A quantitative study of scattering from electromagnetic fluctuations in plasmas

Journal of Atmospheric and TerrestrialPhysics, Vol. 58, Nos. 8/9, pp. 983-989, 1996

~

Pergamon 0021-9169(95) 00129-8

Copyright © 1996 Euratom Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0021-9169/96 $15.00+0.00

A quantitative study of scattering from electromagnetic fluctuations in plasmas Henrik Bindslev JET Joint Undertaking, Abingdon, Oxfordshire, OXI4 3EA, U.K.

(Received in final form 20 April 1995 ; accepted 1 May 1995)

Abstract--A

new compact formulation is given of the equation of transfer for a scattering system for a magnetized plasma. The formulation is based on the low temperature kinetic model and accounts for scattering from fluctuations in density, magnetic field, electric field and current. It is demonstrated that scattering from several types of fluctuations can be significant and that the relative phases of the fields scattered by these fluctuations must then be taken into account by including cross-correlations between fluctuations. It is shown that significantenhancement or cancellation of scattered power can result from cross-correlations, and that this is of practical consequence to existing microwave scattering experiments on fusion plasmas. Copyright © 1996 Euratom Published by Elsevier Science Ltd

1. INTRODUCTION In this paper, we consider only homogeneous and stationary plasmas and neglect the effect of collisions. In such plasmas 'three-wave mixing and Thomson scattering are commonly modelled using the cold fluid model described, ff~r example, by Sitenko (1967). This model is accurate in the limit where the two interacting waves (in the case of scattering, the incident wave and the fluctuations) and the resulting scattered wave all are cold collective oscillations (Sitenko, 1995). The cold fluid model is, however, also widely used for modelling scattering from fluctuations that are not well approximated as cold collective oscillations. Bindslev 0993) showed that this use of the cold fluid model of scattering is not appropriate and proposed a low temperature kinetic model instead. This model assumes that thermal effects are small at the frequency and wavevector of the scattered wave but makes no assumptions about the nature of the interacting waves and hence, in connection with scattering, it makes no assumptions aboul: the nature of the fluctuations. For a general discussion of kinetic and fluid models of scattering and their limitations see also Bindslev (1995a,b). In a wide range: of scattering experiments in laboratory plasmas tlhe frequencies of the incident and the scattered light are sufficiently high that the plasma can be treated as cold for these waves. The phase velocities of the observed fluctuations are, however, typically of the same order or smaller than the thermal velocities of the electrons, and uncorrelated motion

may also have to be taken into account, both of which makes it unacceptable to regard the fluctuations as cold collective oscillations. Here, the cold fluid model is not guaranteed to be accurate, indeed large errors may occur, and the low temperature kinetic model should be used instead (Bindslev, 1995a). In this paper we consider scattering from general fluctuations (i.e. not limited to cold collective oscillations) and therefore base our considerations on the low temperature kinetic model rather than the cold fluid model. First, we give a new compact formulation of the equation of transfer for a scattering system. This formulation is particularly convenient for computational purposes and is also conceptually useful. A brief sketch is given of the derivation to make clear the physics content of the model and to establish terminology. The predictions of the low temperature kinetic model are investigated using the new formulation and it is noted that, when scattering from several types of fluctuations (e.g. r~ and ~) is considered, one must take into account the relative phases of the fields scattered by each type of fluctuation and hence consider not only auto-correlations for each type of fluctuation but also cross-correlations. Particular attention is paid to scattering when the angle q~ between the scattering vector and the static magnetic field is near n/2. This is the regime in which the fast ion collective scattering diagnostic at T F T R (Woscov et al., 1988), and occasionally that at JET (Costley et al., 1988), will operate to take advantage of the signal enhancement derived from the fast mag-

983

984

H. Bindslev

netosonic wave resonance. Here, scattering from fluctuations in quantities other than density is significant. For the parameter ranges relevant to JET and TFTR, and while staying clear of resonances and cut-offs, the assumptions of the low temperature kinetic model are satisfied and we can limit our investigation to fluctuations in density, r7, magnetic field, ~/, electric field, ~, and current, ~, and ignore scattering due to fluctuations in higher order moments of the velocity distributions. It is shown that scattering from ] can dominate the spectrum but, more interestingly, that it can be of the same magnitude as scattering from r~. In this case scattering from all fluctuations must be taken into account and the relative phases of the scattered fields are important. It is shown that the scattered fields resulting from ~ may virtually cancel the scattered fields from ~7, leading to a marked reduction in the scattered power, which is then dominated by scattering from ~ and.]. 2. K I N E T I C M O D E L O F E L E C T R O M A G N E T I C FLUCTUATIONS

Kinetic models of electrostatic thermal fluctuations have been studied extensively (see for instance Rosenbluth and Rostocker, 1962; Lifshitz and Pitaevskii, 1981; Klimontovich, 1982; Sitenko, 1982). Using Liouville's theorem Rosenbluth and Rostoker (1962) established the much-used dressed particle approach in the electrostatic approximation. Sitenko and Kirochkin (1966), and more recently Chiu (1991) and Aamodt and Russell (1992), have applied the dressed particle approach to electro-magnetic fluctuations. Liouville's theorem cannot readily be applied to such fluctuations. It is, however, straightforward to extend the microscopic kinetic discussion of electrostatic fluctuations due to Klimontovich (1982) to allow for electromagnetic interactions. We give a brief sketch of this here, which will serve to outline the model we use, establish terminology and confirm the validity of the dressed particle approach for electromagnetic fluctuations under appropriate assumptions. The microscopic distribution function f M = 5:/5(x-x~(t)) (x = (r, p)) is governed by the dynamic equation (Klimontovich, 1982, Section 24) ( O , + v . ~ + F M "Cqp)f M

= 0,

FM=

q(EM+v×BM) (1)

where (E M, B M) are the total microscopic fields. Introducing the macroscopic quantities as the ensemble averages f=(j.M),

E=(EM),

(2)

we can express the microscopic fluctuations, (~, ~7, ~), as

~=fM__f,

~,=EM_E,

~=BM_B

(3)

Taking the ensemble average of (1) gives (t?, + v" t?,)f+ F" ~?~f+ (~" c~,j7) = 0

(4)

Subtracting (4) from (1) we get (~, + v" ~, + F . O,)f+ ~'. O,f+ ~'. O , f - <~'. ~,f> = 0

O) Neglecting collisions eliminates the last two terms.* Neglecting the influence of other waves on the fluctuation spectrum, f and F simplify to f = f ¢ 0 ) , F = qv ×/~o), where f(o) is the unperturbed macroscopic distribution function and/~0) is the static magnetic field. Equation (5) then takes the form 5¢j~+ ~'" c~rf(°) = 0

(6)

.£~ = O, + v" O~+ q~(v x B ~m) " ~

(7)

where

is the linear operator representing the total derivative along characteristics (or unperturbed orbits) in phase space. Integrating along characteristics we find that

(8) where ~TO)(p,r,t) = f ( p ' ( T ) , r ' ( T ) , T )

~'(~,r,

t) =

; ~(¢(~), ~).

(9)

Gf(°~(~'(~)) d~ (lO)

The characteristics (r'(~), p'(r)) are the solutions to the coupled equations O+r'(z) = p'(~)/m,

~+p'(r) = q(v x B (°)) (1 la)

with initial conditions p'(t) = p ,

* This is a subtle point that is beyond the scope of the present discussion. The reader is referred to the discussion by KIimontovich (1982) for the electrostatic case.

B = ( B M)

r'(t) = r

(llb)

We note that the fluctuation in the distribution function naturally splits into two parts, y>~) a n d ~ ). y>~) represents the evolution of the microscopic distribution~if particle interactions were neglected from

Scattering from electromagnetic fluctuations in plasmas time T. It follows from (9) thatflr°) satisfies the differential equation ~?.~(r°) = 0

a

9 ~ ~

~s = ~ j + ~ } ] ~ a

= k X ~,

)(e> :=)(eO)_i~e0X
~

~(~0 ~' ~'(~o)± ~,(~) 7(~o) '-~ik

/, J~ a¢e

~-~-~ik

Jk

,

a#e

(20) which, for brevity, we will write as <~ifi,'>

~

~~<=)/;,~0),.'~0)\~(~)* ik \ Jk Jk' /Oi'k'

The correlation tensor for the unscreened currents (j~k(a0)~k(,a0)) = q2 f U k Uk' (~(a0)(p)~(a0)(/!') ) d ~ d ~ '

(22)

=W~.t(f(~°)(x, t)fl=°)(x', t')) = 0

+q,a ~) = k .)(e)

( f ~ ° ) (x, t') ~
(24) The latter assumes that initial correlations at time T can be ignored* as ( t ' - T) ~ ~ . The result is m 2 ~ ~ -~ -~ ~q~ /up/~-+p~ = tZ~) ~ d 1=~-~ -

o~i = G, Bi,ji, n

where we have introduced the fluctuation operators it

--i = A~ ~-,

(D~0

Cl¢~ f(a°)~&,Pll)

(25)

a = i,e

where

(16) O) - - l O ) ca

~(Ba)

~k

~

~

(23)

with the initial condition

05)

~(Ea)

(21)

(14)

we have ~

~'~ 2 (j~=o~j~,o~)~,?" ~± ~e<=~/X J~eO)~eO)Ne~)* (ai//~,) ~ e~"~ k dk" / ° ~ k "

(13)

where a indicates charged particle species. In the dressed particle picturc~ ~°) represents the unscreened test particles while~ ~) represents the screening of the test particles. Inverting (13) and making use of the relations ~

The transform of the correlation of any pair of the set of field and fluid variables can now be written

can, for a homogeneous and stationary plasma, be expressed in terms of the unperturbed macroscopic distribution function for species a, jaao), by solving expression (12) in the form

,

AoEfik, m) = - ' ~ t = ° ) ( k , m )

A,s = ~o+N-{~,~-~,s},

(19)

k, ~ , ~ ~oq~

~(kna) =

(12)

which represents the free streaming of particles along characteristics giw~n by (1 la). The particular set of characteristics depends on the configuration in phase space at time T and hence on the choice of T, and we notice that l i m r . _ ~ f ~ ) is not defined. However, the dynamic equation (12) g o v e r n i n g ~ ) is independent of T a n d so is ( f ~ ) > = (rio)rio)>. In the following discussion we will thus consider r i o ) = f ~ ) where t - T ' is much larger than all characteristic times associated with the plasma dielectric response, and only f o c a l l y let T' tend to minus infinity after ensemble averages have been taken, f o > = limr+ ~ f ~ ) given by expression (10) is the dielectric response of the plasma to the microscopic fields (~, ~). Writing the current )o ~ = q ~ @ ~ dp as the dielectric response, -i~e0X~, to the elect~c field ~ and letting )~0~ = q ~ 0 ~ @ be the current associated with the motion of non-interacting charges along characteristics, we have

~0

985

(17)

%+[cj~}~)

ct =

--iv±J~(k±p)

'

Ptl = m . - k~

(26)

vllYt(k±P)

,~
~
--" I~ij • ~jk

~,
(18)

* This point is discussed in some detail in Klimontovich (1982).

Here II and _1_ indicate the components of a vector parallel and perpendicular to B <°), Jl are Bessel functions of the first kind, of order/, and J; their derivatives, p = v±/~oca and ~oca= qaB(°)/m,~.

986

H. Bindslev 3. L O W T E M P E R A T U R E

KINETIC MODEL OF

-.

SCATTERING

J~

From the microscopic dynamic equation (1) one readily derives the expression for the field of the scattered wave (s) resulting from the interaction of an incident wave (0 with microscopic fluctuations (6). The result is (Bindslev, 1993) --"

-- ico%0 ['dkid~oi Ei n~O)

j

x{zT~h~

"

'*, ~ f ) ,

1

~i~ig° i -a

~1 [ ~

ki"+x~,~)

. i -~ / ig 0 t~e0 .~-~ x j ~ - / z ~ qTgX},+za~

cq~Z,~} (30)

.;

A(k ", co') • ~ / ( k ~, o9~) = ~ o J " ( k

(2g) 4

(27) Here Z~and Z~ are the cold electron susceptibilities at the frequencies of the incident and the scattered waves respectively, while

where) ~ is the source current }~ = q~ f v ~ dp

(28a)

1

~ k;

_~

~xZ,~ = Z ~ , ~ V , ~ a ~ ( ~ , + ~,~,) ~.O/a= _ F i ~J~6 •

B'~ ~ f i

(28b)

It is noteworthy that while the incident wave is macroscopic, the scattered wave, like the fluctuations, is microscopic and the ensemble averages of linear dynamic variables associated with the scattered wave are identically zero (e.g. ( ~ ) = 0). Macroscopic equations, such as the Vlasov equation, thus do not rigorously describe scattering from microscopic fluctuations. However, due to similarities between the Vlasov equation and the microscopic equation (1) much of the detailed derivation is similar for the two models. The kinetic expression for the source current, (28a) with (28b), involves the full detail of the distribution function perturbations, ~ and f , associated with the fluctuations and the incident wave. This makes the integrated expression for the source current numerically intractable. Its complexity can, however, be significantly reduced by assuming that the plasma response is cold at the frequency and wavevector, (k ~, ~os), of the scattered wave, or more precisely that ( ) r . ~ ) is significant only for values of ~ that satisfy the inequalities

v

-<<1; C

v,~

--<<1,

¢~0~ + S09 c

s~Z;

vl~:~_

--<<1 O) c

(29)

With these assumptions the kinetic expression involves only the zeroth and the first-order moments of the distribution perturbations associated with waves (i) and (6) (Bindslev, 1993, equation (57)). If we further assume that the plasma dielectric response is cold at the frequency and wavevector (k i, coi) of the incident wave we can use the cold plasma relations to express/~i, ji and n i all in terms of E i. We then find that the source current is given by (Bindslev, 1993)

(3~)

with Fg~ defined by 1 I~i~alb -- ~

4~

[ - i~'~(~]iaCjb "q- ~ia~l jb )

1

- 2f~2{,~{~ + (1 - 2~ 2)r/~,~/jt,] + 1 - ~ x [r/~,~ + ~i~r/~o- i t 2 ( ~ b + ¢ ~ ) ] + ~,~b,

(32)

and ~'~ = ¢-0c/O)~r,

~]ij = 6 i j - - 6 i 3 6 j 3 ,

~ij = eij3,

~ij = 6i36j3

Solving for ~ in (27), taking the ensemble average of the square of ~ and accounting for the transfer of power through an anisotropic inhomogeneous plasma in the WKB approximation (Bekefi, 1966) we find the equation of transfer for a scattering system (Bindslev, 1992, 1993) giving the power received by a diffraction limited receiver Op s --~

~3o9~

1

pio~(2io)2r~en(°) f-- ~

2g

(33)

2io = col~c, re = q~e/4~eOmeC 2 is the classical electron

radius and Ob is the beam overlap, which is defined in terms of the normalized beam intensities, J , for the transmitter and receiver beams and which, in the case of uniform beam intensities and perfect intersection, is identical to the ratio of the scattering volume, V, to the product of the beam cross-sectional areas, A Ob = f~ 3 Ji(r)~Cs(r) dr ~ A iV A~

(34)

Ob will have some frequency dependence due to vary-

Scattering from electromagnetic fluctuations in plasmas

987

10-~ /

G~~, = - t

[e,~,l~, ,. t*'~

\

/(Z~I ~0

l~ o i j~ Xs,s~~ . , ) , ~i ~e i (39) ~ ]~ s*s'

1 0 -1(

The n o ~ a l i z e d flux and field o£the incident radiation is defined by I0 -ll

~ ' = N ' l ~ ' - ~ e ((~"e')(~')*}l,

.

z rad

/

10 -14

\-

'

~, = ~ic/~, '

\ ".,

~ = ~i/l~'l,

~, = , i / , ,

Similar expressions for ~ ' and e' relate to the received scattered radiation. All quantities in E refer to conditions in the scattering volume.

10 -16

60.0

60.2

60.4 vS/GHz

4. INVESTIGATIONOF THE SCATTERING FUNCTION

60.6

Fig. 1. Scattering function, ~ (solid curve) and diagonal elements E,~, YB~, £ee and ~ (broken curves identified by nn, BB, EE and jj respectively), plotted as functions of the frequency of the scattered radiation, v~. Parameters: v i = 6 0 G H z ; B~°)== 5.0 T, n ~ = l x l 0 2 ° m - 3 ; no=nr= 0.48n~; n= = 0.02ne; T~ = 12keV; Tn = T r = 30keV; the alpha particles have a classical slowdown distribution with birth energy E~ = 3.5 MeV and critical velocity v~ = 0.1vr; 0 = L ( k i, k')=20°; ~ = L(]~s-]~i, /~°))=86°; L(ki[ B~°))+L(k~, B~°)) = 180°; mode of incident wave = X, mode of scattered wave = X.

The scattering function, E, consists of diagonal elements such as E,, and Ess and off-diagonal elements, or cross terms, such as E,n. All elements satisfy the symmetry E ~ = E~'~ and consequently E~.~, considered as a matrix with indexes (~, fl), is Hermitian.~" It follows that the diagonal elements are all real while the off-diagonal elements are, in general, complex, but the sums Y~,a+ Ea, are always real. While

10 -6

ing refraction but most of the spectral variation of the received scattered power, P~, is due to the scattering function, E, defined as y~ = ~ E,a, ~

1o4

10-1° ~ : = _ _ _ ~ / - tad

E~

(o~~co~)~

4

-

Op

~

1 ~,

~)~)/,~0),~0)\~:~),&~¢)* ~i

~ i k N J k Jk" / O i ' k ' ~ i "

(35)

.. .. ..... .... .

~1 ~

"-~,Ai

-

O ./ :

with the dielectric coupling operators, ~, defined as ~("~ = (e~)*z~e~ ~s)

=

_ re,-,,., e

~ i,I ~ij jmk

i & e 0 ~i e i qe

~ml I

s * l ,lXUe~me~,,~ /'qe~~'k~ ~X,kl~)~ ~;~ = (e~)

(36)

10 -1~

(37)

10-1~ 140.0

(38)

~"This would remain the case if we further subdivided the vector elements into elements associated with orthogonal vector components, thus replacing, for instance, EB, with the set of elements E~,~ where the index i identifies orthogonal components (e.g. x, y, z).

V I

I

140.8

I

I

141.6 ~'/GHz

I

I

142.4

Fig. 2. Scattering function, E (solid curve) and diagonal elements E.., EBB, Eee and Yjj (broken curves identified by nn, BB, EE and jj respectively), plotted as functions of the frequency of the scattered radiation, v'. Parameters: vi = 140GHz; B~°)= 3.4T; ne=3×1019m-3; no=or= 0.48n,; n~ = 0.02n,; Te= 12keV; T o = Tr = 30keV; the alpha particles have a classical slowdown distribution with birth energy Eb= = 3.5MeV and critical velocity v, = 0.1vr; O= L(k ~, kS)=20 °, ~ = L ( ~ ' - I ~ , /~°))=88°; /_(ki~ B~0))+/(k ~, /~0))= 180o; mode of incident wave = X; mode of scattered wave = X.

988

H. Bindslev

the diagonal elements are positive definite the real parts of the off-diagonal elements may be either positive or negative. If fluctuations in the various quantities were uncorrelated then the off-diagonal elements would vanish and the scattering function E would be equal to the trace, ~ E~, from which it would follow that E was always greater than any individual diagonal element. The fluctuations are, however, not uncorrelated. They are all driven by the same unscreened current (see (15) and (20)) and all have well-defined relative phase relations determined by the fluctuation operators, ~. The same holds for the resulting scattered fields, the relative phases being modified by the dielectric coupling operators, (~. The consequence is that the offdiagonal elements, Z~, ~ ~ r , do not vanish and Z is not equal to the trace. This opens the possibility that E may, in some conditions, be smaller than individual diagonal elements such as Z... In many conditions, scattering from density fluctuations dominates the spectrum of scattered radiation, E ~ Z.., and in some situations scattering from magnetic field fluctuations dominate, Z ~ Z ~ . There are, however, also situations of practical interest, for example in microwave scattering in fusion plasmas, where scattering from density and magnetic fluctuations are of similar amplitude (Z...~ Znn). The relative phase of the scattered fields resulting from each type of fluctuation is then important. If they are

in phase the fields add and the sum of the cross terms Xne+Zs, is positive so X > X~,+Xns. If, on the other hand, the scattered fields are out of phase then they cancei and Z,B + 5:B, is negative so 5: < 5:,, + Znn. This is illustrated in Fig. 1 where the scattering function, 5~, and the diagonal elements, E,,, Xaa, EeE and Ejj, are plotted as functions of the frequency of the scattered light. The curves are calculated with parameters relevant to the fast ion collective Thomson scattering diagnostic at T F T R (Woscov et al., 1988) and show a typical situation for this diagnostic. The structure at low frequencies is due to damped ion-Bernstein modes while the peak at vs ~ 60.47 G H z is due to the fast magnetosonic wave. We note that there is a range of frequencies in which E is smaller than 5:,, by up to two orders of magnitude and that, in this range, E,, .~ Z s s while Z~s+E~, ~ - ( Z , , + E s a ) , resulting in near cancellation of scattering from ri and ] . In this case, scattering from current fluctuations,], is important. In Fig. 2 we investigate a set of parameters relevant to the fast ion collective scattering diagnostic at JET (Costley et al., 1988). We note that here scattering from ~ is most significant but that some cancellation occurs due to scattering from other fluctuations, most notably]. It is interesting that for these conditions, at frequencies above the peak due to the fast magnetosonic wave, scattering from ff is least significant, with Z,, smaller than E and all other diagonal terms by several orders of magnitude.

REFERENCES

Aamodt R. E. and Russell D. A.

1992

Bekefi G. Bindslev H.

1966 1992

Bindslev H.

1993

Bindslev H.

1995a

Bindslev H.

1995b

Chiu S. C.

1991

Costley A. E., Hoekzema J. A., Hughes T. P., Stott P. E. and Watkins M. L.

1988

Klimontovich Yu. L.

1982

Lifshitz E. M. and Pitaevskii L. P. Rosenbluth M. N. and Rostoker N.

1981 1962

Alpha particle detection by scattering off of plasma fluctuations. Nuclear Fusion 32, 745-755. Radiation Processes in Plasmas. Wiley, New York. On the theory of Thomson scattering and reflectometry in a relativistic magnetized plasma. D. Phil. Thesis, Oxford University, also available as Riso report: Riso-R-663(EN). Three-wave mixing and Thomson scattering in plasmas. Plasma Phys. Controlled Fusion 35, 1615-1640. Comparison of kinetic and cold fluid models of threewave mixing and Thomson scattering. Plasma Phys. Controlled Fusion 37, 169-176. Low temperature models of non-linear wave mixing in plasmas. Ukrainian J. Phys. 40, 457-463. Also available as JET preprint JET-P(94)71. Electromagnetic effects in fluctuations of nonequilibrium plasmas and applications to alpha diagnostics. Phys. Fluids B 3, 1374-1380. A Thomson scattering diagnostic to measure fast ion and or-particledistributions in JET. JETReport, JETn(88)08. Kinetic Theory of Non-ideal Gases and Non-ideal Plasmas. Pergamon Press, Oxford. Physical Kinetics 1st edition. Pergamon Press, Oxford.

Scattering of electromagnetic waves by nonequilibrium plasmas. Phys. Fluids 5, 776-788.

Scattering from electromagnetic fluctuations in plasmas Sitenko A. G. and Ki.rochkin Yu. A.

1966

Sitenko A.G.

1967

Sitenko A.G.

1982

Sitenko A.G.

1995

Woskov P. P., Machuzak J. S., Myer R. C., Cohn D. R., Bretz N. L., Efthimion P. C. and Doane J. L.

1988

989

Scattering and transformation of waves in a magnetoactive plasma. Soviet Phys. Uspekhi 9, 430-447. Electromagnetic Fluctuations in Plasma. Academic Press, New York. Fluctuations and Non-linear Wave Interactions in Plasmas. Academic Press, New York. On the applicability of the fluid approach in the description of electromagnetic wave scattering in plasmas. Plasma Phys. Controlled Fusion 37, 163-167. Gyrotron collective Thomson scattering diagnostic for confined alpha particles in TFTR. Rev. Sci. Instrum. 59, 1565-1567.