQ,j/k±. It follows from this analogy that a plasma with an anisotropic distribution function can generate the same instabilities as a plasma with particle beams. The maximum growth rate is given by 2 γl « max
* ζ . n 2ttn2 T
The instability for which k.c.
loss
cone
instability.
(14.124) = nQj is the
resonant
A more refined theory of the loss cone instability distinguishes between the maser instability (convective loss cone instability propagating almost perpendicularly to the magnetic field ωρι » üj, k\\ < k\) and an absolute flute-type instability k^cy » iïj existing in the presence of a radial density gradient
(drift velocity space instability, loss cone flute instability instability, drift cone f Mikhailovskii instability, drift loss cone instability, gradient
loss cone instability /13.25/, /14.138/, /14.459/), requiring df0/dc± > 0 in inhomogeneous magnetized plasmas. This instability is caused by the coupling of an electrostatic ion cyclotron wave with an electromagnetic ion cyclotron wave. The effect of magnetic field inhomogeneity reduces the growth rate, but does not stabilize. Since the instability is highly resonant (ω - ηΩ « Ω) it might be possible to stabilize it by detuning the resonance. In a uniform plasma the loss cone mode is convective. To grow it needs a plasma length of the order of 400 rL /14.157/. The mode may, however, be reflected from plasma boundaries or from magnetic field gradients (negative energy loss /14.158/). At high densities the concone instability vective loss cone mode becomes absolute /14.159/. This mode may be stabilized by magnetic field gradients. The negative energy dissipative mode is a loss
MICROINSTABILITIES
520
cone ion Bernstein wave and becomes unstable if there is a source of dissipation in the plasma (electron Landau damping). Mikhailovskii /14.459/ defines the maser instability as unstable extraordinary electromagnetic waves with kz = 0 (perpendicular propagation) and E1BQ. From the relativistic dispersion relation ε22 = n^ (for ωρ « kc) the equation for the electromagnetic waves becomes 4π£ 2 η Γ ΐ^ c,J|23F/3p. ~2 7 2 c k , o -, -, >^ In *l _ r, Λ 1 + " —T ^ l^l^l^ll *2,-2.1/2-°· o M ω y n=-°° ω-ηΩ(1-£ /ο ) Here pu, p. are the relativistic momenta associated to the particle speed c and F is normalized such that Jp_LFdp|dp|| = 1. Then for spherically symmetric F and dF/dp - F/p the growth rate is given by 2 γ - ωρ/Ω.
This is the relativistic
synchrotron
instability.
There exists also a current driven high frequency loss cone flute instability /14.138/. Also electromagnetic (absolute) loss cone instabilities in homogeneous high beta or mirror devices have been found /14.137/, whereas electromagnetic waves propagating across the field become more stable in the presence of loss cone distributions. There exists, however, a whistler loss cone instability /14.142/ (γ-Ω^/2Ω^/2/β = = ω^υ/β for kc - ωρ^). Other authors who investigated the instabilities of finite pressure plasma with hot electrons and cold ions found that electromagnetic waves which correspond in isotropic plasma to whistler, fast magnetosonic waves and slow ordinary waves become unstable in the presence of the loss cone and are driven by the electron pressure anisotropy /14.143/. For a high beta plasma (g » 1) with electron temperature low compared with the transverse ion -> O, the high frequency loss cone energy TE/mj
+k.c*
(u2E/c2k2)].
PLASMA INSTABILITIES
521
This growth rate has a maximum when k\ - ωρ^/2. The maximum is γ - ω^υ/ΖΙχ, ω Μ is given by (9.113). In the limit ω » k\c* the dispersion relation describes the whistler loss cone instability. For k » ωΡΕ/ο one has the electrostatic approximation and γ * nj,/2fl1./2, kyrLI = (mj/m£.)V2r kzrLI * 1 . For fej_ « ωρ#/ ßj 1 , where ^ 3j = c^/c\. β Also an electromagnetic ion cyclotron loss cone instability in counterstreaming ion beams has been - mj/mE at zero frequency and the found for T\\J/T±J first several ion cyclotron harmonics /14.148/. This is the case when in addition to containing hot ions > 0 one assumes that the characterized by dfi/dcj_ plasma contains cold ions with density n^ = anQ « nQ. We then have a double-humped distribution with a small fraction of cold ions. Assuming kz = 0, r JjI
522
MICROINSTABILITIES
tion instability. Its energy source is the gyratory motion of relativistic electrons. In the relativistic cyclotron instability one has κ·Β = 0, whereas κ·5Φ0 is required for the nonrelativistic case /10.2/, /14.132/. Furthermore, a loss cone type distribution function for the hot electrons is necessary. A decrease in the reflection coefficient of a wave packet at the boundaries decreases *-tie growth rate of the instability. Devices radiating at nüE from relativistic electrons have been called electron cyclotron maser. Other authors look for the plasma wave synchrotron instability, a longitudinal relativistic unstable mode with an approximate dispersion relation zzz - 0 and having a phase velocity close to that of light in an anisotropic magnetized plasma /14.135/. This instability appears together with the transverse wave synchrotron instability. The growth rate of the longitudinal waves exceeds, however, the growth rate of the transverse instability. It should also be mentioned that quite generally also non-relativistic electromagnetic (transverse) cyclotron instabilities are cited in the literature. They are nothing other than unstable electromagnetic waves propagating perpendicularly to a uniform magnetic field. Ring-type and loss cone type distribution functions may make these waves unstable for either zero frequency or near the cyclotron harmonics /14.136/. Although kinetic instability is very often synonymous with microinstability, this term has also been used to designate unstable cyclotron modes excited in a thetatron discharge, propagating obliquely to the magnetic field and due to an anisotropic electron distribution /14.139/, see also pp231, 238, 266 and 335 of volume 1 of this Handbook. The term is also used to designate the Bernstein instability {Bernstein wave instability) /14.140/. The term Bernstein wave instability may be somewhat misleading. If Bernstein waves are excited nonlinearly (as can happen by decay of waves) they can also become linearly (and nonlinearly) unstable in the form of anisotropic temperature insta, cyclotron instability and two-stream instabilbility ity
. On the other hand, the ion cyclotron instability
may be regarded as an instability of the obliquely propagating ion Bernstein wave generated by electrons drifting along the magnetic field. There exist also other temperature anisotropy instabilities. Recently, a lower hybrid temperature anisotropy instability in a uniform plasma /14.144/ has
PLASMA INSTABILITIES
523
been found. (The ordinary lower hybrid instability is excited in non-uniform plasmas, see section 14.5). The new mode is unstable for T^ = 0 and Tj = Tg for almost perpendicular propagation. This instability is in addition to the hose-instability for parallel propagation and to the Forslund ion instability with zero real part and to another electron mode. The latter two grow also fastest for almost perpendicular propagation. In anisotropic magnetized plasmas a self-trapping instability of MHD waves along the magnetic field and of cyclotron waves has been reported /14.152/. In an anisotropic magnetized plasma a microscopic self-gravitational instability appears, whose energy source is the self-gravitation energy /14.151/. The rotation of the magnetized anisotropic plasma modifies the instability criterion when the magnetic field and the rotation have a stabilizing effect on the condensation process. The Hall effect has a destabilizing influence /14.448/. High plasma pressures may stabilize the instability. The kinetic theory (Vlasov-Maxwell equations) of surface electromagnetic waves in a bounded hot plasma may be found in /14.457/. 14.5 MICROINSTABILITIES IN AN INHOMOGENEOUS PLASMA When a plasma is inhomogeneous, instabilities known from homogeneous plasma appear and may be modified. But new modes are also generated. This is especially true for a magnetized plasma, see chapter 13. First of all we will consider an unmagnetized inhomogeneous plasma. The question of stability for inhomogeneous plasma with non-monotonically decreasing distribution functions is difficult because no variational principle has been found. (For a plasma in an external non-uniform magnetic field a variational principle has been given, see later). A necessary and sufficient instability energy criterion for a collisionless plasma in a nonuniform magnetic field has been given /14.226/, but the stability against electrostatic perturbations of an inhomogeneous plasma has also been investigated by other methods and stability criteria have been derived /14.211/. Also differential equations in wavenumberspace have been derived /14.212/ {k-space tecknique) . Local Nyquist diagram analysis may also be found /14.213/. For a homogeneous plasma the equation for the electric potential in Laplace-Fourier space is algebraic, but for an inhomogeneous plasma
MICROINSTABILITIES
524
one has an integral equation. Usually the integrodifferential equation in #-space may be reduced even for an inhomogeneous plasma by the eigenmode ansatz §(x,t) ^ exp(iudt) Φ (x,ω) for the electric potential Φ with the assumption r^ « k~^ . The differential equation for Φ(χ,ω) is solved by the WKB approximation. If the plasma is described by a water bag model, then the stability of special configurations can be discussed by using the Nyquist technique /14.166/ or the normal mode analysis. The water bag model assumes that the phase space particles occupy a constant volume. This single water bag model (one bag in phase space) is equivalent to the closed moment description by the one-dimensional continuity equation (7.2), equation of motion (7.11), Poisson equation (2.3) and water the adiabatic law (4.87). At t - 0 an electron bag
distribution
function
is given by f^{x,c,0)
=A =
= const > 0 within a certain region, see Fig. 49, and fE(x,c,0) = 0 outside. Subsequently, fg{x,c,t) evolves according to 3/ -A _ e g a / g = o + aat dx m^ de
(14.125)
and
|f = -^e{nE
- nQ)
(14.126)
/=0
a
f=0
Fig.
49.
Water
bag
model
(x,0)
525
PLASMA INSTABILITIES
These equations describe the incompressible motion of the electron phase space fluid in phase space. When distort to new values the boundaries £+(#,0)/ ο_(χ,0) c+(x,t), c-(x,t) the electron density remains constant between the boundaries and vanishes outside: f
= A
o_{x,t)
for
< o <
c+(x,t)
rs 4- ·* = 0 outside. According to (6.4), (6.12) we have nE(x,t)
= \fE(x,e,t)dc
(14.127)
= A(o^(x,t)
e_(x,t))
-
(14.128) -f-00
vE(x,t)
+00
= \cfElx,c,t)dc/\fEdc
= (14.129)
1
= -^ {e+(x,
t)
+ o_ (x, t) )
+00
P E = m E\
(o - vE(x,t))
fE(x,c,t)do
= (mE/12)A(c+
= -
cj
3
(14.130)
.
Since the electron heat current (6.17) vanishes in the single water bag model we have adiabatic behavior and may write (4.87) in the form PE(x,t)
= p Q n" 3 n|(x,t).
(14.131)
Using (14.127) - (14.130) we may also derive by the moments method from (14.125) the continuity equation (7.2) in the form
ΎΓ + έ ( V s 1
=
°'
(14.132)
the equation of motion (7.11) in the form dV
and
dv
E
E
e
-rjr
+ v^-—
= - —
at
E dx
do a/
at
_
+ o, +,-
mF da a
1
E
! n
wmjr
'" = -—E
dx
m^
^E
τ^
oX
(14.133) (14.134) '
MICROINSTABILITIES
526
as well as Poisson1s equation (2.3) in the form (14.126) or dE
-r— = -4πβ(η 7Ρ -η ) = -4πβ (Αο^ - Ac E
OX
o
+
-
-n ). o
(14.135) On the basis of similar equations· a stability criterion has been derived for special configurations of inhomogeneous plasmas /14.166/. The equations may also be used to investigate electron plasma waves in warm unmagnetized inhomogeneous plasmas, since chapter 13 has demonstrated that the use of the Vlasov equation for inhomogeneous plasmas is cumbersome. If we assume that ions are stationary and if we linearize V n
V
=
E
E =
η
η
+
ο
ν
+
Eo
=
Ελ
ν
ΕΛ '
=
?E
E =
ΕΛ '
+
?Eo E
o
+
?ΕΛ = ^ 1 E
^ =
Ε
λ
(14.136) then for a given frequency ω = const and for excitation ^exp(ÎQ)t) by a grid the equations (14.132), (14.133) and (14.126) become for the space dependent parts n
. .
(x)
Ei
ννΛ (x) Ελ
n
=
o
dy
- ^ ST βΕ
=
λ
Ελ
- Ί^ α
2 -.
Ε
r
^ p tü)
E
ν
E1
η ^ω
o
dn
o
(14.137)
-3F' άη
Ελ
-τ^,
(14.138)
άχ
where (9.49) has been used, and dE -dF = '^^EV
(14.139)
If we differentiate this equation and substitute for dnE<\/dx into (14.138), we obtain 2 2 -eEa„ d ΕΛ (14.140) vwAx) = r1 + -A s_ ■ E
Substitution of ηΕλ gives the result άΕΛ 1
3
1=
4πβ da:
ax
o
from (14.139) into (14.137) n
di;^
£ω
dx
2
u
dn
ίω
dx'
El _ _£1
2
(14
141)
527
PLASMA INSTABILITIES
Differentiation of (14.140) and substitution of and vE^(x) from (14.140) into (14.141) gives άνΕ^/άχ after an integration the dispersion tron plasma waves in an unmagnetized plasma 9 ά Ε
ω2Ε
= ωρ ( x ) ^ - al
relation for elecinhomogeneous
γ,
(14.142)
άχ
where (2.19) and din η0/άχ « k have been used. ωρΕ = const, For homogeneous plasma aE = const, Ey (x) ^ exp(ikx) one obtains the Böhm-Groß dispersion relation (9.62). For din n/dx « k density gradients enter into (14.142) only through ωρΕ(χ) and may nQ(x) be given arbitrarily. A very similar dispersion relation has been derived by H. Motz using the VlasovPoisson system of equations and assuming the Miller force
(4.29) ("ponderomotive
for E 0 * 0 (hf containment). The Motz
dispersion
2
ω ^
relation
2
= u
pE(x)E^
force11)
of a hf field
reads /14.167/
a2 2 - — | A-L*{x)E
(χ)γ
(14.143)
ω da; where now the density gradient is determined by EQ 2 ωρΕ(χ)
2 = cop5,(0)exp
e
E 2
o
4mEu kBT9
(14.144)
Similar calculations have been made for linear acoustic waves in a density gradient /14.168/. A linearization (14.136) of the five ion and electron equations (14.132), (14.133), (14.126) results, for the assumptions EQ = 0, TE » Tj (ion thermal speed neglected with respect to the ion acoustic speed), Vp# » Vpj, mE/mj - 0, ω <^ωρτ, such that nE<\ ,x i n '" t,-h"-e -e q u a t i o-*n s ( 1'14. ΕΛ -" ^J1 4.137) * in
nj\9
ν
T 1( *{x) *;π > il
= "Ter -— = ^Ζ %u>mT da:
(14.145)
instead of (14.138) and in ηΙλ(χ)
= nQ[eV(x)/kBTE]
(14.146)
instead of (14.139). Calculation of V(x) from (14.146), and of Vj^ (14.137) with substitution into (14.145) gives
from
MICROINSTABILITIES
528
r!+ *<*>ii + kln = °>
(14 147)
·
άχ
where k% = u>2mj/k Τ^ = ω 2 /α| is the square of the wave number of the ion acoustic wave in a uniform plasma, h(x) = [1 /nQ (x)][dnQ(x)
/dx]
= λ/Lix)
and n(x) =
= rij^ix) /nQ{x) . Equation (14.147) may now be solved (leading to the for a number of cases like nQ = const case of a uniform plasma, or nQ = nQ(0)exp(αχ), leading to the Parkinson-Schindler dispersion relation /14.169/ (similar to the gravitational instability) k2 - ak - k^ = 0
(14.148)
of ion acoustic waves in a gravity supported plasma etc. Experimental situations often have nQ(x) = nQ(0) •exp(-x2/a2) resulting in 2 an 2x an , 7 2 (14.149) ö " ~~ô X" + kn = 0, -, 2 2 dx o da; a which may be solved using the confluent hypergeometric function /14.168/. As Budden has shown /14.170/ such an equation has interesting properties. Complete absorption of a plasma wave may take place but also a Budden tunneling factor exists. In general, equations of type (14.147) have to be solved with the WKB method (Wentzel-Kramers-Brillouin method), see p 371 (method of geometric optics, /9.13/, /9.4/). An asymptotic theory has been given for waves propagating in spatially and temporally dispersive, slightly inhomogeneous, and slowly varying media. The resulting eikohas been given /14.178/. For longitudinal equation nal waves (14.150) has been obtained. Geometric-optical methods have also been used for longitudinal oscillations in nonhomogeneous magnetized plasmas /14.234/. Another method for treating electrostatic integral modes in (cold) inhomogeneous plasmas is the equation
method /14.176/.
The theory described so far had a TE/Tj » 1 so that Landau damping could not be an important process. For TE = Tj- Landau damping as well as growth occurs (see p 477) in the relative amplitude n~/nQ, when the mode travels into more tenuous regions. In spite of heavy
Landau damping for Tg - Tj ion acoustic
waves can be
generated in a Q-macnine by electrostatic grid excitation. Calculations and experiments showed /14.171/ that in an exponential density variation, growth oc-
529
PLASMA INSTABILITIES
curs for λ > 2π£. For λ < 2T\L the ion acoustic mode is Landau damped in the same way as in a plasma of uniform density. (conversion
Inhomogeneities
layers)
may have vari-
ous effects: conversion of electromagnetic waves to electrostatic waves may occur /14.172/, see also p 212, which in their turn may produce energetic electrons /14.173/. This is due to the fact that waves of constant frequency ω satisfying (14.142) convect to lower and lower phase velocity as they propagate outward into regions of lower plasma density. The waves always convect outward, for if a wave is initially propagating inward, it will be reflected at the plasma cut-off ω = ω ρ / see p 184 (volume 1). For E^ ^ exp[ik(x)x], (14.142) may be written in the form ω 2 = UpE(x)
Since ω = const, = const de„7_
EÎL
dx
+ a2Ek2(x).
(14.150)
(3.40) gives with (14.150) and aE = =
J7 .
ω dk
7 2 dx
k
=
4πβ <2_,ω-η„ α η 7
""* "pti"PE 72 2
k
~"Ε
dx '
M41R1Ï u
ilDlj
α„τη„
If Cp » aE (no Landau damping), the high phase-velocity waves go to lower phase velocity quickly. Electromagnetic waves incident on a plasma at an angle to the density gradient can also be absorbed resonantly by linear conversion into the electron plasma wave if the incident wave is polarized in the plane of incidence /14.174/. Sources of electromagnetic waves are located outside the plasma or in low density regions. The plasma is inhomogeneous and resonances (k ■> *>, see p 184) appear in the high density regions inside the plasma. If waves are not reflected before they reach their resonance layer (conversion layer) the plasma is said to be accessible for this wave. Tunneling may occur if reflecting and resonating layer occur close together. A rough estimate of accessibility
(accessibility
con-
dition) for the perpendicular ion cyclotron resonance (9.119) may give the answer: the electric field in this resonance layer will be less attenuated by evanescence in a plasma than it would be in the absence of plasma /9.4/. The accessibility condition for the lower ion-electron hybrid resonance may be derived from the full cold-plasma dispersion relation. The hybrid resonance occurs for n 4 + 0, n^ -> °°. For n see (9.18).
530
MICROINSTABILITIES
Accessibility will require n| > 2. A ^nore precise sufficient accessibility condition is /9.4/ n\ > + 2, where D and P are given on p 183. For > \2D2/P\ the Buchsbaum two-ion hybrid resonance the same accessibility condition holds in the form n| > 2(1 + where y Q = k^c2(Ω^ - ω 2 )/ω 2 Ω?. + y0mE/mI), In concluding this part on modes in mhomogeneous unmagnetized plasma we would like to mention some special modes occurring in inhomogeneous unmagnetized plasmas: There has been reported a longitudinal instability in an inhomogeneous -plasma due to periodic alteration of one of the plasma parameters in time. The theory leads to a generalized Hill's equation. This instability /14.175/ in which subharmonics also develop should be termed parameter instability and will be discussed in chapter 17. Electrostatic marginal instability of a one-dimensional inhomogeneous finite plasma has been investigated in /14.177/. Whereas df/dE < O is sufficient for stability, sufficient conditions for instability if df/dE ^ 0 are derived. The spectrum of Langmuir oscillations in an inhomogeneous plasma has also been given /14.181/. In the geometric-optical approximation some calculations on cavitons have been done. A caviton or wave trap is a plasma cavity filled with electromagnetic high frequency (ω » ωρ^) radiation whose radiation pressure is balanced by the pressure of the surrounding plasma with arbitrary density profile /14.179/. Mainly in inhomogeneous plasmas multi-stage instability may appear /10.2/, /14.180/. In this case a certain plasma instability can affect the growth rate of other instabilities if the amplitude of the former becomes large enough to change the equilibrium plasma properties. In such a way conditions for the growth of other instabilities may then become realized which cannot be satisfied, if the original equilibrium remains unaffected. This is a quasilinear effect. Finally, we would like to mention the ambiplasma instability suggested by Lehnert /10.2/. The possible existence of an ambiplasma consisting of a mixture of matter and antimatter containing light and heavy particles of both charges leads to the assumption of new types of inertia and charge separation mechanisms thus, leading to new modes. Magnetized inhomogeneous plasmas exhibit drift instabilities which we discussed in chapter 13. There are, however, other new instabilities, and old instabilities are modified by the inhomogeneities. An inhomogeneity does not by itself necessarily lead to
531
PLASMA INSTABILITIES
instabilities. One must also have a mechanism for transforming energy of the steady-state motion of particles into energy of the instability, see e.g. chapter 13. (Examples are the bump driven drift mode, driven by a bump in the distribution function /14.453/ or the drift-superheating instability /14.496/.) Let us first discuss some general results /14.17/, /1.20/ (p 501), /1.29/ (Vol. 2, p 259), /9.26/, /11.6/, /10.4/, /14.299/ for magnetized inhomogeneous plasmas. To start with we assume that all modes are purely electrostatic, which means ß « 1 according to (14.53). To simplify matters a plane slab of a nonuniform plasma with a magnetic field ÎQ = £0e„(1 +εχ), a gravitational force gmex and a density gradient in the x direction will be assumed. The equilibrium distribution function fQ satisfies the Vlasov equation
Ofo/a* = o) u*B
m
o , s -> V f = 0 (14.152) + —ge + m— os e * x uJ os sL c s and may be constructed from the constants of motion, see p 368. The equilibrium distribution is (for small ε) M V J/
f
os
= nos(ms/2^5T)3/2exp(-msW2/2V' +
gxmjk^) (14
•Γΐ - ε·χ- ε-u / ß l , L
s
s y'
·153)
si'
see (13.40), p 368 of volume 1. Since we have assumed here a gravitational force, ε' is now given by -(din n
/dx)
=
e
s
"
gm /k T
s B
(14.154)
x=0
and not by ε^ =-din fQS/dx, see p 368. The introduction of gravity g permits simulation of curved magnet ic field lines, compare (4.21), (4.95). Equation ^14.154)^results from (14.153) and (9.1d), where 0 = leQfufosdu. From quasineutrality of the equilibs e'j = ε^. and le jfosdu rium? = 0 one obtain 2
s
SJ
* us
s
(14.155) s B s εΒ /4π = sJ nsw s (e'Lr/m - g), so that for g = 0 ε1 > ε (14.156) β « 2ε/ε ι / holds, see (13.44). Our calculations are valid for a slightly inhomogeneous plasma or for waves
MICROINSTABILITIES
532 k » din n/dx,
g « ttj.0
h
.
(14.157)
Physically, this means that the waves are low frequency waves (ω « üj), and that the ion drift velocities resulting from g and BQ are small in comparison to the thermal ion speed c^j. T ^ i s means that only the electron drift vE, see (13.1), in the perturbed electric field E>| vE = -eE^/mjiïj.
(14.158)
is of importance (here eBQ/c = mjüj has been used) gradient drift eu^/2QE and the and that the field gravity drift -g/QE, may be neglected for electrons. Linearization / = / 0 + f<\ , etc see p 369, leads to the dispersion relation for electrostatic drift waves (13.46). Before discussing this dispersion relation we derive another relation. Insertion of (14.158) into the electron continuity equation /7.24/ yields in first order (neglecting nQdvE^/dx) dt
= -v„-^
(14.159)
E dx
or for harmonic perturbations ^exp(-iü)t) ^ r^2. (14.160) nFl = 5Ί m Ω τ dx The drift of the ions due to gravity and due to the electric field E* (given also by (14.158) but with the other sign) gives in first order for the ion continuity equation 3ητ1
I\ dt
U s i n g rij^
9^τΛ
_ J o _ _£_ ~ ~VE dx Ωχ
% e x p ( - i o ) £ + ik η
^ητΛ
ΙΛ Zy '
(14.161)
y) we o b t a i n
-ieE« 9nr τΛ = Ô M J—JT, Γ - ^ 2 · 11 m Ω ω ( 1 - gk /Ω^ω) 8#
(14.162)
This was a macroscopic calculation. In a microscopic theory nlQ has to be replaced by +oo
oo
/K.U. n
Io
2 dv
+ ^ z$ l oiin)fol(uL'uz)n0ldul— oo
u j2
<14·163>
o
Quasineutrality eEnEo
+ eznIO
= 0, neglect of cyclo-
533
PLASMA INSTABILITIES
tron resonances and of all electron terms gives from the Poisson equation the dispersion relation k
= ω
PI. 2
ÙLÎ Ω
ωPI
1
d l n
\j
1 n
Io
dx
rslk
21
U.
8(F
y 1
lo»lo) dx (14.164)
u « du - / + 00
where Fz (u±) = 2π / fr (u,,w eral than (13.4 6) ■°° Using " ' ~~
)dur
This is more gene-
ou
f
_ 2 , λ -x J (a#)e n
o I (#) = i
, -α / 2 _ d# = e I (a 2 /2) 7 (14.165)
(ix) , n where I is the modified Bessel function, see pp 232 and 238, one obtains from (14.164) for a Maxwellian distribution function the dispersion relation /14·17/ # /9.26/ 2
k
y
=
ωPI n2 2 Ω ^
J
2
7
(.-.Io,„ - ι ) + - £ *
31η η Jo dx
(14.166)
-e-*IoU)
= 0.
y
y i 2 2 H e r e z = fc
\
4v
4νΦ
yl νΦ.
(14.167)
For rLI + 0 (macroscopic limit) we have s -* 0, > 0, so that (14.167) cannot be satisfied. For ν Φ Ο
534
MICROINSTABILITIES
the macroscopic theory results always in instability. In the microscopic theory (rL * 0) this is not the case. One speaks therefore of finite Larmor radius stabilization. After multiplication of (14.167) with y2 one may write /14.17/
(y - y+) (y - yj (y - y+) (y -yj
< o for 1 - o + ι/4ν) νΦ > o (14.168) > o for 1 - o + ι/4ν) νΦ < o,
where =
^1 - Φ(1 + 1 / 4 v ) / 2 ± /Ϊ^~Φ
+
2Φ(νΦ(1 + 1 / 4 v ) 2
-
(14.169)
1)
are the curves of marginal stability. For an infinite plasma unstable modes always exist. For a bounded plasma a minimum zm exists making stable modes possible. The curves from (14.169) are shown in Fig. 50, following Mikhailovskii /9.26/ It should also be mentioned that a cut-off at has been found for drift waves in an (ckBTE/eB)k^nt0 inhomogeneous magnetic plasma /14.232/. i,y
100stable 10-
1 -
unstable
io-1-
io- 2
stable
S£-
1—
—»
10
Fig.
50.
Curves according
of marginal to (14.169)
—i
100 stability
*»
535
PLASMA INSTABILITIES
The theory presented here is valid only for z = krL « 1. Without this restriction the dispersion relation for electrostatic modes in slightly inhomogeneous plasma reads / 1 . 2 0 /
- s,nΣ
-z I
w
we - 1 i ω + ηΩ - gk /Ω
ωPs
2 2 Ω rT s Ls -z
y
coe Ί U ) ω + ηΩ -ngk s/Ω
ττ?Ω ω s
-s U ) we + ï/(p)ω + ηΩ - k g/ü s
y*'
i \
B s kOT k B s yt mû ω
s
3In n, os 3# m
s^
IVs
(14.170)
3 In n os 3x
where ω + ηΩs - k .y* g /Ω ' s (14.171) 1/2 fc (2 /w ) || Ve e and W is given by Ζ' (-ρ)/2, where Z(p) is the plasma dispersion function (9.170), p 227. For k\\ = 0 , ω « Ω / n = 0, w e obtain the flute instability and finite Larmor radius effects. Due to W(°°) = 0 w e obtain (14.166). If If in (14.166) w e assume Zj- « 1 (rL -* 0) ω~ » Ω | w e obtain / 9 . 2 6 / , / 1 . 2 0 / for n ( nQ (x) dln n 0 fc feDTT din n r ol = 0 (14.172) 2 2 7 £ J OJ 9dx dx rrijtij ω - ω—*and the flute stability condition (see p 277) in the form 2 2 2 - k ΚΤΔΤ din n T ± y B I OJ (14.173) g 4 2n2 dx P =
m ΛΙ j .
and the growth rate (11.26) for rL ■+ 0. For ω « Ω n = 0, °thE y ωΑ|| > cthl' w e o b t a i n from_(14. 170) the drift Then (14.170) instability (Wz - 0 , WE = -1 +ix/H). reduces to dispersion relations of the type (13.61) , zE < 1, or the like /1 .20/. For ω = Ω τ =fet^feu= 0 , ζχ»λ ω « ÇlE, n = 0 for electrons and only n = -1 for ions, one obtains the Mikhailovskii-Timofeev instability , drift ion cyclotron instability, drift cyclotron re-
MICROINSTABILITIES
536 Qonance instability. For Tß = Tj, lim e - 2 I 0 ( 3 ) = ( 2 π ^ ) " 1 / 2 (14.170)
/4.11/
η% = n j , reduces / 1 . 2 0 / ,
to 1
(1 +fe yVD/(u)("
1 + ω - Ω^ 2 πζ J
\
1
7
2,Λ2 ^ y D
2 λ LE (14.174)
compare (13.77). Equation (14.174) indicates two waves: a drift wave ω = -kyVp and an ion cyclotron wave. If k is chosen properly, instability is possible. The stability condition is m
,2 B
n^T\2
r,dln
2
o i
da:
A-nnm a
(14.175)
) ·
We now return to (14.166). For z « 1, y » 1 one obtains the theory by Rosenbluth, Krall and Rostoker /14.182/. This theory may be derived from (14.152), (14.153), (14.154) etc which lead to (13.46). Expanding the Bessel functions and neglecting λ^ « ν^, this dispersion relation reads kz[
[[Va " ('k„TB s ω
b
Y ~ ω-kg/ü
-
■
(14.176)
2
k zk„Tjmntll\
S
^)
B s' + kezkT}T D
s
S
s. ■dz
/mû S
= 0.
S
A Nyquist analysis shows that there is one unstable mode if ε' > msg/kßT > 0. The frequency of the mode is given by ω
ke'IkBTI = 2
rrijüj
t
mIîlI
)
~A3ZE
. (14.177)
This should be compared with the macroscopic result ω„ = iSge', see (11.26). Now the system is stable if kzlk„T/mTSlT o
l
> 2ω„ l
n
(14.178)
(finite Larmor radius stabilization). The case z « 1, vy « 1 in (14.167) has been discussed by Kadomtsev /14.183/. If FlQ(u\), in (14.164) is a Maxwellian distribuiton with a temperature gradient χ = 3In Tj/dx then (14.164) becomes /14.17/
537
PLASMA INSTABILITIES
4. ^
Μ
"-Tin
"F1
dln n
°
J
(14.179) -
gky/üj
ω
where Q = 1 -χ^(1 - I ^ I Q ) [din n^/da;]" 1 . An analysis analogous to (14.167) - (14.169) gives (14.180) -.1 -Φ(0 + 1/4ν+ (1 -0)/Φ) ±/Q(1 - Q ) a 2 2Φ(νΦ(0 + 1/4ν + (1 + 0 ) / Φ ) - 1) A detailed discussion of these curves of marginal stability shows that a temperature gradient parallel to the density gradient stabilizes modes of short wavelengths (temperature gradient stabilization) . The larger x(dln nQ/dx)~î is, the longer may be the wavelengths where this stabilization is effective. Up to now we have considered stabilizing effects of finite Larmor radius in the frame of the Vlasov theory. The same results may be obtained in a guiding center plasma theory /11.6/, /14.184/ or even in a macroscopic anisotropic MHD theory, in which the stress tensor (6.10) is calculated from a local anisotropic Maxwell distribution. For a magnetic field in the ^-direction one has to write /14.184/ ±
(14.181)
where pi is defined by the usual Π-,*, Π 2 2' an<^ v k =
MICROINSTABILITIES
538
3/2 ffie
m
t+ ^Ι = ns(x't)\2vk„Ttift))
s
^
e X
I1 [v -u(x,t)]2\ m Pr2 s"T^l¥7^—l
ΰ * (14.183) {local Maxwell distribution), then one obtains /14.187/ from the Boltzmann equation (5.8) and (5.33), (5.34) , (6.6) u-v = c, (6.10), (5.36), (6.17) the set of MHD^ equations in the form (6.5), (6.7) and (6.16) with Π not given by (6.21) or (6.22) but by Π = p - r]S
XX
^
XX
yy
zz Π
ffi/
n
X2
Π yz
1 +
= p
^ Ω 2 τ 2 [ yy
'Τ~Τ6Ζ2ΓΊ\ιS 22
üü 1 + Τ9Γ Ω i"T LTJT L
=Π
-η
yx
1 +
= n =Π
zy
Λ 1
+^ti2TT2T{S
9 I I
[«
4^2 2|Sxy 9ΩΙτΙ 4 22
SX
9 I I
A3
5
, 1602τ 2. +-9"Ω L yz
+
yy
z/z/
zz + S
3 I I yz ) -4^
22y
T
S
3 TI IT yz
3ΩΙτΐΛ:ζ
(14.184)
|a I x I xyj - ±Ωττ T(S -5 ) 3 I I yy zz
where η is given by (7.78), τ χ = v j 1 = (5/2/1) •{mj/m-g)^ '^τ E, compare (2.33), τ^ = 4 v ^ , v^ £see (2.27). 5^^ is given by dv
'feZ
k
dv
l
2
(14.185)
In the derivation of (14.184) it has been assumed that the magnetic field is oriented along the #-axis. Staof instabilities can be obtained by the bilization gyroviscosity terms, see e.g. /14.463/. Although in general finite Larmor radius effects orbit instability stabilize, there exists a finite {finite
Larmor
radius
instability
/14.185/).
Thermal
energy of the plasma in the form of drift motions is the energy source of this instability. The inhomogeneity of the magnetic field in toroidal devices causes periodic guiding center drifts {bananas , see p 162) which yield particle displacements larger than rL, see (12.87). Since this is due to col-
539
PLASMA INSTABILITIES
lisions, we will discuss the finite orbit instability in section 14.6. We would only like to mention here that the periodic motion of guiding centers around the field lines leads to another collisionless viscosity guiding center gyroviscosity (due to finite orbits) in the same way as the motion of charged particles in Larmor orbits leads to a magnetic viscosity (gyroviscosity). When discussing finite Larmor radius effects we neglected the diamagnetic current, which is necessary to produce the pressure gradient. Pfirsch /14.17/ and others /14.186/ have shown that the diamagnetic current acts like a destabilizing gravitation and produces drift instability. To see this one has to take into account the relative motion between electrons and ions which is due to the pressure gradient and the gravity. From (13.1) , (2.41), (2.69) we find the electron drift in the presence of density and temperature gradients to be
t>niP = Here again DE eBQ/c
k^T ,~Ί m 31n n T\ B din T ol) (ΛΛ 1 Q £ v r-— — + 1. (14.186) m ti \ dx Sx > T T = m-j-Q-r has been used. Insertion into
the full electron continuity equation results in -ΐωη^ 1 = ~ί1<νΌΕηΕλ
- vEn'EQ{x)
(14.187)
instead of (14.160). So we have ieE^ η
Ελ
- ω/τ? βχ(1 -kvDE/u)
dnEo
~~ΖΊΓ
(14.188)
which replaces (14.160). From the full ion continuity equation -iomj- = -Vjpnj0
- giknj^/Qj-
- Vpj-iknj^
obtains instead of (14.162) the result η
ΙΛ
=
-ieE*
dnT
^ Ω ω ( 1 - grfc/Ω ω - kvDI/u)
Τ^Γ'
one then
(1 4 β 1 8 9 )
This shows that the diamagnetic current -Vpj = +vDE acts like a destabilizing gravity. In accordance with Low's theorem (p 267) drift instabilities are new instabilities and are not found for g = 0, V-Q^ = 0. Since drift instabilities have been discussed in chapter 13, we make here only some remarks on recent literature. When the temperature gradient (perpendicular to B0) comes from the ion temperature one speaks of the ion temperature gradient instability, tempera-
MICROINSTABILITIES
540
ture
gradient
driven
ion acoustic
instability
/14.188/.
When density gradients, temperature gradients and collisions are involved, the growth rate formula may become lengthy. Sweetman gave /15.6/
1/2 *
γ
=
π ' ω k
^ " \\°m
+
2ν π ω* E
*YA^VywSV
(a + e + ß 2 )fe^K^(l + ^ ) - | ε ^ ( 1 + β ) + - ^ i r d + ß)
ak V
l DE Here the first line describes collisionless and the second line describes collisional effects. The first term in each line describes density gradient effects, the second term temperature gradient effects and the last term describes current effects, ω* is given by (13.57). The current driven collisional mode is described by γ - V£
magnetic field the drift ion (electron) cyclotron instability, anisotropie cyclotron instability /14.189/ plays an important role. Ion viscosity and heat conduction clearly have an influence on drift instabilities in an inhomogeneous (weakly collisional) high beta plasma /14.190/. Linear instability of a qua^istatic guiding-center plasma due to non-uniform E*B drift is investigated in /14.191/. Low frequency instabilities in magnetized plasma with non-uniformly compressed flow (div $% Φ 0, ι>Ε electric field drift velocity (4.6)) are considered in /14.192/.
Magnetic
drift
instabilities
due to V£ have been
discussed in several papers^ the plasma-wall interaction through uncompensated B-ripple convection in toroidal devices has been investigated /14.193/ (drift loss cone instability), electrostatic long wavelength instability in a perpendicular shock due to 5-drifts is discussed and reviewed /14.194/, instabilities of drift magnetosonic waves due to the magnetic drift resonance in finite pressure plasma are considered /14.195/ and V£ drift instability in a Tokamak has been investigated /14.196/ as well as low frequency instabilities in a V£, Vn plasma /14.225/. A minimum drift velocity for instability in current-carrying non-maxwellian plasma has been found /14.225/. In spite of their small relative number the high energy fusion products can excite low-frequency drift modes in a uniform magnetic field, a-particles may enhance or reduce the stability in the kinetic or hydrodynam-
541
PLASMA INSTABILITIES
ic regime of density gradient modes /14.461/. In paper /14.196/ a new instability/ the charge separation gradient B drift instability in a torus has been found. A new drift like instability has been detected in a density modulated plasma for λ^ > rLj /14.197/). Cri(density modulated drift instability tical density and temperature gradients required for the onset of drift instability as a function of shear and other plasma parameters are reviewed in /14.189/· Digital spectral analysis techniques have been applied for the experimental drift wave study /14.199/. and of radial elecAlso the influence of non-uniform on drift instabilities has been studied tric fields /14.228/. The two fluid high beta microscopic nonlocal drift instability in a collisionless theta-pinch has been investigated in /14.460/. Several papers have been concerned with the drift mirror instability (drift cone instability, gradient loss cone instability, drift cyclotron loss cone instability (DCLC-mode) , ion cyclotron drift loss cone instability, also Dory-Guest-Harris instability in inhomogeneous plasmas /14.200/, /14.459/. If σ/_|/3/_|_>0, the gradient loss cone instability can be excited. Its growth rate depends on the density gradient. The instability appears mainly for 3 - 0, but can also appear for finite beta. In an inhomogeneous plasma (and the plasma in experimental devices is inhomoge(14.116) generates a neous) a loss cone distribution drift loss cone instability. The main theoretical problem with this instability is its apparent absence in high density mirror machines when u>pE * ΩΕ. Coexisting cold plasma seems to have an influence on the growth rate. For ÎQ{y) = BQ (1 - e<\y)ez , nQ = n0(1 -
lls
2u
\\s4src2ü2s-
=
Hasegawa / 1 4 . 2 0 0 / d e r i v e s the dispersion relation for the drift mirror instability in the form (but see a l so (13.80)) ω =
Wj-ife||W||J/2Ä!T||J/(2'iJß1J)
k2
(14.191)
4n+I(3Is-3|ls)/2)+1+Ißis(1-Tis/T||s) ■ K. .
S
S
MICROINSTABILITIES
542
where ω_ is a modified drift frequency given by ω
*=
Kk u
ï f\s/Çls'
(14.192)
compare (13.57). In a two-^cqmponent ion plasma {h = hot, w = warm ions) with E*B drift the DCLC mode has a more complicated dispersion relation /15.6/, /14.459/ ω ω ΡΕ 1 + - ^ + 2 PE 2-2 n 7
PI , 2
ωPE
utiEk
ω
rp
ΡΙ
7vZ
1
-L
(14.193a)
)-o. ( 2IL·^ 3
Σ ω -ω„Ω T\dciciJî(-itr)
U "-■ -o ^ ""l where r>p is the plasma radius and F(o1) =
2 ft
1
2
+ -5 exp I -
expl-
p«
Stability is possible if ,fec
= ^ , d ClJn(lÇ7 'a o
2
Ω I /ïîkc, c 'h ph w \ "u°J , For large m i r r o r m a c h i n e s isfe_j_l a r g e , Qj/k - βρ^ and Δ ~ Cp^/2c? is n e c e s s a r y for stability. For an ion-electron plasma, for β ->· 0, and ωρ^» Ω 2 the equations n-jj = n-j^, neglect of the electron thermal motion, the hydrodynamic equations of continuity and motion for the electrons and the ion Vlasov equation give in electrostatic approximation /14.459/ the dispersion relation Kfc Ω m i ] 2 J + s ; uJ2{k c /ÏI ) 3 / 1 ο T T ^ I ^ I = 0.(14.193b) 1 +-^H +-i-4l Σ ω-ηΩτ 3 e, 1 1 7 2 7 2J *-^_ Here a g a i n κ = 31η η0/ΰχ.
For k^i/üj
»
1 this
re-
543
PLASMA INSTABILITIES
suits in γ - Re ϋ =ίfijand kr^ - (mj/m^) 1 /^. For finite 3 > (m^/mj)^/^ (non-electrostatic case k < ωρ#/2) and k,c./Ωj » 1 the relation /14.459/ 2 ωπ„ Kk iï^ m-r +~ (14.193c) 1 ~2 7 2 L^ ω - η Ω τ 72 73 3 ti
LI
has been obtained. Then instability is possible for K > 2m^pß/mjo or for g - 1 if KrLI ^ 2 (m^/mj) 1 / 2 /14.459/. The growth rate is γ - Ω ^ 1 / 4 ( 2 π ) " 1 / 2 •(^j3j)"1/4· For 3 > 1, in addition to considering Vn 0 also V5 Q has to be taken into account. The electrostatic and also high frequency instability is due to resonant coupling between drift waves and cyclotron motion in an inhomogeneous plasma with anisotropic loss cone ion distribution. Stabilization by finite beta seems to be possible /14.25/. In some experimental devices 10 4 x the threshold of the instability is reached without the instability appearing. The effect of nonuniformity on anisotropic distributions (e.g. the drift cone instability) may also be (13.73). Midescribed by the drift kinetic equation khailovskii discussed recently the drift mirror instability in a plasma of finite pressure with hot electrons /14.201/. Electrostatic as well as electromagnetic drift instabilities can exist in non-uniform plasmas with loss cone distribution functions in inhomogeneous magnetic fields. The same is true for anti-loss-cone (inverse loss cone, reverse loss cone) distributions/ which are found in toroidal devices and in the magnetosphere /14.202/ and which look like f(u^,u.)
' a.a..)
= 2ηο(π Γ
2, 2
(u„/a..) 2,
(14.193)
2]
exp L" M ||/ a || " w . / a . J . In an anti-loss-cone plasma a new high frequency
ti-loss-cone-ins
tability
(anti-loss-cone
drift
an-
insta-
bility) occurs with a growth rate ^0.1Ω^ /14.491/. Drift mirror instability (instability of a mirror machine) and mirror instability (due to local mirror/ see Fig. 20, p 213) should not be confused. The drift mirror instability (PVC f Ss f vT 1 Bb 1 fE 1 LM'tRpA 1 I) (see p 253 for the meaning of these symbols) is a low beta electrostatic flute-like particle resonant drift in-
MICROINSTABILITIES
544
stability in an inhomogeneous plasma with a loss cone distribution. The mirror instability (PV'C'SsT'Bbf'ELM1t'R'A*I) is an electromagnetic high beta nonresonant instability. It is due to the magnetic mirror action concentrating charged particles in the weak field regions of an inhomogeneous plasma. The increasing transverse plasma pressure in these regions pushes the field lines apart and increases the mirror ratio. At sufficiently large temperature anisotropies and strong pressure gradients a growing mode is generated, whose energy source lies in the T|-T|l anisotropy. By making the magnetic field strong compared to the plasma pressure anisotropy, the instability can be stabilized, see (14.88). The instability condition for the mirror instability may also be written /14.203/ (for a homogeneous plasma) 1 + Eßi8(1 - Ti8/T]l8)
< 0.
(14.194)
S
In an inhomogeneous anisotropic plasma also electromagnetic waves may become unstable /14.204/. When they propagate perpendicular to an external magnetic field, B (x) = BQC\ +ex)
-3/2 - 2 - 1 Γ 2, 2 = ft π α , α,, e x p - u . / a . os is \\s ^ L 1 I s
os
1
0
1-
0
0
0
2/2 1 "U../au II' II eJ
\ i
( 1 4
1 9 5 )
\\
( ε ' - δ . u./a, - δ,, u../a.. ) x + u / Ω )}. is Γ Is | | s W \\sl \ y' sl\ \ s
For wave propagation perpendicular to BQ the dispersion relation sçlits into ordinary and extraordinary mode (K = k~ey, E-jllB, see chapter 9 ) . The dispersion relation in local approximation contains: 1. drift instabilities for ω ρ < nüj « 9,j-r c2k2 » ω 2 , ω^ » kzu\/2tiE. 1.1 resonant electron instability ωρ > 0 1.2 ion drift cyclotron instability ωρ < 2ΩJ 1.2.1 resonant ion instability γ -* 0 1.2.2 off-resonance drift cyclotron instability
2. temperature-anisotropy instability for Ω Ι <<: ω ρ <
545
PLASMA INSTABILITIES
Another general dispersion relation for high frequency electromagnetic drift waves in the range Qj « ω « Ω# arising from an electron stream drifting in a density gradient in a magnetized plasma may be found in /14.205/. Waves in a magnetized inhomogeneous high beta plasma are treated in /14.206/. Helicon waves in a non-uniform plasma are discussed in /14.230/. Reflection and absorption of electromagnetic waves in a hot magnetized inhomogeneous plasma are investigated in /9.4/. The negative pressure instability of the ordinary mode is discussed in /11.60/, /14.207/. In a finite 3 (>m^/mj) inhomogeneous plasma the filamentation instability of high frequency electromagnetic waves may occur /14.208/. Both instabilities_are nonlinear. "Or sufficiently long parallel wavelengths the transverse modulation of the electromagnetic wave gets coupled to drift-Alfvên waves and leads to a modula-
tion
filamentation
instability.
The filamentation in-
stability is a parametric effect and will be discussed in chapter 17. When waves have frequency near the
hybrid frequency the hybrid instability, nance instability, lower hybrid drift
hybrid instability
resomay
appear /14.209/. A theoretical model valid for the of a low beta plasma sheath region (thickness of rLj) has been given for the lower hybrid instability which can be stabilized by high beta effects. The instability may be understood from the fact that the sheath consists essentially of a homogeneous ion beam moving along the sheath at a constant velocity, and of electrons of zero temperature /10.2/. The beam is perpendicular to the magnetic field. It represents the macroscopic flow of matter due to the pressure gradient and Larmor motion at the boundary of the confined plasma. Due to resonant coupling between the ion beam and electrostatic oscillations at the hybrid frequency , a growing wave develops. This mode is similar to the loss cone instability, but the driving force is here due to the anisotropy in the ion velocity produced by the ion diamagnetic current. The beam velocity is the energy source of the instability. When the beam couples to non-electrostatic oscillations an instability appears also in high beta plasma. In addition parametric excitation of the upper and lower hybrid instability by high-frequency electric field is possible. Radio frequency energy can be carried towards the lower hybrid sheath by means of electron plasma waves. In the presence of magnetic field gradients , the lower hybrid wave can be absorbed through
MICROINSTABILITIES
546
collisionless damping at cyclotron resonance. In the low drift velocity regime (i.e. of the order of the ion thermal velocity) and for finite beta a magnetic gradient drift-wave resonance can be important and is stabilizing or destabilizing. Substantial plasma heating and a large anomalous resistivity can occur /14.209/. Plasma inhomogeneities also influence other instabilities. For the E*B-drift instabilities, cross field current instability a temperature gradient increases the growth rate, but Vrc and VB have little effect /14.20/. For sufficiently high beta however, VT is stabilizing (temperature gradient stabilization). A local plasma inhomogeneity may itself also propagate. The spread of the plasma nonuniformity is determined by the greatest diffusion coefficient values along and across the magnetic field. The ion acoustic instability and also instability due to the relative motion of electrons and ions in a magnetic field /14.235/ are also modified by non-uniformities. For a magnetic field BQ directed along the 3-axis, a low beta plasma equilibrium depending on x and perturbations ^expdkyy + kzz ) of the equilibrium one obtains /11.100/ distribution functions fos(x,u) from the Vlasov equation in drift approximation (ω «
Qj)
fJ
= — T 5 Ί / - ^ + — E - τ ^ . (14.196) ω + k ult x Ώ ** ™ * a,, i z II For E - -^Φ the dispersion relation can be obtainThe ed from div j * ikzUz = 0, j z = \eQ\uzfQduz. transverse current components are 3 suppressed by the magnetic field and do not enter into the stability criterion. Then the dispersion relation assumes the form s
u
[
(ok
df
e
df
21
* U-f- 4 ^ + — 4
,
d
" -0.(14.197)
ω + k u \B k dx m du I z z z \ o z s For Maxwellian ion and electron distributions with ^-dependent temperature one obtains
(dT /dx)/emTB * 0 (14.198) ω 3 - 3k2k cknT z y D o o i o {din T0/dx) . In a which holds for k /kz » eB/cy/k^T0mT toroidal plasma confinement device the current-driven ion acoustic instability exists only in the collision-
547
PLASMA INSTABILITIES
> al regime where veff ^bE' the bouncing frequency of the electrons (14.306). In the trapped electron bana< ω ££) t h e i nsta bility is inhibited na regime (veff /14.445/. This inhibition demonstrates that in the banana regime the entire Ohmic heating current is carried only by freely circulating electrons. This behavior of the current driven ion acoustic instability is of relevance because its anomalous resistivity may be useful in plasma heating and may explain the anomalous skin effect in the penetration of Ohmic heating current in large toroidal plasmas /14.446/. Fission driven ion acoustic instabilities in a uranium gas core reactor of finite geometry in inhomogeneous plasmas have also been investigated /14.447/. Bernstein modes also become modified in an inhomogeneous plasma /14.224/; they may become unstable {electron cyclotron drift instability) by interaction with ion acoustic waves. In the presence of weak inhomogeneities the propagation parallel to the drift velocity of the w-th Bernstein mode is described by the dispersion relation (Sanderson /14.224/) 2
7
T
K
/
TP
*
*_ζ.(_ω
v
fe
+ ! +
( ö . 7 » UT)
th
J
E
. Ζ(ξ)
^
+
+ /72Ω-,
E
E Z> (ξ)
°th
uE /Ta
I
2a
=
(14.199) ω
kuT/2*
Linear electrostatic instability of a low beta plasma in a magnetic field has been studied for modulated distribution functions /14.225/ of the type /
= nQf(u
. ,u..) Pi + 2acos(a[x
+ u /Ω^.])].
(14.200)
Periodic magnetic fields have also been considered /14.231/. Ion-acoustic instabilities in magnetic pulses /14.227/ as well as instabilities in helical magnetic fields /14.233/ have been investigated. Besides the instabilities which are modified by plasma inhomogeneities and besides the drift instabilities there exist other instabilities which are excited only in an inhomogeneous plasma. One example is the Kelvin-Helmhotz instability, which occurs only in an inhomogeneous plasma. This instability, which is due to discontinuities of the tangential velocity of a streaming plasma (see pp 74 and 284) has several
MICROINSTABILITIES
548
modes. The macroscopic mode has been discussed in volume 1 of this Handbook, see p 297. There exist, however, two microscopic modes too: these are the microscopic electrostatic collisionless Kelvin Helm/11.49/ and the electromagnetic holtz instability collisionless Kelvin Eelmholtz instability /14.125/ in anisotropic plasma, see volume 1. Velocity gradients may not only generate the classical (magnetohydrodynamic) Kelvin-Helmholtz instability, but also transverse Kelvin-Helmholtz modes in Q-machines and proton heating neutral sheet instabilities in space plasmas, e.g. in the aurora. High frequency instabilities may be suppressed by velocity gradients, low frequency instabilities are responsible for anomalous resistivity. For ω/Ω^- « (w^/wj) ~' 2 one has the classical KH mode, for ω/Qj » (m^/mj)^/2 a n e w mode driven by άν/dx (shear mode) appears /14.462/. This mode is collisionless and may be described by an electron-ion two-component Vlasov theory. In a Q-machine centrifugal force, density gradient and sheared velocity act together and produce a radial electric field, which is stronger at the plasma boundary. If ujr(r) = -mcE/rB frequency depending on the rais the local rotation dial coordinate r and ω# its average over space, the dispersion relation reads /9.59/ ω = | ^ + ;[οω^(1 + 2/(fc*
+2/(k2l
Here δω^ = ω# - ω#. The first term describes the effect of the shear of velocity, the second term describes the centrifugal effect. When we ignore the centrifugal effect and the shear effect, but consider a Gaussian density profile, we obtain ω = ω - ω /2 + ii όω - ω
/4J / .
Here ω* = k V-Q is the drift frequency (diamagnetic . For <5ω2 < ω*2/4 we have a stabilizing effrequency) fect of the finite Larmor radius (here also called density gradient stabilization). For a Bickley jet profile one has ω(ρ) = a)Qsech2 (r - rQ), δω£ = ω|^ αχ /16, / 4 , the stabilization condition is ω^ χ< <ω^> = ω 2 maX <2ω* Another instability due to inhomogeneity of the magnetic field is the tearing instability. The macroscopic collisionless tearing instability has been
549
PLASMA INSTABILITIES
discussed on p 314, the macroscopic collisional (resistive) tearing instability has been discussed on
p 313/ and the collisionless microscopic tearing instability (sheet pinch instability) on p 380 /14.216/. By use of a variational principle based on the Vlasov
equation a plane equilibrium distribution function fos(x,u) = f0s(H0,P0), expressed as functions of the constants of motion H0 (energy) and P 0 (momentum), has been investigated for a magnetic field BQ = = [0, 0, B0(x)]. An energy integral investigation based on the Vlasov equation results in a stability condition and the fact that the real part of the frequency is determined by the gradient of the external magnetic field. A linearized Maxwell-Vlasov investigation for ions and electrons gives the growth rate /13.53/ for a high beta plasma * ~- Gthl('
-
k2 k2
' n)/2d·
( 1 4
·
2 0 1 )
where d is the thickness of the plasma slab and km the marginal wave number ^λ/ά, compare p 313. A study by Galeev /14.216/ showed that the presence of a magnetic field component normal to a diffusive neutral sheet affects the electron orbits near electhe sheet and results in stabilization of the
tron
tearing
instability.
Development of
ion-tearing
instability with a given wavelength is possible only in a restricted range of values of the normal magnetic field component. As with the macroscopic tearing mode due to inertia, electron pressure gradient or Hall effect in Ohm's law, it is a matter of taste whether one calls the macroscopic rippling mode due to the same effects collisional or collisionless, compare p 314. The re-
sistivity
gradient
instability
or rippling
instabili-
ty which is due to inhomogenity in electric conductivity may be resistive (collisional) or may be due to electron inertia, electron pressure gradient or to the Hall effect and may then be called collisionless
microscopic rippling instability /13.53/ or less macroscopic rippling /10.2/ instability
collision(p 316)
since both fluid theory and kinetic theory may describe it. For drift tearing instability see /14.493/. is an electrostatic The negative mass instability collisionless microinstability which can occur only in an inhomogeneous magnetic field. It is found also in particle accelerators /14.217/. When an ion beam, neutralized by an electron
MICROINSTABILITIES
550
background, circulates around the axis of an axisymmetric inhomogeneous magnetic field, then a small (e.g. sinusoidal) perturbation of the space charge density results in particle acceleration (or deceleration) . The particles ahead of the potential bump will be speeded up,and those behind it will be slowed down. Now the average gyration frequency of the particles may depend on their energy, and it may happen that for those slowed down the angular velocity increases. This is the case when the magnetic field strength decreases in the radial outward direction (negative radial gradient). For slowed down particles which increase their angular momentum the angular acceleration is opposite to the force, as if these particles had a negative mass. The result of this behavior of particles is that they tend to move toward the angular position of the potential bump and that the bunching and the perturbation grow. So the negative mass instability leads to an azimuthal clumping of intense beams rotating in magnetic fields. One may see the mechanism of this instability also the following way: if energy is fed into the azimuthal particle motion, it will be brought to an orbit of increasing diameter which has a longer period of revolution. Thus a force in the positive azimuthal direction produces azimuthal bunching of particles which therefore behave as if they have a negative mass producing a displacement in the opposite direction to an applied force. Three known effects give rise to a dependence of the gyration frequency averaged over the period for a particle to bounce between local magnetic mirrors. These effects are: 1) relativistic dynamic effects (E total energy) Ω = Ω (1 - E/c2m)
(14.202)
2) radial variation of magnetic field machine)
(e.g. mirror
Ω = Ω (1 + b\\BQB' /2ü^vBm)
(14.203) %
for u]_ » Un, y is given by (4.14), B = r is the radial coordinate 3) variation of the axial magnetic field
negative
mass
Ω * Ωο Π
instability)
+ (E - \IBQ) /2]iBQ]
.
dB/àr, (modified (14.204)
551
PLASMA INSTABILITIES
Relativistic dynamics and axial magnetic field variations are always destabilizing while the radial magnetic field variation has a stabilizing or destabilizing effect depending on whether the magnetic field has a radial well or hill. By use of a canonical formalism and the relativistic Hamiltonian, a dispersion relation has been derived by Landau and Neil /14.127/. An azimuthal magnetic field tends to stabilize the negative mass instability. It should be pointed out that Maxwell was the first to investigate the mechanism underlying the negative mass instability in his essay on the stability on Saturn's rings /14.218/. He pointed out that whenever the frequency of rotation of particles in a ring interacting via repulsive (attractive) forces is a decreasing function of their energy then the ring is unstable (stable).
Work on the mass-conjugate
instabilities
/14.21/
showed that the negative mass instability and the two stream instability are a little related effects in the limit of applicability of the Vlasov equation. To each and every positive mass system there corresponds a negative mass system which behaves identically except that particle excesses in one case are to be identified with particle deficiencies in the other. The two systems are called mass-conjugate. It seems that the so-called second instability in the AS device is a modified negative mass instability /14.217/. A background cold plasma as well as a thermal spread can stabilize the lower azimuthal modes of the negative mass instability. Mikhailovskii presents the following theory of the negative mass instability /9.26/. If the bounce period τ = φάΐ/u^ or (4.45) is short compared to γ~1, where γ is the growth of an instability, then the ion frequency Qj in a dispersion relation must be replaced by its average over the bouncing period τ
<Ω > = Hü 1
ij
(l)—. 1
(14.205)
U ||
Here
(14.206)
For low density, ωρΕ « ÜE the electrons may be neg-
MICROINSTABILITIES
552
lected and the dispersion relation becomes for monoenergetic ions (F ^ δ-function) and for ω ^ n
(14.207)
One finds that for 91n
space charge
instability.
Other instabilities occurring only in inhomogeneous
plasmas are e.g. the critical
velocity
instability
/14.221/ suggested for a spatially non-uniform electric field in an inhomogeneous magnetized plasma or
the nonequilibrium
plasma instability
/14.222/ which
has been detected in capacitive HF discharges. The pressure gradient and its associated drifts and magnetic curvature effects are the energy source of the
collective
electrostatic
instability
in a two-dimen-
sional field /14.223/. This instability occurs in a low-beta plasma confined in a two-dimensional configuration, such a configuration being the linear analog of a multipole device. The macroscopic field geometry of the system is taken into account, leading to integral equations involving the coordinate along the magnetic field lines. There are periodic particle orbits of both the transit and trapped particle type. There exist several modes depending upon whether the action integrals associated with the motion of ions and electrons along the magnetic field lines are conserved or not: with the bounce frequencies oo^j and ω^^/ there are slow ("magnetohydrodynamic ") modes ω » ω^Ε y> ωΐ>1' fast kinetic modes oo^j « ω « ω^^ and slow kinetic modes ω « a)£j « ω ^ . Stabilization may be possible by a proper choice of field geometry such as minimum average B configurations. Landau damping, finite Larmor radius effects etc. Ordinarily in nonhomogeneous plasmas finite length
effects stability
are important. Thus the negative-energy inin inhomogeneous mirror geometry may devel-
op /14.236/ (there are also other types of negative
553
PLASMA INSTABILITIES
energy instability). Negative energy waves are excited in mirror machines by a removal of energy through the variation of plasma density and magnetic field strength along the magnetic field-lines. At the center of the configuration there develops a negative energy wave which transforms at the ends of the machine into an outgoing positive energy electron-plasma oscillation. Since this outgoing wave carries off energy from the interior, it serves as a dissipation mechanism, driving the negative-energy wave unstable, see p 2 56. The present mode is absolute, is related to the loss cone instability and takes its energy from the deviation from a Maxwellian distribution. It is stabilized by Landau damping (for high density). A similar instability may develop in sheared fields. Other finite length effects in mirror machines or in slab geometry may be found in /14.237/. Another instability of this type is the negative energy instability in shock waves /14.238/. When a collisionless shock wave propagates perpendicular to the magnetic field, particle drifts arise from the magnetic field gradient, the electric field, and the plasma density gradient of the shock. When a negative energy Bernstein wave comes into resonance with an ion acoustic wave or when resonant ions absorb energy from the negative energy wave, an instability is produced. It takes its energy from the drift motion of the particles and may be stabilized by Tj- ~ Tg, since the ion acoustic resonance can only take place when Tj « TE. Effects of cylindrical geometry (including finite Larmor radius stabilization /14.239/) as well as toroidal geometries /14.240/ have also been considered. Mikhailovskii /9.26/ discusses homogeneously magnetized plasma with nonuniform density variation along its length. For electrostatic modes he presents for the electric potential Φ the dispersion relation 2 -, _d_ (Λ ωPE \ άφ -k2 ax V " 2 ) άχ
ω
2
.2
™Ρΐω
ΡΕ
Φ =0(14.208) 2 2 X 3/2 ω E * l o~ ' " compare (14.142). When the plasma is uniform for \x\ < 1/2, and its density drops to zero at \x\ = 1/2 + a) , the solution of (14.208) becomes Φ - cos(kzz where a an arbitrary constant. For ωρπ» ω and Φ 1 (1/2) = 0 one has the dispersion relation 1+
n2
/72
U
MICROINSTABILITIES
554
2 2/
ιΛΛ
<ω
+>?h/™ il
"
= 0. (14.209)
a Lu2o-a2)3/2 2
α\
Similar results have been obtained for inhomogeneously magnetized plasmas (adiabatic traps, multipole traps, stellarators) /9.26/, Vol 2, p 271 ff. There exists a formal analogy between the Penrose criterion (14.11) and the necessary and sufficient condition for the instability of a weakly magnetized (fij « ^pj) inhomogeneous plasma with a monotonie velocity profile v0(x) . It reads/9.26/ for low frequency long-wavelength perturbations of a plasma with constant density X ry
PjJ
Jn £
,(u ,(x). -"*u ,(x) ,2 > °.
#,.0
0
0
X
{TT)-°\ax I λ ' X=X
Various geometric effects as e.g. the influence of gaps on the electromagnetic wave excitation or of the form of the vessel have been investigated /14.241// /14.485/ (Tokamak microinstabilities) . 14.6 COLLISIONAL MICROINSTABILITIES Collisions may provide a damping mechanism for instabilities and may modify known instabilities. On the other hand, collisions may also excite new instabilities. Collisions of charged particles with neutral molecules in a partially ionized gas modify the particle drifts in such a way that a relative velocity between the two charged species appears and a twostream instability may develop. For the cyclotron oscillations collisions provide a new source of damping but also a small destabilizing relative drift. Since the frequency v of Coulomb collisions depends on temperature and density as nT"~3/2 according to (2.28), collisions modify instabilities mainly in dense cold plasmas, see p 389. To see this more clearly we consider (9.153). The partial integration leading to (9.154) was made under the assumption v = const. When we obtain allowing vs = vs(cs) 2
*
fee 8 = - Es - p s *J / o eκω/k — - c -tv—^2k)ά%,' -«> sx s ___^ ? f
-1 - i/k·
dv
(14 210)
'
555
PLASMA INSTABILITIES
On the other hand the dispersion relation for electromagnetic waves with k± = 0 in a weakly ionized magnetized plasma reads /14.459/ ~2 7 2
ww^3f
3
1 - 2 * + Ρ \ - ί £
2
?
äc
(14.210a)
3 J ^ c ω - Ω + ^v (c) o
and gives for a low-density plasma ω = ok and 2
R«"Â
c v (g) L(Ü> - Ω ) 2 + v 2 (a)J
de.
(14.211)
For monoenergetic electron distribution function fo - δ(o - cQ), the electromagnetic wave becomes unstable if {d/dc[a^v(a)]}c - CQ < 0. So for an instability the electron collision frequency has to fall more rapidly than
s
(c)
=
ao
one finds by a numerical evaluation /14.242/ that electrostatic instability may appear for a plasma with rapidly decreasing collision frequency (h ~ 3 ) . Observations of this instability have been reported /14.243/. Since this instability does not need a magnetic field it may be looked on as a bremsstrahlung effect rather than a cyclotron effect (bremsstrahlung
instability
/14.244/).
The full electromagnetic dispersion relation for a dense Krook plasma (5.18) with strong current may be found in /14.245/. A more refined discussion of the effects of collisions has been made on the basis of the Landau collision term (5.10) /9.26/. Since the electron velocities are large compared to the ion velocities the collision integral in (5.10) may be simplified. If there are no external fields in the plasma the establishment of Maxwellian distribution functions takes place within the relaxation time (2.34)(2.36), where (2.36) is the slowest process. Fast ions (beam) with a speed u come into equilibrium with the plasma in a time of the order 3/2 -3/2 · (14.212) V a Uo XII°th
MICROINSTABILITIES
556
For Tjj/ TEE s e e P 2 2 · A collisionless approximation (p 220) is correct only for time intervals much less than binary collision times (2.34) - (2.36), (14.212). That means that the instability with the growth rate γ must satisfy τ > γ~1 that it might be considered as collisionless. Let us discuss some typical instabilities. The hydrodynamic two stream instability has γ - ωρ#, see p489. From τ * TEE (2.35), (2.19), (2.9) equation (2.15) follows immediately as the necessary condition. Therefore this instability is not sensitive to the frequency of collision processes. For an electron beam with and of low densmall thermal spread c-tfo < u0n\'^/n^'^ sity n-j (ni/n0 < 1) penetrating into a plasma of density nQ the condition that collisions may be neglected becomes 2 n«u
o
-γ^^Ε»
°th
1·
(14.213)
The cross section of Coulomb collisions decreases with increasing relative velocity of the colliding charged particles. As a consequence, a group of electrons will run away at increasing velocity when the applied electric field is chosen above the critical value (7.62). This run-away effect /14.464/ may cause the run-away instability {electron run-away instabilelectron instability) /14.465/ since ity , run-away the run-away electrons form an electron beam (compare section 15.2). Such run-away electrons and run-away instability are generated in Tokamak machines and have also been investigated in the relativistic region (suprathermal range). Whereas a non-relativistic theory predicts that an exponentially small number of electrons run away also for fields less than the (7.64), an exact relativistic theory Dreicer field finds that below that critical electric field runaway is smaller for TE » mc^. For large thermal spread collisions are unimportant. In a fully ionized gas the slowing down of electrons as a consequence of collisions with ions and the acceleration in the electric field are balanced is where ^crit 9 i v e n by (7.62). In this for E < Eorttr case a stationary state is possible for frequencies ω > V EE' I n a weakly ionized plasma collisions between charged particles and neutral particles are more frequent than the collisions between charged parti-
PLASMA INSTABILITIES
557
clés, see (5.16), p 81. Then in a three fluid theory a friction term (compare the last term on the r.h.s. in (7.33)) or Langevin force, see p 17, appears which influences stability. In order to show how a collisional background plasma affects the growth rate of the two-stream instability a general method for obtaining the curves of marginal stability of cold streams has been presented /14.249/. The region of absolute instability is shown to be reduced whenever v Φ 0. For v = 0 the plasma background does not change the domain of absolute instability but reduces the growth rate. solid Collisions play a very important role in state -plasmas. Several collision induced instabilities are known. Streaming instabilities in solids described by a hydrodynamic model become resistive and collision-modified. But also new collision-induced instabilities as e.g. in a drifting electron-hole plasma situated in a perpendicular magnetic field may also appear /14.246/. In addition collision-induced surface wave instability can occur in composite magnetic and semiconducting structures. Vural /1.68/ gives the following classification of collision induced instabilities in solids 1. Collision-induced single-stream transverse wave instabilities 1.1 transverse wave collision-induced instability in nonmagnetic semiconductors 1.2 transverse wave collision-induced instability in magnetic semiconductors and in composite structures 2. Collision-induced two-stream instabilities in electron-hole plasmas 2.1 collision-induced instabilities in drifted electron-hole plasmas due to wave excitation 2.2 collision-induced instabilities in drifted electron-hole plasmas due to circularly polarized wave excitation.
In concluding this short introduction we would like to mention that for an inhomogeneous plasma with a periodic equilibrium potential a like-particle collision operator may be derived in the guiding center approximation /14.247/. That paper discusses how collision operators of the type (5.10), (5.11), (5.14) are modified by strong magnetic fields /14.248/ or by non-uniform particle motion /14.247/. Before we discuss the modification of known instabilities by collisions we will discuss the new colli-
MICROINSTABILITIES
558
sion induced instabilities.
The finite
orbit
instability
{finite
Larmor
radius
instability) /14.185/ which we had mentioned on p 538 is due to collisional guiding center displacement in . In an inhomogeneous magnetic field {"mixing effect") toroidal systems the diffusion of trapped particles enhances the influence of collisions by producing displacements larger than those by classical diffu-
sion {"transverse
collisionless
viscosity")
. This
leads to a drift dissipative instability occurring at smaller collision frequencies than the usual instabilities of this type. For small shear (BA « B0) and long wavelength {k ■> 0 is more dangerous) and u\\t «I, I from B{r ,z) = B0[1 - r {cos {2z /1) - h) /R] , t time and kzc^yl <<: ω an<^ τΙΙ = œr λ# « I, Pogutse /14.185/ gives a dispersion relation for the finite orbit instability, which may be considered as a dis-
sipative
trapped
particle
instability.
The maximum
growth rate is reached for ω - ω#, ω| being the electron drift frequency given by ω|» = kv^g or by ω* = -kckBTEn*Q{r)
/nQeBr
(14.214)
compare (13.2). For development of the instability the system must be finite but large enough, i.e. certain conditions must be satisfied. If they do not hold, the longitudinal ion motion becomes important and the instability vanishes. And electron dissipation should be strong enough. Otherwise electrons can reach equilibrium along the field lines, which would lead to a Maxwellian distribution and to the vanishing of diffusion. Systems with small shear or large field modulation cannot achieve the largest growth rates. For small shear a minimum B-configuration does not provide stabilization. The joint action of large shear and minimum B reduces the growth rate appreciably. /14.250/ is another The entropy wave instability instability occurring only in collisional plasmas. When a low beta uniformly magnetized plasma with density and temperature gradients perpendicular to the magnetic field is permeated by electrostatic low-frequency waves k\\ - 0, then for din T/dx « din n/dx a new collisional overstable mode appears which propagates in the ion drift direction. It is due to the combined effect of ion-ion collisional viscosity and heat transport. These entropy waves have neither magnetic field nor overall pressure disturbances, with
PLASMA INSTABILITIES
559
temperature perturbations Tji - -T#i. The energy source of this instability lies in the thermal energy of the plasma. The instability vanishes at low plasma densities. It has very small longitudinal wave numbers and its transverse phase velocity depends essentially on the particle drift in the magnetic field. Another form of the entropy wave instability appears when effects of the order of ω/vjj in the excitation of electron waves are taken into account and when din T/dx « din n/dx or Bin T/dln p > 7/10+3/4 is not satisfied. The terms of order ω/vjj correspond to the (collisional) gyrorelaxation effect. Entropy waves become unstable due to this effect in a certain range instaof 81n n0/81n pQ and for VT 0 = 0. The entropy bility is a nonisothermal effect which is eliminated by parallel ion pressure at high densities. In a partially ionized plasma the collisions of charged particles with neutral particles may excite another type of instability: the electron-neutral atom collision instability and the ion-neutral atom /14.251/. Timofeev has shown collision instability that in an inhomogeneous weakly ionized plasma in a homogeneous external magnetic field a drift instability can arise due to the effects of collisions between charged particles and neutral atoms. (On the other hand drift instabilities can be quenched by ion-neutral atom collisions in a Q-machine /13.11/.) The electron-neutral atom collision instability is due to the dissipative effects of collisions in a plasma of non-uniform density, nE - nj, TE » Tj. The neutral drag instability (p 334) occurring also in a partially ionized magnetized plasma may be considered as a difference effect of the electron-neutral atom collision instability and the ion-neutral atom collision instability. When electric field and density gradient are parallel plasma particles drift in the ExB direction, and because the ion-neutral collisions are more frequent than the electron-neutral collisions {Vjfl/Qj » vEßj/Q,E) the ions lag behind the electrons. The resulting difference in drift velocities gives rise to long wavelength oscillations (ω < üj, ω < ωρ) parallel to the magnetic field. /14.242/ Clearly also the ionisation instability and the recombination instability are instabilities which are due to collisions. A collision induced instability of transverse waves propagating along B in weakly ionized gases (as well as in fully ionized plasmas) due to the velocity de-
MICROINSTABILITIES
560
pendence of the collision frequency has also been discussed /14.253/. Ion-neutral atom collisions may also excite another dissipative instability in a weakly ionized plasma.
The collisions induce supersonic
ion motion
instabil-
ity /14.254/. Here hydrodynamic supersonic flow with friction between charged and neutral components in a plasma is analyzed. If the plasma velocity with respect to the neutrals exceeds the ion-acoustic velocity, an instability develops. In the initial stage the growth rate is proportional to the frequency of ion-neutral collisions Vj^. In the absence of such collisions the instability disappears. Let us now turn to the collisional modifications of instabilities which also exist in a collisionless plasma. A relatively simple example is the effect of Krook model collisions on the electron-electron and electron-ion two-stream instability /14.255/. Here the electron Krook equation (5.18) is solved numerically for a uniform neutralizing Maxwellian ion background. A first order linearized (/# = / 0 + /1) analytical treatment using Laplace-Fourier transforms yields a complicated dispersion relation analogous to (9.153). Such a dispersion relation predicts also collisional damping of Langmuir waves /9.30/, /14.256/. In the investigation of the electron-ion two-stream instability with the Krook model a gradual heating of the electron stream due to Coulomb interaction with the ions at rest was found. It was shown that, as the kinetic energy of the electron stream is thermalized, the resulting enhanced Landau damping is opposite to and comparable with the growth of the instability. The Krook equation is also useful for the investi-
gation of the collisional
current
driven
ion wave
in-
stability /14.257/. Ion waves propagating nearly across the magnetic field in a weakly ionized plasma with cross-field electron streaming uoE become unstable due to electron-neutral collisions in the parallel equation of motion for the electrons and due to ion-neutral collisions. According to p 82 this equation of electron motion (magnetic field in the s-direction) reads for a Krook plasma with electron inertia neglected e
9Φ
1
Έ1 ΎΓ E
^EkBTE
- lOi
E O
9
^1
§5- -
V V
E E^z
a
n
°
MA
01„
(14.215)
compare (9.53). The use of a fluid equation for the
561
PLASMA INSTABILITIES
parallel motion of electrons is justified since v# > (ω - ku0]r) is assumed. Φ-| is the electric potential,rc#-|the electron density and v*z the velocity of the linearized perturbation ^expK (£·? - ω£)] , γ^ = 1 or 3 depending on whether the electrons are isothermal or adiabatic, see (9.51). Neglecting the perpendicular electron motion (in contrast to the analysis of field aligned kz = 0 resistive instabilities where the perpendicular electron motion is essential) and using the electron continuity equation, one obtains n
S1 / n o
=
Îk e
l ^^mEVE{{u-t'toE)
+ Î
^ E W -
( 1 4
·
2 1 6 1
Since vj » 9,j will be assumed, the ions are essentially unmagnetized and the linearized Krook equation may be used in the form df
n
- ot ^ -
-
e
a/
oi
n
n
/
I
+ u-V/V, — νΦ,-τ-^ = - νI Γ \ /l\Γ 1 - —n /J f tolT . I J I\ - m T n I 1I duT o ' Again with the assumption of exponential dependence for the perturbations, /J-J may be calculated. Integration in velocity space and use of the quasineutrality approximation nE^ = nj-| (valid for lowfrequency modes) gives the dispersion relation /14.257/ 2
-1
Γ ^ + ΐν
° thi*- L
k2 2
+ iuFE(kz/k
kG t h I + hI
^
ω+ ^ Ί Γ K k o V K O
*t h Ä II ^^
)vvE ) [(ω \{u -%>u -t*u„voE )y)
χ
+
^IJ" kG
th.I ™m
^I\\
[kG
thI —thi
2 + + ik„ywikk„Tjm 2yEkBwT\E/mEj
+
'-*.( 1 4 > 2 1 7 ) = 0.
For \(Ù + ivj/kctjtfl » 1 t h e F r i e d Conte f u n c t i o n Z may be expanded and f o r s m a l l γ t h e r e s u l t / 1 4 . 2 5 7 / ω - kaq r b (14.218)
γ = -1 ^ ν Γ Τ - KmEl2mx) ( A ^ i v ^ f l -
uQE/aJ
2VJJ is obtained, where α^ is given by (9.51). Here T^»Tj has been assumed in order to avoid the large Landau damping that occurs for TE * Tj. VJJ is the ion viscosity term and the second term arises from electron ion collisions. We see that UQE > aß is destabilizing {supersonic electron motion instability). In contrast to results for fully ionized gases /14.258/, where electron-ion collisions lead to instability, elec-
MICROINSTABILITIES
562
tron-neutral collisions in the electron parallel motion lead to instability in a weakly ionized gas. When perturbations in TE and in Tj are also taken into account, then in the linearized Krook equation for unmagnetized ions (14.217) the r.h.s. term has to be replaced by a more complicated term containing additional terms depending on TJ^/TJQ. Then a temperature relaxation mechanism is found which has a destabilizing effect on ion waves propagating nearly across the magnetic field in the presence of streaming electrons. Experiments on the effect of collisions on the current driven ion-wave instability /14.261/ confirmed an effect on the onset of the instability: the critiparacal wave number kQ decreases, when the collision meter ^pjTj increases. The minimum drift velocity for the instability is called the critical drift velocity vc corresponding to the critical wave number and is defined by [dvD(k)/dk]^
= 0.
The effect of ion-neutral collisions and of more sophisticated collision operators on the ion sound wave have also been investigated /14.292/. Boundary effects, thermal conduction and relativistic effects have been included /14.262/. Relativistic treatment of the electrons reduces the growth rate (relativis-
tic
stabilization).
When oscillations with frequencies low compared with the collision frequencies are to be investigated, it is necessary to solve the Boltzmann equation. Braginskii's method /14.263/ with Maxwellian distribution functions (7.1) yields after some algebra (moments method, see p 92) a system of equations of the type (7.2), (7.11), (7.16). For vanishing magnetic field Braginskii writes these equations in the following form 9n -TT- + d i v ( n v ) = 0 at
SS
(14.219)
dv + ^ m n -ττ^- = -Vp - ν · Π + e n E + R s sdt s *s s s s s dT 3 n 2 sdT
+
PsdLv
% = "
d i v
*e"
(14.220)
t V(V#V+«s(14-221)
s
where now the transport RE = -O.MmEnE\>E(vE-v
phenomena
are given by
) -0.71n VT^ = - £ χ (14.222)
563
PLASMA INSTABILITIES
compare (7.33), for friction, and qE = Ο.ΊληΕΤΕ(νΕ for electron heat
electron thermal
- v χ)
ouvrent,
- K^VT^,
(14.223)
compare (6.18), where the
conductivity
is given by
κ^ = 3^6nEkBTE/mEvE, compare
(6.20), Qj =
and
-
(14.224)
(7.76b) .
Furthermore
K
KT =
j
V
ef = v w
V
-QI' I Q
for the dissipation The stress tensor
3.9nTk BTI/mIvI
3m
=
(14.225)
EnEkB
(14.226)
function. is given by ,(*vk'kl
_,_
II
2Χ
,.
->
(14.227)
where η1 = 0, compare (6.23), electron viscosity has been neglected (see p 129) and η χ = 0.96nTknTTT_ 1
1
compare (7.78). For the ion
D 1
acoustic
(14.228) 1
instability
(TE »
?j)r
γ - œpj/m^/mj ) (see p 477) the condition for applica-
bility of the collisionless
approximation
ωτ^, « 1, or γτ^ « 1,
λ
ΙΕη0ΐ71Ε/ηιΙ>>
is 1
' 04.229)
since the electrons are slowed down during τΕ. Now in Jhe refined Braginskii theory a static electric field EQ gives rise to a relative motion u0 between ions and electrons in a fully ionized plasma which instead of (7.62) is now expressed by = 0.51eE
£ o
/m^vr o' E El
(14.230)
according to (14.220), (14.222). This relative motion can excite an ion-acoustic instability. Under the assumption k\D « 1, nE^ = nj-| , ToE = TQl, with neglect of electron inertia, linearization of the equations yield for frequencies in the range τ E k^T Q /m E » ω » » τ 7 ^ 2 Τ 0 /^ χ the result /9.26/ ω^ = k(8k
T /3m ) ^
/ 2
(14.231)
MICROINSTABILITIES
564 and
γ =
m ku £-(4-—2) -|(0#96 ωρ ; 3 mITE\2
+
3 3β 8
k^k
T τ 5_o^ f ^
(14 .232)
where uQ is given by (14.230). Equations (14.231), (14.232) now replace (13.218). Ion acoustic modes are In contrast to a colliagain excited for UQE > c^j. sionless plasma, instability is possible even for ToE = T 0 J· T ^ e results so far are valid for a fully ionized plasma. For a weakly ionized plasma collisions between the charged particles and neutrals have to be taken into account and R according to (14.222) has to be modified. Then one obtains the result that and that for modes grow for u > (kßTE/mj)^/^ < k(kBTE/mj)*'2 one has ω ρ = k {kBTE/mI) 1/2, vEmE/mI and for longer wavelengths (vEmE/mj > γ - mEvE/mI compare > k (kßTE/mj) î'2) ω ^ = kuQ, γ = k u^rrij/m^, (14.2187. Ion acoustic modes dominate the Bernstein modes for homogeneous and inhomogeneous plasmas in the re< v/ü « 1 gime of small collision frequencies (krE)~^ for high frequencies /14.259/. This leads to a collisional destruction of Bernstein waves in the high frequency regime in which the modified Gordeyev integral may be reduced to the plasma dispersion function /14.260/. Since drift instabilities have been extensively discussed in chapter 13, we content ourselves with some additional remarks. In /14.264/ the structure of
the oollisional
drift
instability
has been examined
in several density regimes. For kz\E > 1, where λ^ is the electron mean free path, the drift mode is isothermal and should be treated by a kinetic theory. In the finite heat conduction regime kz\F < 1 the threshold is reduced at low densities (vj_ < 0.1νμ) and increased at high densities (v_^ > 0 . 1 V M ) . In the energy transfer limit (k2XE < (mE/mj) ^'2) tne instability is adiabatic in a fully ionized plasma and isothermal in a weakly ionized plasma. The current driven oolli-
sional drift instability (and the hydromagnetio current driven Alfvên wave instability) have been studi-
ed /14.265/ on the basis of the Braginskii equations (14.219) - (14.226) written for a magnetized plasma. The strong destabilizing effect of an axial current has been demonstrated theoretically as well as experimentally in a Q-machine. By use of the Krook model /14.266/ a dispersion relation for the drift dissipa-
PLASMA INSTABILITIES
565
tive instability in a non-uniform electric field has been derived. In /14.267/ the effect of longitudinal electron heat conductivity and electron-ion heat exchange on the drift instability in a strongly collisional finite beta plasma has been considered. The dissipative effects considered result in broader regions of instability than those found when these effects are not taken into account. The stability theory of drift waves has been extended to include dissipative contributions from electrons in various collielectrons and sional regimes and also from blocked inverted gradient profiles (VrcVT < 0) /14.268/, as well as to include the effect of collisions on impurity modes /14.269/. Inverted profiles, which are realizable by introducing cold gas, show in a certain collisional regime a substantial gain in stability over normal profiles (VnVT > 0 ) . Cross-field gradients of guiding-center drift velocities excite /14.270/ several high-frequency (ω > ßj) electrostatic velocity gradient instabilities {longitudinal electron drift velocity gradient instability, cross field velocity shear instability, cross-field two-stream inguidstability with ion velocity shear, collisional ing center drift instability, etc). These instabilities can enhance electron-electron and electron-ion collisions, which in turn destroy the velocity gradient. In pulsed turbulent heating experiments this effect may appear as anomalously rapid current penetration {anomalous skin effect). The theory of collisional drift waves in two-ion(low beta) plasmas /14.271/ shows that a high species ionic charge and a TE > Tj have destabilizing effects. In /14.27 2/ the influence of ion-neutral and electronneutral collisions on the drift dissipative instability in an afterglow plasma is investigated. An interesting effect has been found /14.273/: while the ordinary electrostatic drift wave is stabilized by either high beta effects or an admixture of cold plasma, a compressional drift instability is shown to be excited under the same conditions. When the cold plasma density exceeds a certain threshold, the compressional Alfvên wave is destabilizing either by inverse transit time damping of ions or by inverse Landau damping associated with resonant particles diffudrifting due to magnetic field gradients. The /14.274/ appears even if the sional drift instability conditions for drift dissipative instability are not fulfilled. Clearly, all theoretical predictions de-
MICROINSTABILITIES
566
pend on the model which is used: when the effect of electron-ion collisions on drift instability is investigated it is found /14.27 5/ that the Krook model gives for small collision frequency a destabilizing effect, whereas the Fokker-Planck model indicates stabilization. For large collision frequencies, the growth rates increase for collisional drift waves, but the growth rate obtained from Fokker-Planck model is not nearly as high as in the Krook model. Growth rates tend to vanish in both models for very large collision frequencies. Another mode which is also modified by collision is the whistler wave. The collisional whistler instability is an electromagnetic microscopic mode /14.276/ which draws its energy from the deviation from the Maxwellian distribution due to an electron peak. In a partially ionized collisional magnetized plasma electrons are released, e.g. by photoionization. The generated monoenergetic peak in the electron distribution function is assumed to survive until it is destroyed by an instability. Due to a phase shift and the feedback due to collisions, the whistler wave splits into two branches, one of which becomes unstable under certain conditions. When frequent collisions destroy the non-maxwellian peak of the distribution function before the instability sets in, then the growth rate is zero. Collisional cyclotron frequency phenomena like the gyro-relaxation effect or ion cyclotron resonance heating are important in plasma physics, see chapter 21. Here we will be concerned with the collision-
al Alfvén
cyclotron
resonance
instability
/14.277/
and the collisional cyclotron instability /14.278/. The first instability is of interest for mirror devices. Various problems have been discussed. The stability of electromagnetic ion cyclotron waves propagating obliquely to a parallel current in a fully ionized collisional plasma has been investigated. It was shown that when a certain threshold current is exceeded, instability results through parallel resistivity, thermal conductivity etc. /14.279/. Longitudinal (electrostatic) ion cyclotron instabilities in a magnetized collisional plasma are discussed in /14.280/. Of physical interest may be the collisional cyclotron instability in a Ramsauer effect ionized gas /14.283/. A collisional Weibel instability is treated in /14.281/. The guiding center kinetic equation with
567
PLASMA INSTABILITIES
Fokker-Planck collision term is used in /14.282/ to
study the microscopic oollisional
tearing
instability.
It also has been shown that toroidal effects can stabilize the tearing mode, whereas contraction of the current channel is destabilizing. An electric field
driven
collisional
flute
instability
has been report-
ed in /14.284/. A Fokker-Planck type investigation of plasma oscillations in a constant electric field is presented in /14.285/. If the solution of the kinetic is written in the form equation df/dt = {df/dt)0 f (e ,\i,r ,Q ,t) , where ε the kinetic energy, μ the magnetic moment, the drift kinetic equation (13.73) may be written in the.form df/dt +3//3Θ·Θ + d f / d e · έ = Here Θ = u\\Be/rB, ε = -eEu\\BjB, ufj = = iîf/*t)ç.
= 2 (ε - \iB) /m. Then pitch sion)
angle
scattering
is described by the Fokker-Planck term
{\y-dvffu-
where ν(ε) = (<ε>/ε)3/2<ν># if collisions are included into the Bohm-Gross dispersion relation (9.62) reads /14.301/ (9.62a) where γ^ = 1 or 3, compare (9.158). 14.7 KINETIC THEORY OF MACROSCOPIC INSTABILITIES Macroscopic instabilities have been defined as those instabilities which can be described by a simple fluid model (p 251). Since the fluid equations (chapter 6) can be derived from kinetic theory (chapter 5) it must also be possible to investigate macroscopic instabilities within the frame of a kinetic theory. Macroinstabilities are such slow processes that the displacement current may be neglected, and the gyration radii of the charged particles of both signs may be assumed to be zero. Hence the discussion of macroinstabilities is usually based on magnetohydrodynamics (chapter 11) or, for a collisionless plasma, on the drift approximation of the Vlasov equation (quasi-magnetohydrodynamics, see pp 88, 104 and 381). For plasma motion across the magnetic field, the drift approximation may equivalently be replaced by the CGL-theory, a dissipationless magnetohydrodynamic description with anisotropic pressure (see pp 103 and 272) .
MICROINSTABILITIES
568
The comparison theorem (see p 271 and /11.40/) . < 6WnrT, (14.233) δ^ < δ^ CGL9
quas%
which says that a system is stable both in quasi-magnetohydrodynamics and in CGL-theory when it is stable according to (11.22) in ideal magnetohydrodynamics, warns us that stability in CGL theory does not assure quasi-magnetohydrodynamic or magnetohydrodynamic stability. On the other hand /1.20/, if collisions are taken into account in kinetic theory, one has 6W ηη > &W (14.234) coll
so that stability is not destroyed by collisions. What now is the collisional domain in which low frequency (ω « Ω) modes may be described by ideal magnetohydrodynamics (chapter 12), and when do these modes have to be described by appropriate low frequency kinetic methods? In order that we may neglect resistivity and assume an ideal plasma, according to (6.51) we should have \i0avQl » 1. Furthermore, to be able to assume that the plasma may be considered as a fluid, (2.56) or v » ω must be valid, see also (14.212). With ω - vQ/l, a given by (7.43) we obtain 2 e n„v l/m„ » v » v /I, E o E o
or
_„ (14.235) 9 3x10 nl » 1 (cgs units) for the condition that ideal MHD may be used for low frequency modes (ω « v, ω « Ω) propagating in a plasma of characteristic length I and velocity i>0. In a typical fusion plasma (14.235) is not satisfied. Since we have discussed MHD theory in chapter 11 we will here investigate the collisionless microscopic theory of low frequency (ω « v) phenomena. To do this we consider the order of magnitude of the terms in the Vlasov equation (5.6). They are written for 3 = 1 (MKS system). The terms whose magnitudes are to be compared are ω ^
v
0/1"
°thl^
'
eE mc
/ thI'
and
e#/m = Ω. (14.236)
For slow motions vQ < c+^j- and the second term e thl/T- y> ω · T h e ra"tio of the third term to the second / (ctj2j/l) . These terms are of the same is (eE/mcthI) - ^BT ~ eE^ ' a n d w e assume this to be order if mo\}lI the case, see also p 14. When the last three terms are
569
PLASMA INSTABILITIES
approximately equal and larger than the first, we have with v0 < c-thjr £ » λ^, ω « Ω, eB/m = Ω » ω or rL « I (14.237) (MHD-ordering, see p 101). For finite Larmor radius magnetohydrodynamics (drift approximation, adiabatic theory, quasimagnetohydrodynamics) one has the low frequency ordering ω/Ω « r2L/l2«
1
(14.238)
(finite Larmor radius ordering). A third possibility is the CGL-theory (double adiabatic theory) comprising ω ^c+hk-r infrequent collisions and large magnetic fields (expansions in powers of 1/5) In order to derive some general results /1.99/ we now reduce the Vlasov equation (5.6). For 3 = 1 and drift velocity u = c + vD, where V-Q is the electric defined by (4.6), (4.82) we obtain
|| + s 7/+ s.v, - ^ . f dt
D
dt
(14.239)
33
£ - (îDn+5).vyDnde || + mH..-ML II 83|, + m[5χ5]·|Ι de = 0.
Here (dc/'èt)u = -ivO/ît, (dc/dx)u = -ZvD/dx and vD according to (4.82) (so that E^ may be eliminated) have been used. We now introduce cyclindrical coordinates (9.180) and define D
= TZ + £ n # v + ^ιιΤ^-/ dt
D
(14.240)
II OX..
where x\\ is the coordinate in the direction of the magnetic field. Since the plasma and magnetic field are both inhomogeneous, the unit vector î = Î/B along B is not constant (as in chapter 9 ) . Therefore in c,. = (
c. = ~c - c..î
(14.241)
b has to be differentiated \£hen the transformation (9.180) is made. Since £· (df/dc^) = 0, (14.239) then becomes
-1·
-1
"
(14i.242)
MICROINSTABILITIES
570
where β\ = (cicos φ,
2 1
-y
->
2
m OX,.
2
and 3/3ci = 2cj_3/3e|. The condition for single valuedness in φ (see p 230), (integration over φ from 0 to 2π) plus an expansion in powers of rL/l (Eu «E\, E\\ - TjEi/lt - see also (4.13)) gives in rirst^order Ej_ - ~E, t>£ = (Ε*Ϊ>)/Β - v the equivalence of (4.82) and (6.70). Use of (6.43) and of
at'1
=
= -—'dt
~*'νΎ - °\\-2' P
(14·244>
- compare p 73 -, finally results in
2
s + $.^2l II
c v
2-
(14.245) +c
hl)M- _ y JB. _i£= o
ΊΙ- 2p/ 3 o M
u
m 8x„ 8c,,
instead of (14.242), since / depends on ej only through μ, so that dy = -yd£/£, and 2 3//3cJ = 3//53y, compare with (13.73). For a stationary (3/3t = 0) and static (v = 0) plasma one has a !£° + % . . ^ - ϋ ^ ^ 2= ο Il dx,, m II 3c.. m dx.. 3c..
(14.246)
with the equilibrium solution
fQ - f 0 ( » . K ) .
x
= I e 2 + M-£J^^ 2 4 7 )
This solution may be verified either by inserting it into (14.246) or by showing that y and K are concharacterstants of the equation of motion (Lagrange istics equations) associated with the partial differential equation (14.246). The characteristics equations of P(x,t,c,f)ft
+ Q(x,t,c,f)fx
-Tlx.t.cf) are 76.3/, /14.286/
+ R(x,t,c,f)fc
= ( 1 4
·
2 4 8 )
571
PLASMA INSTABILITIES dt P
=
dx _ de Q R
df Te
(14.249)
Therefore the equation of motion associated with (14.246) is (compare (4.15)) m- d*
-H
35\dx dx,,'dt
μ eE „ - m
(14.250)
and has the solution K according t o ( 1 4 . 2 4 7 ) . I f v does not vanish but is small, we s o l v e ( 1 4 . 2 4 5 ) by the linearization / = / o + / v £|| = £||o + £v 5 = 5 0 + s1 and obtain for f^ 85 -
_3_ 3 * + c,H3as. = —o. ■ v / 0 -
Il 3x ,\mJ
111
Λ
3e„
dx„ - ^ B II
dt
(14.251)
,) - *ç]
9/, O
Here 3/0/3* = 0 and (3fo/3J0 (3*/3ÖN) = 3/0/3tf|| have been used. The structure of (14.251) is very similar to that of (9.151). We may use for its solution the method of integration along the unperturbed particle trajectories c\\, see section 13.4. This gives with ν^ = ξ, see p 271, f
Λ
(x,c..
, t)
= f.
{x
ΊΙο
f-vf
,0)
(14.252) ΊΙ 3tf..\m 3a:, J
||1
II
m 1/
II
P
9/. 2 J 3X
compare with (13.30). If ξ η is the projection of ξ on the normal to the drift surface ψ = const, the term containing f*Vf may be written Jdt'ξ η 3/ 0 /3# n = = /dt'f·νψ3/0/3ψ. If νψ can be neglected, then the disturbances are determined only by ξ (magnetohydrodynamic
limit).
In the low
beta
limit
,BQ
being a
vacuum field has a minimum energy. Low energy modes do not change the magnetic field and they are therefore electrostatic modes. If J = §>c\\dx\ = i>cf[dxu/c , is introduced, we obtain from (14.247) dx. J =
2(K-
\xB/m) + 2eJE
dx
/m
(14.253)
MICROINSTABILITIES
572
On the other hand, if f<\ varies slowly in comparison with orbital periods τ we may average^over the fast orbital variations. We then have for / = f^(x^\,c^,t) ~ fîteoiCWoit) o n a drift surface (νψ = 0) the result da:, dt
f
Tjdt
τϊΊΗ
— {\ m J ~ N 1 îa?NM \ 'H3a?
(14.254)
8
Λ>
2% + ox II rnM II p 2 ZK
By use of (14.253) it then can be shown that and that (8.142) follows. The MHD energy f = f(K,\i,J) principle and a comparison theorem may now be derived. For low frequency instabilities with weak electric field, the electric drift is small. Then after some algebra and a second order expansion the Vlasov equain /7.5/, /1 .99/ tion results for f(K,\i)
If + V v '
+c
n^
=
°'
(14 255)
·
where now V
D
=
ΊΓ + T^V
+
-Ω-<μ>·
(14.256)
Ωρ Then in the denominator of the dispersion relation resonance, see p 370). So the ω ± κ·νρ appears (drift modification introduced into the mag netohydrodynamic by a refined kinetic treatment (orbital eftheory denominators fects) is the appearance of resonant /14.480/. Another which may produce drift instability Larmor radius stabilizamodification is the finite tion, In chapters 11 and 12 one finds that macroinstabilities are generated by pressure anisotropy (Pu * PD 9 by external forces (gravitation in connection with density gradient, Coriolis and centrifugal forces) and by gradients like Vp, ν·Β, In, VT, Vv, Vn, \7Φ. Thus mag ne tohydro dynamic instabilities are fast gradient instabilities (see p 350) and are due to inhomogeneities. They are therefore closely related to drift instabilities /14.495/. After these general remarks we will look more flute closely at some special macroinstabilities. The (k u = 0) may be discussed on the basis of instability (14.166) /1.20/, /4.11/, /11.11/ or of a canonical flute treatment /14.287/. For the gravitational
573
PLASMA INSTABILITIES
(11.26) the well-known result (p 277) is recovered that for instability the gravitational field must be directed toward the direction in which the plasma density decreases. This can be shown explicitely in a kinetic theory as follows. Consider a plasma slab suspended against gravity by a magnetic field along x· Since the constants of motion are K = mu /2 + mgz and py = muy + eAy/c, and since y is a cyclic variable, f0(K,py) the equilibrium distribution function / 0 = for A = -BQz is given by (see p 368) / \3/2 2 fr, = oUTπ / exp[-a(u + 2gz)] ·[ 1 - ε ' (z -y u /Ω) 1, ° ° (14.257) where a = m/lkWI and pnl/m = unj - Qz has been used. Now n
n = \f
du = n (1 - ε'z)exp[-2agz]
(14.258)
gives , ε' ~ , , din n -ÔT = 02ag + T ^ 7 r 7 - 2ag + ε' which goes over into ε' for g = 0, see p 3 68. For perturbations kz » ε and low frequencies this yields a dispersion relation in local approximation which reads (n = 0) /9.26/, /1.20/ for ions 2
fc =JL compare
e
~
2a
k r
H - 1 + «V
(13.54)
L
£
'e
17 2 2 2 L
Here ω = k
8ε = Im ε = γ-τ—
T\k
y
k D v,
y i (lk2r2)-z(
rT L
V
ν^. F i n a l l y , 7
ε' exp
)} ,
z
th' (14.259) from (13.56)
22
JL
D
z
th
72 2
(14.260)
If the plasma is supported by the magnetic field against gravity, then (dnQ/d;s) < 0 f ε1 > 0 and we have growth of the instability, ε is the (dimensionless) dielectric function. When the flutes are due to thecurvature of the magnetic field, we may replace mg -* F, see (4.21) , (4.95) (pp 58, 73) and get (4.96) for abrupt change of density, or /4.11/, /11.11/
-r-
±n n 2ΚΒ(ΐΕ + ΤΙ) dx rrtjp
(12.261)
MICROINSTABILITIES
574
for continuous change of density. According to (11.30) a flute instability can develop if the field curvature is convex outward. For a self-gravitating plasma with stratified density the dispersion relation has also been investigated /14.448/. Neutral gas friction and viscosity are stabilizing , but finite resistivity is destabilizing. If the plasma is bounded by conducting walls, flute modes are absent (line tying stabilization) and also modes withfe..* 0 should be considered. Such modes grow if φ ? < \gz% \ , see chapter 16. Since feII - π/Ζ, where Ϊ is the distance between the conducting walls, we obtain the instability condition Ζ2β > π2ρα.
(14.262)
Here p is again the curvature radius of the field lines, a is the transverse distance, coil diameter or plasma radius (e.g. the distance from the axis of a mirror device to the Joffe rod, compare pp 149 - 150). So for sufficiently long plasmas instability occurs (ballooning instability). Since the curvature can change along the magnetic field lines, the growth rate γ (14.261) and also 3 become functions of the distance along the field lines, andfe,.varies too. This makes local adverse curvatures important, if particles can resonate locally in bad curvature regions. Since a bounce resonance can alter J (14.253), for stability df/dJ > 0 (14.263) has to be added to (8.142), see also p 176. If the radius of curvature p depends strongly on position, p in (14.262) must be replaced by 1/i/1, where U' = 6U/U, see p 283. This gives /9.26/, /14.288/ the maximum pressure for which the plasma can be stably confined as 3^ * π2α2/Ζ2ί/' ,
(14.264)
compare (11.43). This β is called the depth of magnetic well. The ballooning instability which is localized in a region of unfavorable curvature develops with a growth rate /9.26/ Ύ - cthl//^
(14.265)
which could be derived from (11.32). We now turn to other macroinstabilities of very
575
PLASMA INSTABILITIES
low f r e q u e n c i e s a s w e l l a s l o n g w a v e l e n g t h s ( » r^) i n high b e t a p l a s m a s . In t h e magnetohydrodynamic approxtensor imation Ckietfaj/Qj - ω/Çlj <& 1) t h e dielectric of a m a g n e t i z e d a n i s o t r o p i c plasma ( 9 . 1 9 4 ) becomes / 1 0 . 1 4 / by e x p a n s i o n of t h e B e s s e l f u n c t i o n s and f o r n = 0
s
ε
xy
Ω
= -ε
I
L*
= { ) L"L
yx
ε
= ε
Λ
ε
=
-ε
^
[ where
„
ί Ω2
*
u
2
2 ^
1
ω2
ω^
/kt
L
ω„ fe^
2
+°°
=
>
s/
k,
2 β " + ^ < >ιs σ ω 2fe Is
7
2^ 2 ^
ω π ,k,
S
(14.266)
IIM
s ss
z
,2, 4
4ω 2
= -il-g.{ ωΩ V
υ
/
^ LU 2ω
.2<
S
n
ωω
ι-, 2
S 06 /Ci · s Ω ^,, β II
2S
^
kn
ωπ_ / ,, Λ ωΩ ωΩ \V
2 ,
2ω
7
>
2^
7
2 ^ - i
k.
>\\ J
2π
f dc||f2uc1deicn/'os(ei,c|1)
(14.267)
and /
Λ Ο
<°Ι'
dc..l27rc.dc.—
IIJ
1
β
±
ΙΙ>
ü_y.
l ( e -ω/fe )
2
(14.268)
Here Ω 8 includes the sign. When we try to redetect waves which we know from section 9.6 we find that 52fc2/u>2 = εχχ
(14.269)
MICROINSTABILITIES
576
describes the incompressional shear Alfvên wave (which becomes the firehose instability) and that 52^2/ω2 = ε
+ ε 2 /ε (14.270) yz zz yy describes an ion acoustic wave and the compressional magnetoacoustic wave (which becomes the mirror insta. Substitution of εχχ from (14.266) into bility) (14.269) gives, for a Maxwellian distribution function, for u>/k\ic.f-foj « 1, for ω2/&2
= 1 -ΐΣ(β„β
- e
i s
),
(14.271)
\\°A
where (14.272) One sees that (14.271) is equivalent to (14.85). Similarly for large ez (decoupling of the ion acoustic wave) one obtains from (14.270) the result for ions
» = -
i^'-fc)}
+ 1 + s Instability (Im ω > 0 for ^expi-iut) (nearly parallel propagation) if
-ΊΙΚ-*Ι.)
< 0
(14.273)
occurs for ^ » k . (14.274)
or for A:,. «:fej_(nearly perpendicular propagation) , if 1 +
p l s ( 1 - ß la /ß lls) < °·
(14 275
·
>
Condition (14.274) describes again the firehose instability, see (14.271) and (14.275) describes the mirror instability, see (14.88).
577
PLASMA INSTABILITIES
In addition to the comparison theorems mentioned earlier, a comparison of unstable modes in ideal magnetohydrodynamics and guiding center plasmas has been for plasma inmade /14.289/. Also an energy principle stability against low frequency electromagnetic modes for Vlasov plasmas in the magnetohydrodynamic approximation has been given too /14.290/. An energy principle for two-dimensional resistive instabilities leading to a necessary and sufficient criterion for stability has also been given /14.450/, /14.432/. A consequence of this criterion is that the current density in a plasma with arbitrary cross section should not increase to the outside. Otherwise the plasma becomes unstable against resistive instabilities (cur-
rent
density
gradient
instability,
current
gradient
instability /14.451/). A special optimal current density distribution stabilizes this resistive kink mode. The oscillations driven by current gradients have sawtooth form and have a peak at radial location rQ. Ideal MHD-theory predicts a singularity at rQ. Inclusion of resistivity and boundary layers creates a reconnecting mode /14.502/ which reconnects the magnetisic field lines at rQ and which generates magnetic lands. Kelvin-Helmholtz modes are studied in /14.489/. 14.8 TRAPPED PARTICLE INSTABILITIES
Trapped particle instabilities /13.49/ have been mentioned on several earlier occasions, compare volume 1 of this Handbook p 384 and section 14.2 in this volume. They may be classified according to the trapping mechanism as follows: 1. Trapping by electric fields /13.68/ 1.1 deviation from Maxwellian distribution /14.6/, see p 463 here, 1.2 trapping in a moving strong electrostatic wave /14.16/, see p 464 here. 2. Trapping by magnetic fields 2.1 collisionless plasma /13.40/, /14.291/, s^e volume 1, p 385 2.2 dissipative (collisional) plasma /13.40/, /13.64/, see volume 1, p 390. Let us first consider the basic electric trapping process. If we replace fix) in (3.4) by -eE or by βάΦ/άχ, where Φ = 0cos kx for an electrostatic periodic potential (or potential of a travelling electrostatic wave in the wave frame of reference) we obtain as equation of motion for a negatively charged particle
578
MICROINSTABILITIES
mx - -βΦ fesin kx - -βΦ k x (14.276) o o which has for small amplitudes kx « 1 the solution bounce frequency ωβ x = a;0sin u>et, where the electric is given by ω = fe/οΦ /m = lekE /m. (14.277) e o o For a sinusoidal travelling electrostatic wave - ut), (ω - ωρ^) in the limit of zeE(x,t) = EQsin{kx ro amplitude EQ + 0, the electron is free-streaming For EQ Φ 0 a transformation to the x(t) = x0 + u0t. u1 = u - ω/fe brings us back to wave frame a: ' = x-bst/k, (14.277) with EQ = Φ0k. For particles moving near the phase velocity ω/fe of the wave {resonant -particles) thewith Ifex1 - 2ηπI « 1, n = 0,±1,±2..., (linearized ory) , the replacement of the sine by its argument is justified. For E0 = const exact integration of (14.276) gives .,2 ^ βΦ(χ·) = J/ = const. (14.278) It is clear from (14.278) and Fig. 51 that a particle with a kinetic energy W in the interval -βΦ0 < W< +£Φ 0 is trapped by the wave. Hence we have u' = ±/4βΦ /m (14.279) max o for the maximum possible velocity of a trapped particle in the wave frame. Particles with u1 > u^ax or W > βΦ0 cannot be trapped. In phase space (14.278) defines a separatrixf see Fig. 51. The distribution function f(x\u\) has peaks in phase space wherever there is a potential trough. Trapped particles exhibit periodic motions in the wave frame. An untrapped particle decelerates and accelerates when passing over the potential, but does not reverse the direction of its motion. When E0 is slowly varying in time particle orbits may still be determined from (14.278) as long as the adiabatic condition din EQ/dt « ω β is satisfied. In the linearized theory (kx1 « 1, u' - 0, u - ω/fe) all trapped particles are resonant particles (cf pp 223 and 232). If particles are trapped by an (inhomogeneous) magnetic field, the equation of motion of the guiding center is given by (4.15) and the bounce frequency is given by (4.45). We assume that the positive ions form a stationary neutralizing background. The damping of the oscilla-
579
PLASMA INSTABILITIES
(χ')
-βΦ
u n t r a p p e d p a r t i c l e , J/>+e
-βΦ
o
\u* ( χ ' ) ί/>+£φ
ί/=+βφ
W=-e$
max Fig.
51.
Partiole
trapping
tions of negative particles is described by averaging over one ensemble of particles having velocities close to the wave velocity {linear Landau damping). Particle trapping is often said to be a nonlinear effect since strong waves act back on the particle distribution function, see p 583. Then also the damping process is nonlinear. The linear theory breaks down when the condition for validity of the linearization approximation is no longer satisfied, i.e. if τ^.ρ« τ L = = γ"1 /14.292/, /14.293/. This breakdown occurs also see p 220. Here γ is the Lanfor dfs<\/du « dfso/du, dau damping rate given by (9.169) and τ+r is the peritrapping time scale given by τβ or τ^ {bounce ods) . In such a situation ( τ ^ « τ^) trapped particles strongly influence the distribution function long before the wave damps or grows. According to (9.169) for k\DE = 10"1 and τ * ω-1, = lO"10[s] one has TL = 10^[s]. For τ^ v - τ^ or xe we have either - τ or τ ^r » τ, then the wave makes many oscillaτtr tions during one bouncing period of the trapped particles. No waves exist with ω~1 - τ » τ ,_. For feXX n77 - 1 Ό ΡΕ
tr'
DE
MICROINSTABILITIES
580
(9.169) is no longer valid and τ^ - τ, τ^ν » τ^: the wave damps out long before a particle can finish an oscillation in the potential well of the wave /14.294/. From τ * ω"^, τtT - ω~1 we obtain from (14.277), (2.19) the estimate τ 2 / τ ? * kE /4i\ne « 1 (14.280) o tr so that T^r » TL is satisfied. The inequality x^r » τ assures that the field amplitude EQ is sufficiently small to guarantee validity of the linear theory. For times > τ t r the linear theory breaks down anyway. For larger amplitudes the monochromatic electron oscillations exhibit strong distortions due to nonresonant wave-particle interaction and the adiabatic condition is no longer satisfied. Furthermore, (14.277) and (14.279) become invalid. We discuss first the integration for EQ = const. From (14.278) which is exact, we obtain dx
'
t = I
— + const.
J S2(W + βΦ cos
Defining the trapping τ|
(14.281)
kx%)/m
time by
= m/e$ok2
(14.282)
and the modulus κ of the elliptic Λ
functions
by
W + βΦ
1 _
(14.283)
O
2 " 2βΦ K O we may write (14.281) in the form
j
kx'/2
t = κτ^ tr
I
J
kx'/2
dz
rV1 - K sin—s Λ
r
^ 2
By use of the normal elliptic kind /14.296/ z d2
integral
(14.284)
O
of the
first
' (14.285) l/i 2 . 2 , o V 1 - K sin s' this may be written as t / κ τ ^ = F (kx* /2; κ) -F (kx* /2; κ) + + t0/
= f
+ t .
7T~ 2
J
581
PLASMA INSTABILITIES
which are not oscillating in the potential trough. For K 2 > 1 the particles are trapped and oscillate. To see this periodicity clearly we introduce (14.286) v = sin η = KSin(fea;f/2) into (14.284). Introducing κ' = 1/κ we obtain dV + const. (14.287) * = τ. tr\ V λ - v V 1 - κ' ι; The integral in (14.287) is given by F(n,Kf) In order to eliminate the integration constant__we integrate (14.287) over v for such a time span τ that sin η makes a quarter of a full oscillation, i.e. η from 0 to π/2 or v from 0 to 1. We then obtain the complete normal elliptic integral of the first kind
π/2 Κ(κ)
=
dz
f
J
ΙΛ
O M - K s
Κ(κ)
=
= F(TT/2;K)^
2.2 sin z 2
^d 1
4
+ ^ - + ···) dt>
f
J
+ X
ΐΛ
2 1/Ί
(14.288)
= F(7T/2;K·) .
2~~2
Then (14.287) becomes 7 = τ
Κ(κ') = F(TT/2;K')
(14.289)
since F ( 0 ; K ! ) = 0. Now the bounce period is given by ^4τ > 4i£ r . For other arguments and again #' = = x% U = 0 = 0 (14.287) reads o (14.290) * - tQ = TtrF(n(Ä") ;κ· ) which gives the solution x% (t) of (14.278). It is useful (but not necessary) to introduce the inverse function of the normal elliptic integral of the first kind, i.e. the Jacobi amplitude function η = amU/τ
;κ·) Ξ F ( ~ 1 ) .
(14.291)
Then (14.290) may be written sin am[(t-t o )A t r ;K']E sn [( t - t ) Λ ^ ; κ '] =KSin(fcc'/2)
MICROINSTABILITIES
582
where sn i s t h e Jacobi x% (t)
Then
sine.
= l a r c s i n ^ ' s n [(t - tQ) A t r > ; κ ']] .(14.292)
This satisfies the exact equation of motion (14.27 6) xQ = x% (0) is determined by tQ and for Φ 0 = const. the energy W since κ depends on W. In plasma physics the equation of motion is the characteristics equation associated with the kinetic equation, such as e.g. the Vlasov equation, see (14.249). In the wave frame x% , u* = u-w/k, ti = t the Vlasov equation associated with (14.276) reads, for electrons in an unmagnetized plasma (ions at rest) , ^dt Ξ dt ^
+ u'-^Ç— E sin fcs'|^ = 0, όχ' πΐτ-, o du
where / = f(x',u',t').
If the solution of (14.276) is
g i v e n b y x% = x* ( t 1 , x ,uQ), 1
(14.293)
E
i>' = v% (t% rXQrUQ)
where
xQ = x (0), u0 = u (0) then according to Jeans1 theorem, the solution of (14.293) is given by an arbitrary function of xQ and uQ namely f{xl,u*
l
,tl)
= f(xQ(x%
,u% ,t%),uQ(x*
,u% ,t%)
,0) (14.294)
since / is constant along particle trajectories. The function xQ (x* ,u% ,t') (and u ) have to be calculated or from e.g. (14.292). If / depends from x% (t% ,x0,uQ) is on time, the travelling wave E(x,t) = EQsin(kx1) modified according to the Poisson equation dE(x',t%) 3a:1
= -^e[jf(x%
,u% ,t')du'
- nl(14.29 5)
where nQ is the neutralizing ion background at rest. If the arbitrary function in (14.294) is a Maxwellian this leads to Landau damping, and for τ t r « τ L to a slow variation of EQ or Φ 0 with time. In order to investigate this slow variation we assume τ « τ^r « τ^ (here τ is again -ωΰΐ). τ « τ^ ρ guarantees that EQ is small and that the distortion of the monochromatic wave is slow, while τ. « τ^ assures that the rate of charge in EQ is small on the characteristic trapping time scale. Having obtained in (14.292) the exact solution of (14.276) and in (14.294) the solution of (14.293), both for EQ - const, we now determine the (slow) variation in E from the conservation of field energy plus particle energy, since the solution of (14.294) using (14.295) is practically impossible /14.297/, /13.293/. Similar results have, however,
583
PLASMA INSTABILITIES
been obtained by a perturbation analysis of the nonlinear Vlasov-Poisson equations /14.298/. For strictly monochromatic waves with spatial periodicity 2i\/k, the energy conservation f+TT/fc _d_
at
ÇL·2(x,t)/8Έ
+-γ
\u2f(x,u,t)dujdx
= 0
(14.296)
Ι-π/fe
is exact and may be derived from the Vlasov-Poisson system of equations. The calculations based on (14.296), (14.294), (14.292) are quite cumbersome /14.297/. Therefore we content ourselves with exhibiting the results. For a Landau damped electrostatic wave E(x,t)
= E {t)sln{kx
- u)t)
(14.297)
(14.296) gives for an initial condition f (x* ,u% ,0) = / (w1 ) + f 1 (uf ,0)cos kx% (14.298) and after some tedious algebra the approximate result dE ( } d tV
= 2y(f)E(t')
(14.299)
for τ « τ ^ , u'2 » eEQ/kmE = ω|/&2 (fast particles) is obtained. E is the spatially averaged electric field energy density E^(t9)/16π and y(t') is the nonlinear Landau damping rate, or better: quasilinear Landau damping rate, which is expressed by an integral over the normal elliptic integral of the first kind and over a sum of sine-terms. For tf « τ+r the untrapped electron terms dominate and y(£f) reduces to the linear Landau damping rate. In this sense it is often said that particle trapping is a nonlinear process /14.301/. For t' » τ t r the integral gives oscillations phase-mixing y(t') - 0. So for t'« x^r we have linear Landau damping, for tf > τ^ν there is an oscillation of E with a period of approximately τ^Γ and for t1 » Ttr, E tends to a level E(«>) < E(0). The trapped electrons support the wave: Landau damping disappears for waves in trapped particles. For γ > 0 a linear instability appears which later on has an oscillatory structure and saturates for t 1 » ! ^ at a level E(o°) > E(0). In both cases (y > 0, γ < 0) the time-averaged electron distribution function forms a plateau /14.299/, /14.300/ in the region where ω/k - u+r. The formation of such a plateau is charac-
MICROINSTABILITIES
584
teristic for the wave-particle interaction, see Fig. 52. The presence of collisions, which always tend to reduce the distribution function to a Maxwellian, disrupt the plateau formation and play a decisive role in maintaining the linear Landau damping for long times. For this reason not only the linear collisional Landau damping
/14.302/ but also the
dau
dam-ping
nonlinear
collisional
Lan-
ω/fc
/14.300/ has
Fig. 52. Plateau formation been investigated using a simplified one-dimensional Fokker-Planck-Chandrasekhar term (5.20). It is shown that weak collisions (3 « γ) have a negligible effect on the initial (linear) Landau damping but are effective in preventing plateau formation and in maintaining Landau damping for times close to the decay rate of the linear theory. Experiments /14.304/ with k2\2 « 1 confirm, however, the prediction of a decrease in the damping rate from the linear value. Experiments also show the growth of sidebands of the fundamental wave. The appearance of these sidebands (a trapped particle instability) cannot be explained by a monochromatic wave. It is a typical nonlinear effect /14.342/. Trapped particle dynamics is important mainly for monochromatic waves. If there are several waves (wave packet) or if the field is localized the situation is changed. For a wave packet the trapping time does not have a precise definition /14.292/, /14.299/. For the weak electron beam instability {gentle bump on tail instability) the transit time may be much shorter quasilinthan T£r. This fact forms the basis for the
ear turbulence
theory
/14.303/. The nonlinear
Lan-
dau damping involves the beating of two waves /10.13/. Two high-frequency electron waves ω-|,^ι and ω2/&2 could beat to form an amplitude envelope travelling at a speed (ω2 - ω-j ) / (^2 " &1 ) ~ άω/dk. This velocity may lie within the ion distribution function and can generate energy exchange with the resonant ions. These ions see an effective potential due to ponderomotive force, and Landau damping or growth can occur. This damping provides a mechanism for heating ions with high-frequency waves (which ordinarily do not exchange energy with ions). If the ion distribution function is double-humped, it can excite unstable electron waves (modulational instability).
585
PLASMA INSTABILITIES
Up to now we have considered trapping in electrostatic fields. We will now discuss magnetic trap-ping. This occurs in the space between two magnetic mirrors. Magnetic particle trapping may also be used to visualize the containment properties of magnetic fields /14.481/. (Then uLR - const for toroids is used; Ü)£ = eB/m = eBQ/mR) . Examples of magnetic trapping may be found in mirror machines (p 64) , in toroidal traps (p 161) or in the magnetic dipole field of the earth /14.305/. According to Liouville* s equation (5.3) the phase space density is constant along the dynamical path of the particles /14.305/. In coordinate space this has the consequence that particles can be detrapped only by collisions, by interaction with electromagnetic waves or by nonadiabatic changes of the trapping field. On the other hand /4.6/ conservation of energy requires for static fields that a particle injected with a given initial energy must return to the initial position with equal kinetic energy and therefore be capable of escaping the confining magnetic field. The motion of trapped charged particles resp. their guiding centers in an adiabatic (see p 54) toroidal magnetic field is given by (4.15) or by u Ιί + u*B)-U: 8t + »¥II ds - —(% mE du = 0.
(14.300)
Under the assumption of small aspect ratio, we may assume that ε = r/RQ is small (compare p 340). Then the magnetic field components analogous to (12.67) may be written in the form /8.48/, /8.13/, /14.291/ B.φ = Bo (1 - ecos Θ) B
= β
β
(1
£COS
(14.301)
θ)
Q ο θ "" 2 When ε «
1
^ 7 = ^ — ( 1 + s c o s 0)uM o^ dp = mc 2 2 dî 2eB R ^U + u | | ) s i n o o
(14.302) θ
(1
+ e c o s
θ
) (14.303)
MICROINSTABILITIES
586
me — {u + u N ) (qcos Θ + 2eBR-
at
° ° + ^2·Θsin Θ) (1 + ecos θ) ,
(14.304)
where r in ε in (14.302) may be considered as constant. Thus (14.302) can be solved independently of the other equations which describe the departure from the line of force. After some algebra (14.302) can be transformed into d|
at
=
±
|!d|/
2 K
2_
1
_cos
(14.305)
R q on Here κ^ = (u^-uf)2eu^. Trapped particles have a small longitudinal velocity uy = u.. A turning point occurs for the particle at 0 Q defined by 1 + cos θ 0 = 2κ 2 , which is only possible for κ < 1. Thus κ = 1 distinguishes again the trapped particles from the transiting particles. From (14.305) we obtain the magnetic bouncing period Tb = 4-2-1 \
αθ
=
_2l_
K(K)
(14 .306)
Θ V 2κ - 1 - cos Θ compare (4.45). Here K is given by (1 4. 288) . For u.. « u (14.302) - (14.304) give /8.13/ the displacement Ar
=±
m5uc
^ 2 κ 2 - 1 - cos Θ.
l
eB
(14.307)
/ε
o In its motion along the magnetic field (un > 0) an ion drifts outward and inward in its reverse motion (u|| < 0) . The minus-sign is for electrons. The radial see (12.87). displacement is of the order rLqe~^/2, The combination of (14.307) with (14.305) gives the banana orbits Θ{r), see p 159. The displacement Δξ of the trapped particles along φ during one oscillation period is given by (14.308) where J is the longitudinal invariant (4.16) given by (14.309) where Ε(κ) is the complete
elliptic
integral
of
the
587
PLASMA INSTABILITIES
second kind. Since the motion of a particle in equilibrium is now known, the kinetic equation ^1 4.300) _^may be solved by linearization / = fQ + f<\ , E = Ε0 + Ε<\, where fQ is the equilibrium distribution function which can be written as a Maxwellian in which density and temperature depend on r. The perturbation f^ can then be determined from (14.300). Long and tedious calculations demonstrate that the presence of trapped particles may lead to the appearance of flute instabilities (trapped particle instability). The trapped particles in a force tube between local magnetic mirrors (in a toroidal device) will in general execute an unfavorable magnetic drift if the magnetic field falls off to the periphery. A small perturbation thus leads to charge separation which in turn amplifies the initial perturbation. Whereas in a mirror machine all particles are trapped, the trapped particles in a toroidal trap are immersed in transiting particles which can neutralize to a certain degree the charges of the trapped particles. By use of the expressions (14.306), (14.307) etc it is possible /8. 1 3/, /1 4. 291 /, /14.306/ to derive the growth rate γ2 * ω * 2 ε 3 / 2 where
or
ι
γ 2 * /Ξω ω*/2, ι
(14.310)
m ρ
ω*
=
B*L reBn ar
(14.311)
ω
= 2lqeckOT/eBr2
(14.312)
and where m
^
B
ωn
- ω *!τΗ·
(14.313)
p din n It is shown that the instability develops mainly in the external region of the torus and that its amplitude is small for |θ| < π/2. Since there is an effect of q% = aq/dr on the instability, it may be stabilized by decreasing q% , and the stabilization criterion reads dln ? < -1 (14 3-14) din r
2
{
I*.Ji4j
(stabilization by shear). There is also a critical density gradient such that stability occurs for /8.48/
MICROINSTABILITIES
588
"4·315)
-3E-; < itfl·· r
O
The strongest trapped particle instability develops when TE = Tj /14.291/. If the plasma is not isothermal (Τ^ Φ Tj) the instability range is broadened. Other growth rates are given in /11.11/, /9.26/ by where yf is the growth γ - yfkirLI·(ntrap/nuntraç)î/2, rate of the flute instability. When particles are trapped in the magnetic field of the earth, a similar instability occurs /14.307/, /10.14/. For trapping near the equatorial region of the mirror, the magnetic field may be assumed to be BQ{z)
= £ Q (1 +bz2),
b * 9/2R^,
(14.316)
where RQ is the equatorial distance to the field line. Then the magnetic bounce frequency is given by
r
^y-^f
(14 317)
·
where μ is the magnetic moment. The Vlasov-Poisson system of equations gives after some approximations the dispersion relation (14.318) where ωρ 0 is the plasma frequency at z = 0 and (14.319) y 0 being the magnetic moment for BQ. From (14.318) it can be shown that a beam of trapped electrons causes an instability at ω ^ ω^ 0 when it is mixed with the background plasma. Recently various details of trapped particles have been investigated. Trapped particle orbits with bounce frequencies close to the frequency of electrostatic standing two-dimensional toroidal modes can amplify these. Quasi-bananas have been found /14.308/ with radial width significantly larger than that of the known bananas. Their centers are displaced from the magnetic surface on which the modes considered are on collisionless excited /14.308/. High beta effects and collisional trapped particle instability, also in non-circular cross-sectioned Tokamaks have been considered /14.309/. Finite beta effects have a stabi-
589
PLASMA INSTABILITIES
lizing influence on collisional instability. A stability condition for the trapped electron instability which is valid in both the collisional and collisionless regimes has been derived from wave-particle interaction between the perpendicular phase velocity of the waves and the gradient B drift velocity of the electrons /14.310/. As far as electrically trapped particles are concerned , *the following research has been done: evolution of the electron energy distribution due to trapping in the potential of an electron plasma wave /14.311/, the generation of trapped-particle modes by injection of electrons in a plasma wave /14.312/, instability of particles trapped by an electrostatic wave field in a constant magnetic field /14.313// electron trapping in a growing electrostatic field showing that the average field at trapping is given by ekE0/m - π 2 ω 2 /16 for growth rates γ < ω/4 /14.314/. A quasi-linear treatment of the time evolution of a large amplitude electron plasma wave showed no instability at wave amplitudes given by βΦ0 < 0.1 kBT in contrast to experimental results /14.315/. Finally an oscillation-center formulation of particle trapping in the beat potential produced by the superposition of two coheren waves has been given /14.316/. For magnetically trapped particles many details have been discussed recently: the interaction of particles trapped in a magnetic field with coherent electrostatic waves has been investigated /14.317/, trapped particle instability in axisymmetric toroidal sys-
tems /14.318/ and the electron
drift
trapped-particle
instability have been investigated /14.319/. Several papers have been concerned with the destabilization of trapped-electron modes by unfavorable curvature of the magnetic field {magnetic curvature drift resonances, electron temperature gradient drift trapped instability, trapped electron drift instability caused by bad
curvature
/14.320/). It has been shown that
the stability of trapped electron modes driven by the magnetic curvature drift does not improve appreciably, when their frequency becomes comparable with the electron magnetic drift frequency. A new trapped-electron instability has been found. It is driven by a radial gradient of the electron temperature or by the electron drift in the unfavorable curvature of the magnetic field lines for high-temperature confined plasmas in which a considerable fraction of the electrons is magnetically trapped. The mode stands along the field, is symmetric about the point of minimum field
MICROINSTABILITIES
590
and has relatively short wavelengths across the magnetic field. Trapped electron modes have also been investigated in cylindrical geometry /14.321/. Reversed density and temperature gradients may stabilize trapped particle instability /14.322/. In connection with drift trapped particle instability Coppi recently found a new mode driven by drift which he called ubiquitous instability /14.323/. These modes have frequencies in the range oo^j < ω < ω£#/ where ω£# is the electron bouncing frequency, ω-^χ = TjU>^/TE and ω| is the electron density gradient drift frequency ω* = = -kcTßkß (din n/dr) /eBQ and where ω^j < ω^-j < ω ^ < ^tE (spectrum of standing waves). For the standing electrostatic toroidal potential perturbations the expression Φ 1 = Φ (Θ ,r) exp[-i(ßt /
where q(r)
(q
= 1/ε
/ n
- q (r) )n (safety
+ in
(q (r) Q - ζ)
ip/AN
(14.320)
]F(Q) ,
factor),
qQ = mQ/nQ
and
m0,n0 mode numbers has been used. Shear in a toroidal trap reduces the growth rate of the electron trapped drift instability /14.324/. Generally, particle trapping by electromagnetic waves is not possible, however, if the phase speed of such a wave is low enough trapping becomes possible. So whistler waves may trap particles /14.325/. A second whistler wave may detrap these particles. The de trapping effect is, however, also possible in sheared oscillating magnetic fields /14.341/. Whereas for an electrostatic wave and a magnetostatic field the resonance condition reads ω = ωβ and ω = ω^ respectively, it reads ω = kzun - ηΩ for electromagnetic waves. Finite amplitude whistlers propagating in a nonlinear /14.327/ dispersive plasma give rise to self-focusing and self-trapping /14.328/. These effects are due to the dependence of the plasma dielectric properties on the amplitude of the wave fields and produce the selftrapping modulation instability (whistler modulation instability) /14.329/, which is a typical nonlinear effect. In Alfvên waves particles may also be trapped /14.326/). Trapped par(cyclotron trapped particles ticles which do not leave a restricted domain are very sensitive to collisional effects, see section 12.3. Thus the trapped particle instabilities (dissipative trapped particle instabilities /13.64/) and the plasma transport properties are influenced by collisions. Collisions may expel trapped particles into
591
PLASMA INSTABILITIES
the loss cone and the related perturbation can be The (approximative) disdamped at a frequency veff. persion relation is then (based on a Landau collision term) /8.13/, /8.48/, /14.291/
ω=
/- *
2
*2
iv r
2- + t X " ^ 7 " ~Γ'
(14.321)
The maximum growth rate is γ - /ε"ω* /8·48/. Small scale oscillations are excited most rapidly.
Very important is the dissipative
stability
dissipative
trapped
ion
in-
/8.48/, /14.330/, of fewer importance the
trapped
electron
instability
/14.331/.
There are not only linear eigenmodes of the dissipative trapped electron instability, but also nonlinear effects on the dissipative trapped ion instability, including anomalous scattering due to the development of an unstable loss cone ion distribution function. The electron mode is important for high density Tokamaks and may be stabilized by shear. It may exhibit a two-dimensional structure /14.476/. The low frequency electron modes have been identified in a simulation model to be low frequency dissipative drift instabilities. The dissipative trapped ion instability may be stabilized by mode-coupling through nonlinear saturation. It also may deplete trapped ions by anomalous
diffusion {trapped ion depletion sipative trapped electron drift
/14.469/). The disinstability has been
investigated experimentally and theoretically in cylindrical geometry (Q-machine with special mirror coils etc) /14.321/, but is not so important as the dissipative trapped ion instability. Whereas collisionless trapped particle instabilities are driven by magnetic curvature drift, dissipative trapped particle instabilities are similar to collisional drift waves in that the excitation of both depends on collisions. In addition to the usual dissipative trapped ion instability (which is basically an electron diamagnetic drift mode) there exist ion diamagnetic trapped ion inmodes of this type /14.332/ (residual
stability)
.
There exist many cross effects between trapped particle instabilities and other modes. Ion trapping may influence microwave tubes /14.340/, electrostatic modes influence trapped particle modes in toroidal plasma /14.333/, impurities may stabilize the trapped ion dissipative instability /14.334/ (but collisionless trapped ion modes are not stabilized by impurities /14.470/, if the impurity density gradients are
MICROINSTABILITIES
592
too weak), lower hybrid waves may excite or suppress dissipative trapped particle modes /14.335/ etc. On the other hand trapped particles also affect other modes. For example the thermal spread of trapped a-particles or the trapped ion mode influence the Tokamak cyclotron instability /14.336/, trapped electrons transfer energy to the ion acoustic instability, trapped particles have a strong effect on the characteristics of kinetic modes driven by an electron current along the magnetic field /14.334/, and electrically trapped particles produce new resonances due to the interaction with a whistler /14.338/. Impurities have various effects on dissipative trapped ion modes depending on whether the parallel wave phase velocity is above or below the impurity thermal velocity. In general it can be said that when the impurity density gradients are in the same direction as the electron and ion density gradients, the impurity ions are stabilizing /14.339/. On the other hand with impurities new modes may appear in the banana regime. Impurity ions can produce substantial variation of the electrostatic potential within a magnetic surface. The resulting electrostatic trapping and electrostatic drifts may alter significantly /14.477/. the result of transport theory It should also be mentioned that collisional trapped ion instabilities and non-resonant trapped particle instabilities (ω < ω^) can be stabilized by nonlinear saturation /14.340/, see chapters 16 and 19. In concluding this section we would like to mention that there exists another instability connected with the family of trapping instabilities /10.2/. This is the Hirschfeld instability, negative radiation absorption instability, nonrelativistic mode /14.343/. The energy source of this instability is a deviation from a Maxwellian distribution. Only in the case of a Maxwellian distribution does the plasma radiation temperature Tr which for non-maxwellian isotropic plasma is defined by oo
k
BTr
=
-§cr(u)f(u)u3du/
oo
^or(u)^£^u2du
(14.322)
o o equal the electron temperature TE. or is the cross section for spontaneous emission of energy Τζω by an electron of velocity u. A system in which stimulated emission exceeds stimulated absorption has a total
593
PLASMA INSTABILITIES
negative absorption and thus by Kirchhoffs law a negative radiation temperature. In this case the radiation intensity can exceed greatly the intensity from a system that has a positive temperature, because now the electromagnetic waves generated at some point within the medium are amplified in traversing it. So synchrotron radiation negative absorption is possible. It can be shown that negative radiation temperature arises at microwave frequencies when df(u)/'du > 0. A second necessary condition is obtained from integration by parts of the denominator in (14.322). When the radiation cross section ar is expressed by the absorption cross section a (u), the condition reads 3 (oa(u)u4) fèu < 0. This condition leads to a class of "trapping instabilities" by which radiation takes place in the form of bremsstrahlung and cyclotron radiation in a partially ionized gas, partly as a result of pressure broadening. The instability may be stabilized by suppressing the deviation from the Maxwellian distribution or by choosing certain stable ranges of the radiation parameters. The Hirsehfield radiation instability should not be con/13.54/, see fused with the radiation instability pp 321 and 361. There exists also a negative radia-
tion
absorption
instability,
relativistic
mode
/14.343/. This mode belongs also to the class of "trapping instabilities". In the relativistic case, the conditions for negative absorption of synchrotron radiation can also be satisfied in a fully ionized plasma, in a manner without non-relativistic analog. 14.9 INSTABILITIES IN CHARGE EXCESS PLASMAS
A charge
excess
or nonneutral
(Breit-Darwin)
plas-
ma /14.492/ is a collection of charged particles, satisfying (2.11) but not (2.7). To be in equilibrium it must rotate. Such magnetically confined nonneutral plasmas are found in electron ring accelerators /14.344/, /5.31/, in intense high current relativistic electron beams /14.345/, in nonneutral electron and/or ion clouds in toroidal and mirror magnetic fields / 14.346/. We consider first the electrostatic instability of a nonrelativistic nondiamagnetic nonneutral plasma /5.31/, /14.347/. The fluid equations of species continuity, of momentum conservation and Maxwell's equation give for a cold constant density plasma in linearized approximation the dispersion relation foi electrostatic waves. It reads, for a plasma column of radius rO,
MICROINSTABILITIES
594 K7 (k r ) I I ( f e i - J £ g g l z P k r. z'PKAk r )I7(fe P J Z a g Z 2 P 2
- 1 - Σ
- K' (k 3?„)I-(fe r ) I z P l z o - K 7 ( k 2» ) I - ( f c r ) Z z P I z o
\
ω„ (ε Ω + 2ω ) lr
S V
/
(14.323)
PJt(Tr
i l
)
v
2 g
Pg
g
g
g
(ω - fe z; 2
OS2
-Ζω ) g
r c is the radius of the confining conducting wall, Kj, Ij are the modified and Jj the usual Bessel functions,
Illkzrp ω
) =[άτί(χ)/άχ)]
s
=
ε Ω s s --
1 ±
, and zP 1/2 4πβ e n g P r o2 wΩ
χ=]
.1-21
(14.324)
g g is the rotation rate (angular velocity) of species s, is the s-component of the zero orε 0 Ξ sgn es, vosz der streaming velocity of species s, Z = 0,1,... is the order of Bessel functions and
2
2
T Z = -kZ z
Ps
1-1
s (ω - k v
where 2 v = [ω - k v g \
- Ζω )
z osz
s
- Ζω I
z osz
s)
I
- (ε Ω ss \
1-1
s
ωPs
, (14.325)
v
(14.326)
+ 2ωs )V
If the wave guide is completely filled with plasma, one has rp = r> . For a very thin plasma beam in a wave guide one has rp « rQ. For rp -* i» , (14.323) be= 0. Introducinq the effective perpencomes Ji(Tre) dicular wavenumber fef = p? /^J, where p^ m is the m-th zero of Jj(^) = 0, one may write the dispersion relain the form ^ι)Ί ω -η / (ω - Ζω -k v I 1L\ Ps/ \\ s z osz) I ίω - Ζω - k v 3/ \\ s z
Here the Cortex frequency by
) osz)
- ω
sv
h
4.327)
72
1
x
72
z
(Pötzl /14.347/) is given
+ ωsv, = - \[εs Ωs + 2ωs)) = ±ίω - ω"). (14.328) \ s s) One of the instabilities described by (14.327) is
595
PLASMA INSTABILITIES
the electron-electron two-rotating-stream instability which occurs when there are two electron components
(η#1 'nE2 w i ^ h rotation velocities ω-] = ωΕ, α)2 = ω£, defined by (14.324), see also centrifugal flute insta-
bility, rotational ponderomotive instability, p 283. ^ e For equi-density electron components (n^- = nE2) instability condition reads /5.31/, /14.384/ Αω
ΡΕ
72
ft
72
ft
, > 1 (14.329) 2 2 2 2 4 (ω ω ) Ε' Ε ftZZZ ftZ(ZZ - 4 - 1 4 6 | Z | 2 ) for I = ±λ , ±2, 6.. is the Kronecker symbol and ωρ^ = Ai\e2(nE^
+ nE2)/mE·
(14.330)
On the other hand an unneutralized rigidly rotating electron plasma is absolutely stable /14.365/ if the distribution function is a monotonically decreasand if rαω < c. EQ and Lz are ing function of EQ-uL2 the single particle energy and angular momentum around the 2-axis and ω is the angular velocity. A hot electron plasma with angular drift frequency comparable or larger than the ion cyclotron frequency can be stabilized by a small amount of cold electrons /14.475/. When (14.323) is examined in the limit of long axial wavelengths [k2 = 0) the modes may be called surface waves /14.349/. An unstable mode is the electron-ion two-rotating-stream instability, electron ion stream instability. sities are related by
η χ = fnE,
f Φ 1
If ion and electron den-
(14.331)
then the allowed values of equilibrium rotation velocity may also be calculated from (14.324) and depend on /. More details may be found in the specialized literature /5.31/. When perturbations of the electrostatic potential in a low-density, pure electron gas equilibrium with diocotron u>pE(r) « Ω2;, rij (r) = 0 are investigated the instability, also electronic instability, electron beam instability etc (p 283) is found. It is assumed that kz = 0, |ω- lu>E(r) I 2 « tiE and that the electron column is_rotating slowly with angular velocity It can then be shown /14.350/ that the ü)£(r) = b)E(r). equilibrium is stable if the radial density profile is such that
MICROINSTABILITIES
596 dUpE(r)/dr
< 0.
(14.332)
A necessary condition for instability is that db)p^(r) fèr change sign in the interval 0 < r < z» . An application of the diocotron instability is the hol-
low electron beam instability /14.351/ and the resonant diocotron instability /14.352/. Normally, the
diocotron instability has a growth rate comparable to the oscillation frequency {strong instability). It has a small growth rate (ω^ « tor) when the density gradient is weak in a low-density nonneutral plasma. The resonant diocotron instability occurs when there is a bump in the density profile. In plasma diodes large amplitude oscillations in the diode current arise at low pressures {plasma di. These oscillations occur at frequenode instability) cies ranging from a few hundred kilocycles to as low as a few hundred cycles per second and have waveforms suggesting a relaxation mechanism /14.24/. The period of the oscillation depends on the transit time of ions in the diode and the energy stems from the energy of the free particles. These free particles can be emitted from and absorbed by the electrodes. There are in addition trapped particles moving back and forth between the electrodes indefinitely. The stability properties of this system differ from those of an infinite plasma, because the latter does not contain free particles. Thus, a coupling arises between the free particles and the longitudinal plasma oscillations /10.2/. Resonant and trapped particle effects do not alter the situation as compared to that of an infinite plasma. Other authors are of the opinion that these plasma diode oscillations were initiated by high-frequency electron space charge instabilities /14.24/. The latter cause a re(Pierce instabilities) duction in the electron density within the diode but no substantial change in current. Then, on a slower time scale, the resultant positive space charge drives the positive ions out of the diode. Due to the ion inertia the process goes too far and the ion density falls below the electron density. The negative space charge inhibits the electron flow and reduces the current to a small value. On the slow time scale, the positive ion perturbation builds up again and the current swings back to the initial high level. The process repeats itself. If the growth time is short compared to the transit time of the bulk of the free particles the coupling becomes weak and the dynamics ap-
597
PLASMA INSTABILITIES
proaches that of an infinitely extended system without electrodes and the instability becomes stabilized. When a stream of charged particles is sent through an unmagnetized plasma with isotropic temperature, a bunching of the beam structure produces an induced magnetic field which gives rise to a Lorentz-force and to further bunching. As a result, a transverse unstable electromagnetic wave arises {cross-stream in-
stability
in an unmagnetized
plasma /11.56/). The in-
stability is macroscopic and absolute and resembles the Weibel instability of an anisotropic plasma.
The current
chopping
instability
{current
inter-
ruption instability) /14.25/ is related to the spacecharge and Pierce instabilities. The phenomenon is a gas discharge effect which occurs when a critical current density is exceeded and involves one or more rapid reductions in the current and sometimes complete . A decrease in plasma denextinction {arc starvation) sity or a constriction of the cross section requires an extra voltage to keep the current constant. The necessary acceleration of electrons within this region produces two double space-charge sheaths. The sheath at the cathode causes electrons to enter from this side as a beam having a mean energy equal to the sheath potential plus the mean thermal energy. The potential hump thus formed causes the ions and electrons to be accelerated in a way that lowers the density even more: the initial disturbance is enhanced /10.3/, /10.2/. It has also been suggested that a local magnetic compression of a discharge could induce current chopping. The instability can be stabilized by a thermal spread of the distribution function and by collisions. The instability of an E-layer (see p 149) is also a nonneutral instability. In a layer of mono-energetic non-relativistic electrons having no spread of guiding centers rotating in a uniform magnetic field, oscillations are unstable for an arbitrarily small average density. The growth rate is much longer than the period of rotation of an electron /14.468/. On the other hand, the sporadic equatorial E-layers in the ionosphere may be explained by the cross field in-
stability
/14.353/. The E-layer
instability
/14.354/
depends sensitively on the density of the background plasma, but is essentially independent of its kinetic structure. The real part of its frequency is determined by the gradient of the external magnetic field. Increases of the magnetic shear of the equilibrium
MICROINSTABILITIES
598
tend to stabilize the configuration at moderate wavenumbers. It has also been shown /14.366/ that charge sheets of ions parallel to magnetic field lines are unstable whereas charge sheets of electrons parallel to magnetic field lines can be stable. In connection with studies of ring currents of relativistic electrons contained in a magnetic field
(electron
ring
particle
accelerators
/14.355/) inves-
tigations have been made of the relativistic electron ring instability /14.356/. Density profiles, thermal spread and the Budker parameter n^e^-/2i\r-pm^c^ /1 5.1 93/ play a role in its theory. Also a dipole instability of a relativistic electron ring loaded with ions has been found /14.357/. Pure electron plasmas show instabilities of their
own: electron
gas instability,
instability
of
layered
electron gas /14.358/, electron plasma oscillations in the presence of Brillouin flow (see p 99) /14.359/,
electron gas losscone instability /14.360/, gas two-stream instability /14.361/, ordinary magnetic instability (plasma electromagnetic
electron electroinstabil-
ity) in electron gas /14.362/. In connection with the theory of these instabilities a thermal Larmor radius in[c^h/(ω# - ω#)2]1/2 has been defined /5.31/. Flute stability may also be found in a nonneutral plasma /14.364/. The stability of a degenerate electron plasma in a strong magnetic field (degenerate electron plasma instability) depends on the equation of state used and on the collision frequency /14.367/. In grossly nonneutral plasma an ion resonance instability has been found /14.363/. It may occur when ions of high energy are added to an electron cloud of low density (ωρ < Ω^) contained in a magnetic field. When rLI exceeds the transverse dimensions of the configuration and the ions are trapped in the strong electrostatic potential well produced by the electron cloud, the oscillatory ion motion becomes coupled to the azimuthal E*B drift of the electrons thus producing growing modes. The energy source of the ion resonance instability is the electrostatic field of the electron cloud. The instability may be stabilized by choosing the magnetic field strength between two limits determined by the electron density. Another instability related to the diocotron instability is the magnetic instability occurring in magnetrons. The properties of an inverted ion magnetron in which a confined ion cloud was produced have also been studied. Above some critical magnetic field a strong drift instability with frequency of the or-
599
PLASMA INSTABILITIES
der of 100 kHz was observed. This instability has been called inverted ion magnetron instability /14.368/. It is characterized by a spiral character, and by the electron acceleration in the electric field of the runaway regime. The instability generates intense rf emission and may be used for wave generation. It also seems to be of interest in astrophysics and planetary physics. Several other instabilities in nonneutral plasmas have been discussed in the literature. A review of glow discharge instabilities {attachment instability, recombining plasma instability, decaying plasma instability, recombination instability, thermal instability, dissociative attachment instability in partially ionized molecular plasmas etc) has been given /14.369/, /14.370/. A nonlinear rotary wave in ion plasmas /14.371/ and the transformation of space charge waves at plasma density discontinuities /14.372/ have been mentioned. Instabilities in negative ion plasmas are important for negative ion sources. Severion al papers appeared recently treating the negative plasma instability, hydrodynamic instability of negative ion plasmas /14.373/. A fast ion mode exhibiting spatial growth in the presence of a fast ion beam has been found. In these multicomponent plasmas with one negative ion species crossover frequencies (the frequencies at which the refractive index of two modes of wave propagation are equal and are of the order of ion gyrofrequencies) play an important role. External electric fields may cause an electron-ion plasma to behave like a charge-excess plasma and modify longitudinal oscillations /14.374/. Similar effects appear in rotating non-uniform plasmas /14.375/. The neutral deviation drift instability which is due to small deviations from quasineutrality has been discussed in volume 1, p 391. Also the ion space charge /11.28/, /14.376/ has been mentioned instability there (p 284). Quantum-statistical systems like an electron gas have also been investigated /5.9/. If the distribuis continuous tion function (e.g. Fermi statistics) and has a single maximum, the system is stable. A Fermi gas in its ground state (zero temperature) is able to sustain a permanent collective oscillation ω = ωρΕ for k ■> 0. For shorter wave-lengths the dispersion relation reads for neutral systems 2
ω
2
j . -37 2
2
/ c
2
= ωρΕ + 3k uF/5u>p
_ 2
7
4
+Π k
/A
2
2
/4mEupE.
MICROINSTABILITIES
600
Hereft= Α/2π, where h is Planck's constant and UF is the Fermi velocity uF = (37z32/87rm|) 1 ' 3 . This result point shows that in the ground state part of the zero energy comes from collective modes in which a large number of particles oscillate in phase. 14.10 ANOMALOUS TRANSPORT For the operation of laboratory devices the transport properties of plasma - mainly diffusion coefficients and electric conductivities - are of paramount importance. Transport properties depend not only on the force law between plasma particles (see p 120) but also on geometric factors (see p 336) /14.378/, on microinstabilities (p 348), and on turbulence, as well as on other nonlinear effects. Abnormally fast transport across a magnetic field not due to collisions is called anomalous transport. It is a phenomenon closely related to instabilities. The transport caused by microinstabilities is in general in excess of and relatively unrelated to classical transport, compare (7.85) and (12.95). Collective effects on e.g. the high-frequency conductivity (ac resistivity) of a fully ionized plasma have been investigated by Dawson and Oberman /14.377/. They found an enhancement of the classical resistivity around the plasma frequency. The effect is a nonlinear parametric effect. Other candidates for the source of anomalous transport instaphenomena are the current driven ion acoustic cyclotron bility, the current driven ion (electron) of various types and instability, drift instabilities other phenomena like magnetic effects, electric effects, fluctuations etc. It should also be mentioned, that trapped particles and untrapped particles have quite different effects on field perturbations (instabilities) . If trapping occurs, the plasma behaves like a two species plasma. Untrapped particles are free streaming with a Maxwell distribution along the field lines. Trapped particles (temperature ^/εΤχ) have a Maxwellian distribution with an additional term. (For ε see p 340). The current driven ion acoustic instability of interest here has a frequency near the ion plasma frequency. It is unstable if the current drift velocity ^ο = J/ne i s large enough that uQ > as(mI/mE)U2(TI/TE)3/2exp(-TE/2TI)
,
(14.333)
601
PLASMA INSTABILITIES
where as is given by (9.51). In many Tokamak experi< 2, so that the ments one has TE/TZ < 4 and uQ/as modes are stable. But even if the mode is stable and weakly damped, it might cause an anomalous electric resistivity. For fast ion acoustic waves ion Landau damping decreases (p 211 /9.4/). Even recently many papers have been concerned with anomalous transport due to ion acoustic instability /14.379/. On the other hand, the interaction of ion acoustic waves with a slightly inhomogeneous magnetic field generates an electric current which can produce intense and rapid changes in the magnetic field {ion/14.467/). acoustic distortion of a magnetic field Anomalous electron transport coefficients have also been derived and it has been shown not only that effective collision frequencies can be larger than in classical theory but also that there are differences in the structure of the transport equations as well (Dunn /14.379/). If it is assumed that an electron drift driven by an applied constant electric field causes the ion acoustic instability and that a steady state is formed in which ballistic clumps of plasma behave like dressed particles and collisionally scatter each other the electric conductivity is given by (Kadomtsev, Pogutse or Dupree /14.379/) σ * -[OupE/k\D.
(14.334)
The full theoretical treatment of ion acoustic resistivity must, however, comprise a nonlinear theory
(ion acoustic
turbulent
resistivity),
see later. Plas-
ma heat conductivity also differs greatly from the classical values when ion sound turbulence is present /14.379/. The current-driven ion cyclotron instability may also induce anomalous transport /14.380/. This instability may be excited in Tokamaks by finite amplitude trapped-ion modes. It is, however, found that although ion cyclotron instabilities are likely to be present, they are not likely to modify the diffusion rates originally predicted by Kadomtsev and Pogutse for anomalous transport due to trapped ion modes, see later. In recent Q-machine experiments it was found that the high frequency electrostatic (and electromagnetic) ion cyclotron and ion cyclotron drift instabilities generate anomalous electric resistivity which inhibits the destabilizing current when the plasma is stable to ion acoustic modes. Finite heat conduction effects on the ion cyclotron and drift
MICROINSTABILITIES
602
cyclotron instabilities have also been investigated and found to be destabilizing. The nonlinear development of the electron cyclotron drift instability may also generate anomalous electric resistivity. The anomalous skin effect (and the anomalous electron viscosity) may be explained by an electron plasma mode driven by a gradient in the current drift vedrift locity u0 over the plasma cross section (current velocity stability
gradient instability, current gradient in/14.381/) or by the ion acoustic instability.
Another important class of microinstabilities generating anomalous transport are the drift instabilities (mainly as trapped particle instabilities). When trapped particles are taken into account the drift waves are modified /14.393/ due to a change of the dispersion properties. In the presence of trapped particles the resonant interaction caused by particle bouncing between the mirrors is reduced by (ω/ω^)^. Then a short wave (k±rLj » 1) electron drift instability ω > u)f arises with a growth rate γ - u>i>i?/}/k±rLI, +Tj/TE). This instability is inif ω*/ω££ > SK\rLIC{ dependent of curvature and can arise in systems with an average minimum of B. Also ion drift instability may appear for (ω > (Jj) with γ - ^^jkir^j k
>1
l^LI^i'/^bI
+Tl/TE-
Since the wave number k§ in the θ-direction has a minimum value of κ (p 352) and a maximum of rj^ , the range of drift frequencies ω* (ρ 372) is given by ω* . = νηκ < ω* < vnr~\ rmn
D
= ω*
D LI
.
(14.335)
max
When some processes in the plasma generate a phase difference between the electrons and ions (having the same £x£-drift - see p 52) then the drift wave may grow due to particle-wave energy and momentum exchange. The type of drift wave generated depends on the collision frequency v, the bounce frequency ω^ (4.45), (14.306) and the curvature
drift
frequency
ωβ = kßVD = ω*, see also p 386. Drift waves will therefore be classified as follows /14.387/: 1) dissipative
drift
v * ω*, 2) dissipative
wave /14.382/
v > ω* . > ω,-,
untrapped
rmn
bE
trapped
v > ω, ρ/ v < ω*
.
particle
(14.336) mode
(14.337)
Trapped particles before they bounce and bE' collidemax
603
PLASMA INSTABILITIES
hence do not know that they are trapped. trapped electron mode, type I /14.384/ 3) dissipative ( 1 4 3 3 8 ) vE < %E' VE >
VE
K
%E'
V
E < °-'
(14
'339)
The DTEM (dissipative trapped electron mode) may also be characterized /14.391/ by ω | < νΕ/ε and ω | > νΕ/ε and u)£j < ω < ω ^ , Vj < ν^/ε. The collisionless mode mode with ω < b)fo£ is sometimes called the ubiquituous /14.378/, /14.323/. This mode does not require the ions to be in the banana regime and is unstable with and without the effects of collisions /14.415/. trapped-ion mode /14.385/ 5) dissipative V
J
K< ω
2>1'
{ V
EK
b)
bE f °
rΤ
Ε= Τ Ι ) Λ
(14.340)
The growth rate is highest (γ - /εω*) if ν^/ε^/εω*. Electron collisions increase the growth rate, while ion collisions stabilize. Landau damping by transiting ions stabilizes short wave modes. The drift wave is supported by just the trapped particles. The DTIM(dissipative trapped-ion mode) may also be characterized /14.391/ by Vj/ε < ω < ν ^ / ε , ω
< o)j>j.
6) collisionless trapped-particle flute (interchange) trapped particle mode /14.386/ (CPTM - collisionless mode) v < vDE(rR)'^/2.
(14.341)
Estimates of the diffusion coefficient due to these instabilities may be found from (7.80) by replacing τ~1 = v by γ and rL byfcj^.The growth rate γ of the dissipative drift wave is of the order of ω^ and hence (for ^m^n > ω^) (Kadomtsev formula) roughly D * yfe~2 * ^*fcj2.
(14.342)
This resembles the pseudoclassical formula (12.95) /12.86/, /12.72/, /12.97/. More refined calculations give the following results: When (14.337) is valid, one obtains /14.383/ for the electron heat conductivity
MICROINSTABILITIES
604
«E *
(14 343)
·
°Λ*ΕΑΕ.
where Cj, are numerical coefficients chosen empirically and D*
C VT 2 \E
°r
^"1/2r"1/4f
(14.344)
compare (12.95)/ p 348 {-pseudoclassical theory). Losses decrease with increasing TE (like in classical theory). When (14.338) is valid, one has /14.384/ again γ - v - ω* and ~ 3/2
E
2 *
rmn
-1
TErmn
(14.345)
El
(where κ is defined on p 352, ^^^min *-s ^ e minimum electron-tempe rature gradient drift frequency (c/eBr) (dT/dr), compare p 381. κ^ increases with TE. The diffusion coefficient is D * εκΕ.
(14.346)
Transport due to this drift regime (dissipative trapped electron instabilities) is very unfavorable since it increases with plasma temperature as τ|./14.494/. Stabilization is possible by magnetic shear. When (14.339) is valid, k is not determined by v - ω*, but by the magnetic shear /14.384/. Now the growth rate is 3/2 * -1 (14.347) γ - eJ/ ω ω ^ and the electron heat conductivity is K
E
s C 2
2*EAE>
(14.348)
where C 2 - 10 , depending on the magnetic shear strength, and D ^ V F 2 > ? J · T ^ e trapped-electron instability plays a major role in a non-isothermal plasma {TE > Tj). In the nonlinear regime of the trapped electron instability the expressions for D and κ are again modified /8.48/, /14.394/. In the nonlinear domain also, increased radial transport is possible due to magnetic island formation generated by the tearing instability /14.395/ or due to convective cells. When (14.340) is satisfied (dissipative trapped ion mode), the mode frequency is lower than the ion bounce frequency. As with the dissipative trapped electron modes, the trapped electron collisions have
605
PLASMA INSTABILITIES
a destabilizing effect (trapped ion instability /14.384/)· However, trapped ions and ion-bounce resonances have a stabilizing effect /14.385/. For collision frequencies lower than 0. 4ω^/ (mET^,/mjT^) 7/1 8 the threshold for this instability could be raised by elliptical shaping of the minor cross section or by artificial ion detrapping (ion depleting) by magnetic oscillations /14.388/. The mode is unstable and D
»
0
+V2V2)'(14-349)
where the numerator ^T 2 5~ 2 . Transport grows as T J ' 2 . In a two component plasma a new trapped ion instability may occur due to the presence of the hot ion component. The new instability leads to enhanced diffusion /14.389/. The effect of trapped ions on transport phenomena is intermixed with geometric effects. Let us recall some of these effects. In the superbanana regime (pp 162, 347) occurring in a stellarator or in a slightly corrugated Tokamak the bananas are displaced across the magnetic surfaces. This displacement is independent of the magnetic field strength so that even for very strong fields, diffusion and thermal conductivity on the bananas may be very intense in a rarefied plasma /14.392/. These effects are only observed for v < ε 3 / 2 £ £ ^ 0 (kQ is the wave number of modulation of the field BQ). The two types of field inhomogeneities in a steltarator or a corrugated Tokamak (non-uniformity of the winding and inhomogeneity due to torroidal curvature) or a bumpy torus like ELMO are described by /8.48/ (h refers to helical, t to toroidal): Bz = so M - eh 7cos(Z0 - k z) o
- ε+ t cos
θ] ,
(14.3 50)
where Θ is the (small) azimuth, z the coordinate along the torus axis and I is the number of winding depressions (e.g. I = 3 for 3 winding gaps). Nonaxisymmetric toroidal magnetic field ripple caused by discrete coils can form an additional trap for particles /14.474/). This may occur in a flux (ripple trapping conserving Tokamak /14.428/ (FCT which offers a true beta limit of 10 - 30 % instead of 3 % as previously assumed for Tokamaks). Such a toroidal field ripple can lead to anomalously large particle and energy losses /14.471/. The fact that not axially symmetric toroidal systems like stellarators or bumpy torus exhibit a substantial increase in the diffusion coeffi-
MICROINSTABILITIES
606
cient and in thermal conductivity is known since some time. So it has been shown /14.479/ that either for low collision frequency or large electric fields transport coefficients in not axially symmetric toroidal systems do not depend on the magnitude of the magnetic field. Transport of plasma across a braided magnetic field has been treated in /14.466/. Such a magnetic field occurs in a region through which magnetic field lines meander in a stochastic fashion and in which the magnetic surfaces are destroyed. This might happen due to asymmetric magnetic perturbations. Diffusion due to magnetic braiding and magnetic field reconnection (leading to disruption) are responsible for enhanced electron heat conductivity, plasma dis-
ruption
and sawtooth
oscillations
/14.497/.
For a stellarator ε+ « ε^ « 1, for a Tokamak ε^ « ε^ « 1. In a toroidal stellarator the main role
is played by localized
-particles
{blocked
particles)
trapped by the inhomogeneity due to the helical winding. This field inhomogeneity is weaker and covers a larger distance than the (stronger) inhomogeneity due to mirrors. Particles not trapped by local mirrors will be blocked (trapped) by these large scale inhomogeneities. The fraction of blocked particles is /ε^. The bounce period of blocked particles between the mirrors of the helical winding depends on the pitch angle ψ and the energy. Averaging over the energy gives ck T
^bh
" —^2 ε 7ζ^ ( ψ )
= ω
// ( Ψ } " ωΗ'
(14.351)
eBr
since /(ψ) - 1. r is the small radius. For veff - ν/ε/j > efockßT/eBr2 Ξ ω^ one may neglect ω ^ « /8.48/, and obtain for the diffusion coefficient /14.392/
v
eff
2 D
- 1 ^ 7 Z7 2 ^ ' h
For veff
v
>w
(14 352)
·
< Wfofo one has z2 ck T
D - ^ΓΤο -Â-> £.1/2 eB h
In a Tokamak
(14.353)
with slight corrugation (eh « ε,) the
607
PLASMA INSTABILITIES
main inhomogeneity is due to the toroidal curvature. The corrugation produces only a small fraction / ε ^ of blocked particles. The diffusion coefficient is now given by /8.48/
Dα
1ieh\3/2
ι(τι) 0
Ό
V
1
-vDo> D
= c
(14 354)
·
r
R
a
eau
where v^ = ^ ^ ^ / ^ ' o th î^^ o ^V^ ^ diffusion) . Transport coefficients are modified by trapped instabilities /14.392/. The trapped-particle effects are strongest in highly rarefied plasma, when the electron collision frequency is smaller than the drift frequency. At low collision frequencies but V E > ok-βΊIeBv^ the main role is played by the dissipative trapped-ion instability. From (14.342), (14.349) and then D - T^E^DQ/TEZV
(14.355)
is obtained /8.48/, where Tj- < TE, ε = r/R, v ^ = v-j diffusion for c-f-fo = ctfo-g. We see that at Tj - TE the at v^ = v ^ exceeds the classical value DQ coefficient by a factor ε"^ and decreases as vE increases, see also p 345. When (14.341) is valid, i.e. at very low collision
frequencies v^ < ck^T/eBr*, the dissipative trappedion instability is modified into the trapped-particle
flute instability, and D and κ are no longer dependent on v#. For ω < ^hE' oscillations in the trapped electrons with ω > ω| may be excited. The trapped particles are then a source of oscillations. For ω < b)foj the main instability source is the resonant interaction between trapped ions and the mode ω > u)j /14.393/. Such flute-like oscillations (quasi-flute modes /14.378/) with frequencies ω « OJ^J have been considered /14.396/ taking into account temperature anisotropy and other effects. The instability is called flute-like because it resembles the ordinary MHD flute instability in a mirror machine, in that it is driven by the combination of the radial pressure gradient and the unfavorable drifts of the trapped particles, and in that it has nearly constant perturbed potentials along the magnetic field lines. The flute mode takes place with a frequency /14.384/ ω 2 - ε 1 / 2 ω*ω π ,
γ 2 = -ε 1 / 2 ω*ω η ,
(14.356)
MICROINSTABILITIES
608
where ω η = kv„ = k ckOT/eBR. u u y D
(14.357)
This mode can give rise to transport similar in scaling to DBohm (7.85) /14.387/. Stabilization of the collisionless trapped particle mode is possible by finite beta and cross section shaping /14.400/. For magnetic wells radial plasma transport due to low frequency instability (ω « üj) is described for dB/dr > 0 by the diffusion coefficient /14.399/ D]L - rLIckßTI/eBR(TE/TI
+ m^m^.
(14.358)
Odd modes /14.378/ for which the perturbed potential changes sign somewhere along a field line have been found in collisionless and collisional regimes /14.398/. Doubts have however been raised /14.455/ if these odd modes exist in reality. Coppi showed /14.397/, /14.378/ that a periodic dependence of magnetic drift on particle motion along a field line gives rise to flute-like perturbations. It is characteristic for this instability that the phase speed changes sign according to the sign of (1 - TE/Tj). For TE > Tj the mode propagates in the direction of the Larmor drift of electrons, whereas for Tj? < Tj the mode propagates in the opposite direction. Shear is effective in reducing the anomalous particle diffusion due to the collisionless drift wave instability. There are some other instabilities which may influence plasma transport properties. An instability for which the toroidal configuration is insignificant temperature instability. It yields a is the drift thermal conductivity K * 1θ"2ε""2κ ,
(14.359)
where κ 0 is the classical ionic thermal conductivity /8.48Λ Historically, the screw
instability
in a
partially
ionized plasma (see p 292) seems to be the first instability connected with anomalous transport. It has also been investigated recently /14.401/. The onset region instability causes an increase of the cathode of the wall ion flux in the plasma column. Also the two-stream
instability
(two-stream
ion
cyclotron instability) has been investigated as a candidate producing anomalous resistivity by ion
609
PLASMA INSTABILITIES
bunching /14.402/. When the electron drift speed is larger than the electron thermal speed, the electronion two-stream instability occurs. Then the electronion collision frequency v#j is said to be increased instability, to the growth rate of e.g. the Buneman see p 472, such that V
eff
* V(V
W
) 1 / 3 i
I
(14.360)
This increases momentarily the effective electric resistivity, but since the beam loses its energy, the ion acoustic instability will appear. Sagdeev reviews electric conduc/7.7/, /1.34/, /1.20/ the anomalous tivity
produced by dc current
driven
instabilities.
The following table summarizes the situation: instability threshold frequency growth conductivity Buneman / 1 4 . 4 0 3 / ( c o l d e l e c t r o n - u n > c,l7P . , v D thE ion two-stream) von
sound
,
^ V^ >a
Drummond-Rosen-
bluth /14.405/ ,. ' (two-stream . v Ί , ion cyclotron) fied Buneman)
Bernstevn
/14.407/
<ω
^
σ
1/3
-1 1/3 ù} PFÎmT\ p 4 7 2 —^—(—) * 2 xm^l E 2 o)-Tvn ne a0TT PI D SI
Q
„ v^> a D
electron sound / 1 4 . 4 0 6 / (modi- vn < e . ,
imw\ ω^ί— PE\mT) I
n
ü-r
T
I
« Ω_,
Q
*Λΐ
**£
ID cthE ,7_
g £>
c ft£,
The de resistivity values described roughly by these growth rates are, however, based on the nonlinear stage of the respective instability. Anomalous transport processes associated with the lower hybrid instability have also been discussed /14.408/. There appears an enhancement of the electron transport of heat and momentum due to electron turbulent motion in the field of thermally excited lower hybrid waves. Nonlinear and parametric effects are included. Anomalous heat transport may also be due to tearing instability /14.409/. The appearance of trapped particles increases the resistivity as a consequence of the fact that only the circulating particles can carry a current and that in
610
MICROINSTABILITIES
the presence of a dc electric field parallel to B the trapped particles lose momentum to the confining magnetic field and therefore provide a frictional momentum sink for the circulating electrons /14.410/. Also a localized collisionless instability that occurs in a nonuniform or skin current flowing along an extervisnal magnetic field is shown to produce anomalous cosity giving cross field diffusion of the current. This instability occurs under conditions where the ion sound wave is stable /14.416/. Trapped particle instability theory /14.411/ is also important for describing the effects of impurity ions. The trapped ion modes may stabilize or destabilize impurity /14.412/. Impurity ions, also α-particles, may excite special drift modes. Since in laboratory plasma °th-imp « Gthl <
611
PLASMA INSTABILITIES
ω - = cE./reB el
(14.361)
1
appears /8.48/. For u>ei > ω^ (for ω^ see p 606) the drift trajectories are strongly distorted. Then the bananas in non-symmetric systems will be untrapped. This leads to a diffusion coefficient D - e?v1/2(feDT)25/u) ,e2EBr t
for v < ε^ωβ£.
B
F o rv > ε
ΗωβΙ
eL
o n e
(14.362)
°btains
D - elel/2c2klT2/ve2B2r2. t h B
(14.363)
Particles trapped in strong electrostatic waves may also undergo trapped particle instabilities /14.417/. Nonlinear waves with electrically trapped particles can display new physical properties. An electric field externally applied along a magnetic neutral line results in a redistribution of plasma and excites instabilities /14.418/. Other efrefects which have been investigated are anomalous
sistive
instability
/14.433/, conductivity
sum
rules
island
for-
in two component plasmas /14.419/, transport coeffi/14.420/ and effects cients in relativistic plasmas magof strong magnetic fields /14.421/. Very strong netic fields not only alter the kinetic equations /5.9/, /14.421/ but also the transport coefficients directly. Another magnetic source of anomalous transport is the break-up of magnetic flux surfaces by
kink
and tearing
instabilities
{magnetic
mation) . Across magnetic islands transport is enhanced. If there are adjacent island structures with different periodicities magnetic braiding occurs /14.422/ (see p 606). This means that large radial excursions of field lines and even contact of the plasma with the vessel wall occurs. Break up of flux surfaces modifies trapped particle instabilities. Besides instabilities there are other processes which might be responsible for anomalous transport. Due to long range polarization fields arising from scattering convective instabilities a quasiclassical process /14.423/ may occur. This process can be regarded as being the transition between a stable collisional plasma and a fully turbulent plasma. When a microinstability grows, the fluctuating fields produced by the instability scatter plasma particles and in velocity space /10.14/. As we will cause diffusion see later, this (nonlinear) process can be described
MICROINSTABILITIES
612
as induced scattering of a particle with momentum p conby a wave with momentum %Kt where ft is Planck's stant. (Real quantum effects are however negligible.)
The trapped
ion scattering
instability
is also de-
scribed in a review /14.378/, which classifies nearly all other collective modes in confined toroidal plasmas as follows: slow modes (ω < CÜ^J « ω ^ ) trapped ion scattering instability impurity sound instability impurity instability interchange trapped particle instability dissipative trapped ion instability quasitrapped ion instability intermediate modes (ω < ω ^ ) circulating ion instability electron drift instability dissipative electron drift instability ubiquitous trapped electron instability trapped electron scattering instability fast modes: ω > u t r a n s i t I circulating electron instability current electron instability. Another process leading to enhanced transport are fluctuations. The spectrum of fluctuations is quite different in an unmagnetized three-dimensional plasma from that in a two-dimensional (i.e. magnetized) plasma /14.424/. In the limit of extremely high level of fluctuations Böhm diffusion ^λ/Β appears. Also the fluctuation spectrum of a low density (u)pj < üj) twodimensional plasma can produce Böhm diffusion
("vor-
tex diffusion", "two-dimensional diffusion"). Studies of plasma diffusion across a magnetic field in two and two and one half dimensional models /14.429/ show that at large magnetic fields collective transport by convective motion can dominate over diffusion due to particle interaction. Plasma transport properties transverse to a dc magnetic field can be dominated by fluctuations in the electrostatic field whose time dependences are slower than the ion gyro-frequency. In twodimensional turbulent fluids (interaction of line vortices) and in two-dimensional guiding center plasma the negative temperature instability may appear/14.425/. Fluctuations and turbulence are basically nonlinear phenomena and will be discussed later (chapter 19). Fluctuations of an enhanced level can be thermal turbulence. Periodic fluctuations average out and give no contributions to anomalous transport.
613
PLASMA INSTABILITIES
kB.
/
jo
Fig,
53.
Geometry
in Stringer's
theory
A general theory of fluctuation driven anomalous transport has been given by Stringer /14.426/. The geometry used is depicted in Fig. 53. Low frequency potential perturbations are described by plane waves Φ ( ί , £ ) = $(x)exp[i(k
z - ut) ] .
y+k y
(14.364)
%
The assumptions ω « üj, krLI « 1 and 1 « kzX « « (βΦ/kßT)"3/2f where λ is the mean free path, are made. If collisions are ignored, the kinetic equation in guiding center variables reads df au df d u f s J J s z , s 1 = 0. + («·ν)/ + Λ dt du dt 2 dt z J
(14.365)
du
Expansion of the species distribution function / in s powers of e^/k-nTs gives B
f
=
** s
f (0) + f J
s
(1) + f ( 2 ) + *s
•'s
(14.366)
Using = 2knT /m , ω- = ω - k u 'ths B s' s 1 z so as well as the drift velocities
(14.367)
MICROINSTABILITIES
614
T an dT 1 V = Dns vin~-d' DTs 7~ß ~d# s s s a n d u s i n g γ « ω-j^, ω<\ = ω ^ + ί γ , S t r i n g e r / 1 5 . 5 6 / t h e f u n c t i o n -2Z1 = 1 + # Z U ) V
=
— I —
/FJ
— CO
'
expands
X
dz; = 1 - 2*e * l e * du + ifuxe
VX
(14.368)
(14.369)
J
for real arguments x and then for small x, i.e. for th<< ^/k-z' which holds for most instabilities. With the definition of heat current (6.17), the heat flux in ^-direction may be defined as c
1/3ΦΓ1 2.(1),3 \ „n. (ΛΑ d U /· (14.370) «ax " S \ ^ J 2 ^ ?s This gives the heat flux and the thermal conduotivity for electrons and ions as a function of the observable quantity Φ =
H
/π y 2 c , k
E
E l eB\
-¥υυ0ΤΕ)\ΐψ-Ε\ ^J
' ,y J / 5 ω eß U
"tftia ω
7
r T
z oE
7
y DnE (14 371)
-
E
3. I 4 y\vDnI
βΦ
DTI
' T. r
Vi'
The particle flux and the electric current ^";s " σΙΙ^ο^ Z-^tf c a n a ^ s o ke found. The transport coefficients satisfy the Onsager relations of irreversible thermodynamics (except in turbulent state and for diffusion anomalous processes). For drift waves the coefficient
is
n
β
25yk
v _(fe v _ + 2 a l k 2 / ω l) . 2 y nDnE y nDnE S ζ' r \ βΦ \ 2
Ύ 25,
^Φ
fc2 Τ ' ^ Τ ;
Since experimentally γ 1 ^ * £ 1 40
2
'fr Φ
k 2(/
,2 u 1 #Where U
e$/kßT
Vs
7
S
r VM
- 0.03,
one
I
-
(14.372)
has (14.373)
615
PLASMA INSTABILITIES
which should be compared with Kadomtsev's formula (14.342). The results for drift waves may always be compared with the results for ion acoustic waves, since from Fig. 39, p 3 54 (volume 1) it can be seen that the drift mode becomes an ion acoustic wave for large ^2 >:> ky For the diffusion driven current {bootstrap current) , which has the definition
(14 374)
·
h = Kv*> y
where Yz is the particle flux in the ^-direction, the result is
h - *o-r^r·
* ^i$-J
·
(,4 375)
·
All these values of transport coeffficients will be enhanced by toroidal trapping. It should also be remarked that in experiments, drift waves have been detected which did not give rise to anomalous transport. During the growth phase of a finite amplitude wave Φ the particle flux Tj = -< θΦ/3ι/) nf>/B is outward, during the decay phase of the damped wave the flux is inward. The outward flux may be responsible for losses of plasma occurring in a stellarator {pump-out). The pump-out rate is defined by 1 = /din n\ _ /din n\ \ an J no instability* τ \ an iinstab. pre sent (14.376) Pump-out is possibly connected with enhanced loss due to Ohmic heating current in the collisional and intermediate regimes /14.430/. Pump-out should not be confused with burn-out, the rapid decrease in the density of neutral particles in a plasma due to the fact that the mechanisms of neutral particle formation are slower than the ionization, charge exchange etc. In concluding we would like to mention that anomalous transport processes have also been investigated recently using numerical codes /14.431/. In this confor a viscous, nection a newly found energy principle resistive, heat conducting, finite Larmor radius electron plasma is of interest /14.432/. Runaway transport in Tokamak has been discussed in /14.472/. Increased electric resistivity may also be due to
MICROINSTABILITIES
616 ouvrent
density
gradient
instability
/14.473/. Cross-
field energy transport by plasma and ion waves has also been found, even in thermal equilibrium, because wave propagation is uninhibited by the magnetic field. Fusion reactions also modify transport /14.478/. 14.11 SCALING LAWS AND DISCHARGE REGIMES
Plasma transport properties decisively control the scaling laws for the construction of larger and larger devices and dictate also their operational regimes. Authors mainly use either the electron heat conductivity /14.435/ or the diffusion coefficient /14.436/ to define various regimes. For given density, E = const, and with a constant toroidal magnetic field Mercier writes for the electron heat conductivity as a function of r, T, BQ the general equation (for K Q see p 608) K
E
=
K
(/(r)Ta(r/Vß
(14.377)
and then specializes it for the following regimes classical (collisional, p 124, Sp = Spitzer) K
Sp
s
pseudoclassical
T5/2
'
(empirical law - see p 348)
K
ps * n2/Bl
or -nßZeffI~2Tß^/2, (Artsvmovich
plateau
scaling
V^E'
(14.379)
where I is the total current /14.443/).
(p 346) (empirical) K
pl
banana
(14.378)
nT3/2r/BQ/
*
(neoclassical, K
p 346)
2 2 2 3/2,7,1/2 - n r / DB~r ' T ' , Θ
nc
(14..380)
(14..381)
Böhm (p126) (assumption not proven) κΒ trapped
(14.,382)
T,
«
particles Ktp
*
nT3/2r/BQ.
Recent Tokamak experiments have shown κ7 •z1£2n-1T-3/2 /14.442/. *
(14..383) 1-3/2
617
PLASMA INSTABILITIES
as function of Based on the diffusion coefficient the collision frequency one may classify seven microinstability regimes /14.436/. They are Böhm Dn = cTj\$eB, (14.384) D
b
see (7.85), p 126. 22 Neoclassical diffusion is - q rLE\)gj see (12.89), (for q see p 164). Kadomtsev-Pogutse (intermediate collisional) 1/3 2 1/3-4/3 4/3 EI E I I \ D K= 2/3 1/3 1/3 4/3 4/3 \1 + ~ / ' (14.385) (r Θ) ' « / n;/ e ' B ' E n
i
t
where r^ = din n/dr, Θ = (rr /B)/d(B§/r)/dr. Έseudoclassical (collisional drift)
V
-e " » e * V 8 î , 1 + W '
(14.386)
where the const is 10 or 5, see (12.95), to be determined experimentally. For decreasing v one has DB > DK > Dps. If v decreases further, four "collisionless" regimes appear. They are: collisionless drift regime (minimum of D and K«) (14.387) trapped
electron,
high collision
regime
(14.388) ^ _ \ Q -2 3/2-3/2-2^2 -2 D -2 // c T^e B I/ \v^+10v , or: 3^n r ' R E E o) where v Q = trapped
rT]/201/2/Rrnml/2,
electron,
T E o see (14.395))
low collision
regime
/14.437/ (for
3/2 (14.389) £> 0 = ε ' 21- v^/5 or = const. This formula is valid when the bounce frequency exceeds the effective collision frequency of trapped electrons. Finally, we consider for very small v the trapped ion collision-less drift instability regime
MICROINSTABILITIES
618 0
D
+r
2ε
a
5/2~2„2 oT 9 9 o
!
5·
(14.390)
Furthermore, other regimes like the Tokamak empirical diffusion regime /14.438/ and the steZZarator empirical diffusion regime or the convective cell regime have been defined. Actually, the stable operating regime of Tokamaks is merely restricted by instabilities: the disruptive instability sets in both at high and low currents and at high density. So we have a low-current operating limit, a high-current ing limit and a high density limit of about
operat-
10^2 one has the run-away
for
10^4 [particles cm~3]. For densities lower than about limit
(runaway
regime
E > Ecritt where Ecvit ~ n/TEt s e e P 1 2 1 ) /14.490/. Sometimes the various plasma regimes in a Tokamak experiment have been characterized by 3 dimensionless
parameters. These are the streaming ξ
=
u
/a
U
D\\E thE'
D\\E
=
parameter
V
n
e
(seep 387) (14.391)
and
γ^ = V ^ o " Φ | | / η 5 '
(14.392)
where λ„ is the electron collision mean free path, and « τ\'2/ητΕ,
Δ = rE^/TE
(14.393)
where T ^ J is the characteristic time for thermal energy exchange between electrons and ions (when Tj«TE)
and is given approximately by T^/^A/n-rln aspect ratio and τ ^ is the energy
Λ. A is the
replacement
time
(the time required for a plasma to lose a quantity of energy equal to its average kinetic energy). Other authors /14.440/ prefer observable parameters like the effective electron
by
energy
EE
3 o go 4 E||
or the energy
replacement
T
τ X
E
=1^EolhoL 4
E..<3,.>
replacement
time given
( 1 4 3 9 4 )
IH.JS4J
time
'
(14395)
U4.J»bJ
where o indicates the peak temperature (at the center
619
PLASMA INSTABILITIES
of the radial profile) and where En is the induced longitudinal electric field.
*PB-
2
™oVßp·
(14 396)
'
These parameters may be expressed in terms of TEo, n Q and the total current I: these parameters may be used regime, the to define regimes like the low density slide away regime /14.488/ (5x1012 < nE < 1.5*1θ13, - 0.25^, the transitional 0.7 < n0/I < 2, i>icrit (2 < n0/I < 4) or the quasilow collisional regime (nQ/I > 4, 1 0 ^ < nE < 5x10^4, resistive regime T%0 - lV2#1/2) i n which the heating of ions is collisional. Here^nQ is the density on the axis measured in 10^ 3 cm"37 τΕο is the dimensionless peak electron temperature, and I is the dimensionless total current. I is measured in 100 kA. Other authors /14.441/ define regimes by equations of state: banana regime —Ό 9ft
plateau
(14.397)
nT u '^° = const, regime
ηΤλ · 4 6 = const, Pfirsch-Schlüter regime
(14.398)
nT0'58 = const. (14.399) A parameter expressing the influence of the collicollisionality, sion frequency on Tokamak regimes is defined by C(r) = Ra(r)R3r/2/r3/2\rT ° ° C(r) = vEIA/ubE,
t2
or by (14.400)
where A ^ RQ/r is the aspect ratio and ω£# the electron bounce frequency. Small influence of collisions is defined by low collisionality (C < 1 ) . (p 4) is important for Since the Lawson criterion the construction of fusion power reactors, it is of
MICROINSTABILITIES
620
interest to calculate ητ for various regimes. For a total current I in [MA], temperatures in [keV] and a containment time calculated according to p 126, B in 50 [kG], one obtains for a given $pE the results pseudoclassical 14 2 1/2 ητ = 2χ1θ' I τΙ/Δ/Ζ _. (14.401)
E
eff
Here Zeffe is the effective charge, compare (2.73) trapped electron I 3 / 2
I 4
■ ^VEW ητ = 6x10'"l"£*e;:^Z ^ J / V ? /V ^ , trapped
electron
II
ητ = 10 1 3 Ι 2 τ1 / 2 Λ 1 / 2 /Ζ trapped
(14.402)
t.
ion
_,
(14.403)
ejj
ητ = 3 x 1 0 1 6 J 4 5 2 ß 2 E Z ß / / ^ 5 / 2 ( 1 + Τ Ε / Τ 7 ) 2 / 4 Τ ^ 1 / 2 . (14.404) Here A is the aspect ratio and the temperatures reactor are profile averaged values. A Tokamak fusion is expected to operate in the trapped ion regime and will require a current of about 10 MA or more for ignition (A = 3, B = 50 [kG], a = 360 [cm], q (a) = 3 ) . For a D-T reactor with 0.1 % molybdenum impurities one expects ignition at η#τ = (2-5)χ1014, Τ = 8χ10 7ο Κ (ignition device). In February 1977 at MIT ητ =2x10^3 at Tj- = 1θ7°κ has been reached and on PLT (Princeton Large Tokamak) 1 0 1 3 and 2x10 7o K are expected soon. As a general tendency the performance improves with increasing ε and decreasing A. Energy confinement time increases with plasma density and radius /14.482/. The plasma current I is ^eaB^/q. It is now of interest to understand how ητ and T scale when device parameters change (scaling laws). One important scaling law describes the dependence of the electron energy confinement time on the minor radius of toroidal machines. One obtains from experiments /14.440/, /14.442/ 3/2 τ
*~
3
eff
A l s o τΕ = 3 x 1 0 " 1 5 n ^ 1 j / 2 a 2 / j
1 0
·
(14
[ s ] h a s been g i v e n .
'405>
621
PLASMA INSTABILITIES
A maximum of ητ# as a function of nB% appearing at 10 3 e 11 /7( a # t ) 3/7,4-2/7^-12/7 has been found. For the electron temperature experiments yielded <ΤΊΡ> = Ζλ/2Ιλ/2/λΟα t
or * Iq^2,
(14.406)
ejj
(or TE s jV2#1/2 i n dimensionless units). More sophisticated formulae may be found in /14.384/ - MATT 1136. Βτ scaling laws are discussed in /14.486/. The scaling of plasma beta in a Tokamak has also been investigated /14.444/. The scaling law for Tj in the neoclassical plateau regime is given for hydrogen by /14.387/ (plateau regime) in the form T [eV] = 6x10~ 7 (I£,R 2
(14.407)
~U
and ^/2^1/2a-1/2 ±n the banana regime. A very wide range of present experiments seem to satisfy the scaling T „ [ S ] = 2χ1θ" 3 (<η>α) λ / 2 Βα,
(14.408)
or (
(14.409)
where I in [MA], B. in [kG], A = R/a (aspect ratio), ellipticity (e = b/a). Then the current density is given by j = I/i\a2e
(14.410)
(14.411)
and
In many experiments (see Tokyo Conference /11.57/) plasma density varies linearly with the plasma current. When instabilities with growth rate γ are involved, from D - y/k2 and p 126, formulae of the type e.g. (collisionless current driven drift mode) t
k2Bn Ξ τ * kT ^
U
y E
(14.412)
MICROINSTABILITIES
622
have been found. Stellarators obey scaling laws similar to those for Tokamaks. The Kurchatov τ
ΕΕ
scaling
law
( 1 4
~- **™~**\ol'
·
4 1 3 )
where Q = \η{ττ
+ ΤΛ V
(14.414)
is the energy content and V is the volume, seems to be satisfied by high current stellarators. The relationship T2?£ - na?- seems to be very well established for all devices. The particle
containment
Tp - 3.5τ# is proportional to the energy time
time
containment
τ^-
When writing scaling laws, some authors use the
Artsimovich
parameter
ζ = (l p B t
(14.415)
to formulate dimensionless equations. An accurate calculation of the neoclassical parallel conductivity σμ is essential for obtaining an estimate of the effective Ί Ä
Ση8Ζ1/ηΕ
charge
(14.416)
s*E
and hence the impurity content of a Tokamak /14.483/. Since the neoclassical conductivity relates the parallel current density j \ \ to the electric field averaged over a magnetic surface j'
= o..
(14.417)
the impurity content can be estimated from an approximate analytic conductivity formula /14.484/. In all collisionality regimes, neoclassical transport theory predicts a toroidal flow driven by radial gradients of temperature, density and the electrostatic potential /14.499/. Scaling laws must be invariant under transformations leaving the plasma equaignitions invariant /14.500/. Scaling laws for the tion parameter Τητ have also been given /14.501/.