Nonlinear plasma oscillations

ANNALS

OF

PHYSICS:

28,

478-499

Nonlinear

(1964)

Plasma

W. E. DRUMMOND John Jay

Hopkins

Laboratory for General Dynamics

Oscillations* AND D. PINES?

Pure and Applied Science, Corporation, San Diego,

General Calijornia

Atomic

Division

of

The nonlinear limiting of unstable plasma waves is developed by expansion in powers of the field energy. The lowest order nonlinew terms lead to a diffusion of the background distribution and this in turn reduces the growth rate of the instability, and a stationary “equilibrium spectrum” results. The corrections to the lowest order theory are calculated in detail and it is shown that these “mode coupling” terms are negligible except for very long times for which they lead to a slow decay of the equilibrium spectrum.

I. INTRODUCTION The collective behavior of a fully ionized plasma, in which the number of particles in a sphere of radius a, the Debye length, is very large compared to 1, is governed by the collisionless Boltzmann or Vlasov equation. In an infinite homogeneousplasma of this type it is well known that certain velocity distributions lead to unstable (growing) oscillations. The frequencies and growth rates of these oscillations are obtained by linearizing the Vlasov equation about the unperturbed distribution function, and this leads to exponentially damped (stable) or exponentially growing (unstable) solutions. After a sufficient time the unstable solutions evidently grow to such an amplitude that the nonlinear terms become important and the linearizing of the Vlasov equation is no longer valid. The question then arises as to the ultimate fate of such unstable oscillations. It is this question that we wish to consider. It will be shown that the development in the nonlinear regime for certain types of unstable modes can be followed in considerable detail for long times. This is illustrated by unstable electron plasma oscillations. The result is that those waves which are initially unstable grow in a short * A somewhat brief summary of this work was given at the Sal&burg Conference, paper CN-134, and the present document is being issued in order to give to those interested a more complete account of the nonlinear theory. This work was performed in late 1969 and early 1961 and written up in the present form at that time. The work was carried out under a joint General Atomic-Texas Atomic Energy Research Foundation program on controlled thermonuclear reactions. t Permanent address: Departments of Physics and Electrical Engineering, University of Illinois, Urbana, Illinois. 478

NOXLINEAR

PLASMA

OSCILLATIONS

479

time to a quasi-stationary spectrum (in 1~space) and then decay slowly to zero. The limiting of these waves to a quasi-stationary spectrum is a result of a change in the velocity distribution due to nonlinear effects, and cannot be obtained in the magnetohydrodynamic approximation. The decay of this spectrum is, however, essentially a magnetohydrodynamic effect. The method of solution is to divide the nonlinear terms into two groups. One of these, combined with the linear terms, yields the nonlinear dispersion relation, which leads to the establishment of an equilibrium spectrum. The other group of nonlinear terms provides coupling between the different modes and it is this coupling which leads to the eventual decay of the spectrum. In Section II the linearized solutions are discussed and the nonlinear dispersion relations are developed in Section III. Section IV contains a description of the decay of the equilibrium spectrum and the results are discussed in Section V. II.

The collisionless

Boltzmann

LINEARIZED

or Vlasov

THEORY

equation is

(1) where F(r, v, t) is the distribution function, i.e., the number of electrons per unit volume, per unit velocity, v is the velocity, e and m are the electronic charge and mass respectively, and E( r, t) is the electric field. We set F(r, v, t) = Fe(v) + f( r, v, t) where F,(v) is the unperturbed distribution function which is independent of space and time and is made up of a spherically symmetric distribution of density nl electrons/vol which is a monatonically decreasing function of energy, and which has a mean square velocity 7,’ plus a beam of electrons of density nJvo1 with mean velocity l)b >> z/VI2 in the x direction and a mean square Velocity r&tiVe to vb of fit,‘. The parameters are such that $/au, > 0 in the neighborhood of 2)b. It is assumed that nl >> nb , nG%* >> nbub*, nb@b*, and that there is a uniform background of positive charge such that the over-all plasma is electrically neutral. Figure I illustrates the projection of such a distribution function on the v2 plane. In addition we have V.E = 4?rp = 4?re

I

f(r,

v, t) d3 v.

(2)

The charge density 4?reJFo(v) d3v is cancelled by the uniform positive background of charge. We expand f( r, v, t) and E(r, t) in Fourier seriesin space to obtain (3)

480

DRUMMOND

AND

PINES

7 F,,(v)

FIG.

1. Initial

velocity

distribution

of electrons

‘b showing

the gentle bump at vt

where fk(V, t> = +, j” e--ik’rf(r,

v, t) d3 r

Ek(t) = k 1 ePik” E(r, t) d3 r f(r,

v,

t)

=

cfk(v,

t)

eik”

k

E(r, t) = F Ek(t)eik” and 2k.E = 4?Tpk= 4?re s

fk(V,

t) d3 0.

(4)

Further we restrict ourselves to the Coulomb interaction for which Ek is parallel to k. We wish to solve the initial value problem which is: givenfk(v, 0) and Ek(0) [self-consistently], what is fk(v, t) and Ek(t)? We shall assume that the initial data is sufficiently small so that the linearized theory is initially valid, and further we shall assumethat 1Ek(O) 1is roughly constant for ka < 1 [a is the P2 as in the case of random noise and no assumption will Debye length, d-1 be made with respect to the initial phases. Integrating Eq. (3) along the unperturbed orbits yields (for f > 0) fk = -

; 6’ dt’ Gk(t - t’) E&%‘~

Fo (5)

-tg

'd( 0

where Gk(t) = exp (-zkqvt].

Gk(t

-

t')Ek-&t')*&fk(~)

+ Gk(t)fk(v,O)

NONLINEAR

PLASMA

OSCILLATIONS

481

We wish to restrict our attention to modes which are unstable, i.e., o/k E vb . These modes grow exponentially and after several e-folding times the lower limits can be set to - m and the term in fk (v, 0) can be neglected. Thus we obtain

fk = -; 11dt’Gk(t

- t’) E&‘)-Vv

Fo (6)

- ; F jt di Gk(t - t’) E,&tl)V&(tl). -cc Integrating

over v and making use of Eq. (4) then yields t

Ek=-4&

s

d3v

s--m

dt’G,c(t

- t’) E,&?%

Fo (7)

-- t’;tz

9

jt dt’ G,c(t - t’) E,+q(tl)VJq(f). --oc

jd3v

In the linearized theory we neglect the second term on the right hand side of Eq. (7) and set Ek(t) = Ek(0)eskt where Sk = -iwk + Tk, Re yk > 0 to obtain

(8) or Ek(t)e(k,

Sk) = 0

where 4re2

e(k,Sk) = 1 - fc2m

ik.V, Fo

s d3v Sk+i k-v.

e(k, Sk) is the dielectric constant of the plasma. Sk is determined by solving the equation c(k, Sk) = 0.

(9)

To find e(k, Sk) we must evaluate the integral in Eq. (8). As discussed by Landau (1) this yields (here for simplicity we consider only a one-dimensional plasma. The three-dimensional case is contained in Appendix A.)

(10) (we have assumed that yh/k ) where

aF!aVk

varies only slightly in a velocity

increment

482

DKUMMOKD

AND

PINES

24 =- 2a2e2aFo(v) I k2 1n au v=wk Wk

<< 1 ‘I;

5’ = ; /- dv t? Fe(o) n= w

s dv Fdv)

,2 = 4me”

m

The solution of Eq. (9) we denote by Sk0

(ka)* << 1

(11)

insures that the wave is t.raveling in the fc direction. For future reference we that E(k, Sk) g 1 + (~p2/.$). It should be noted that the frequency Wk does not depend on the details of the distribution function whereas the growth rate, yk , is proportional to the slope of the distribution function at the phase velocity of the wave and is positive if IlOtC!

af/av > 0. III.

NONLINEAR

DISPERSION

RELATIONS

From among the terms in c, Ekmp(a/aV)f, on the right hand side of Eq. (3) we select the term with q = 0, i.e., E&( a/av)fo and define g(v, t) = F,(v) + ,fo(v, t). The nonlinear one-dimensional Vlasov equation thus becomes

and for

k = 0, since afo/at = as/at

where the prime on c,’ denotes that the q = 0 term is to be deleted. In the term Eo(a/av)g in Eq. (13), E. is determined by the boundary conditions. If the different boundaries are held at different fixed potentials, E. is nonaero; however, if we apply periodic boundary conditions (to the potential) Eo = 0. We shall take Eo = 0 for the sake of simplicity. The nonlinear terms in c, represent an interaction between different modes, whereas the nonlinear part of Ek(d/&)g gives the nonlinear part of the disper-

NONLINEAR

PLASMA

483

OSCILLATIONS

sion relation, and this leads to a slow variation of the frequency and growth rate with time. Our procedure is to first solve Eqs. ( 12) and (13) neglecting the mode coupling terms and then to treat these as a perturbation. The justification for this is that the solution without mode coupling does not grow indefinitely as in the linearized theory, but instead comes rapidly to an equilibrium spectrum, the amplitude of which is small. Since the equilibrium amplitude is small, the nonlinear mode coupling leads to a rather slow decay of this spectrum. Dropping the mode coupling terms and integrating Eq. (12) along the unperturbed orbits yields

fk = -p L; dt’G&- t’)&Jt’) ?&g(i) where again we have set the lower limit to - 00 and neglected fk(v, 0). Integrating over velocities then yields

l&(t) = -4z;/dv/-t --m dt’Gk(t

- t’)&(i)

;g(t’).

We expect the solution of Eq. (15) to have the same general form as the solution of the linearized problem except that the frequency and growth rate will be slowly varying functions of time. Then we take Ek(t’) to be of the form

Edt’) = Ek(O) exp { d t’ Sk(r) d7j

= = Ek(t)

(16) exp (-.sk(t)(t

-

t’)}

1 - f 2

(t - 0’

i

+ **.

1

where we have expanded sk( 7) about r = t. We shall assume and verify later that So changes slowly in a period of oscillation and that the change in frequency and of growth rate are of the same order of magnitude. In particular we assume

1--- a-f 1 yauw'

--l aw1

7 at w

<
(17)

Using Eqs. (16) and (17), Eq. (15) yields

The term involving

Fe(v)

is the same form as in the linearized

theory

and thus

484 we

DRUMMOND

AND

PINES

have

I&(t)

{

1- f G

;

1

4/c, 4t)) (19) =

1 - ; 2

-$}

‘$

j dv jk;j;$?;

.

i NOW

( sk( t) -

sk,,)/skO << 1 and thus we can expand ~(k, &(t))

about

Sk0 to

obtain &, Sk(t)

]

i%

E(k,

Sk01

+

-

(Sk

-

sko)

ask0

= $

(Sk -

&q,) i% -2$

[Sk(f) -

~

2

Sk01

0

ko (Sk(t)

-

Sk,,)

%

since c( k, sk,,) = 0. The correction terms on the left hand side of Eq. (19) of the order of since c E 1 + (Op2/sk2), ask --

af

i --

%dp2 Sk4

-

wp

wp2 at

(

ask

-

1

7 at w >

<<

are

32

wp

and can thus be negIeeted. It will be shown that fo(v, t) is nonzero only in a small neighborhood Au, where (Au) is of the order of z/g about v = 2)band thus (the three-dimensional case is contained in Appendix A) a.fdav

I = !- dvSk(t)+ ikv=

”

(20)

s

a21/as2 is thus of the order of I/(~Av)~

and the correction terms are of order

~i!G!-= at (kAv)’ since ( v~/Av)~ ‘; < 1. Thus the correction terms can be neglected, and the result is 27re2

+ iP 1 dv (;ff;v) v=4k

Combining this with Eq. (11) we obtain

(21)

NONLINEAR

PLASMA

485

OSCILLATIONS

Sk(t) = -iWk(t) + Yk(O

1

dv

Wk(t) = h Wko 2?r*e* ag(v, k%z &I

Yk -=Wk

t)

(22) (23)

v=ofJk '

The change in wB(t) is of the same order as the change in 7k and since yk/wk << 1 we can neglect the correction to wk. The change in yh is, however, significant. The energy in the kth mode thus grows according to

al ~ Ek I* = 21 Ek /‘-fk(t) = 1Ek 1%(k) ‘+ at

1u=aklk

(24)

where

In the same way Eq. (14) can be integrated fk

=

-k

$&

to obtain 2

and thus Eq. (13) becomes

=(:Jt zP1Eq (2 {&i !liqv)} 2 P +(s--, I2 (wq - ;Gz + 0; 2$F1Eq

ZZ

(26)

e

y

I &/a I2; 2 = $ PIJ&/u I22 where 6 = (2?r/v2)(e/m)2. If we define k = Wk/u, ( Ek j = 1E(u) 1, and 1E(u) 1’ = rL(u), Ew and (27) can be written as

(27)

(24)

(28)

(29)

486

DRUMMOND

AND

PIXES

The nonlinear dispersion relation thus yields the result, Eq. (28), that the growth rate of the uth mode at time t is determined by t,he slope of the distribution function at u and t, a not unexpected result. Equation (29) is a diffusion equation in which ,8$(u) plays the role of a diffusion coefficient. Thus the presence of energy in the uth mode leads to a diffusion of the velocity distribution at U. The temporal behavior of this pair of equations can be described as follows. If 8flau is positive at u then #(u) increases. However, as #(u) increases so does the diffusion of f(v) and this tends to reduce @/&A. Thus the behavior of the pair of equations tends to limit $(u). In order to determine the asymptotic behavior we combine Eq. (27) and (28) as follows

and therefore

(30) We assume that (p/a)(~?#/&)

is negligible at t = 0 and thus ! t!! = g - F,, . a au

We either to be ul- as

au

now seek a solution as t -+ 00 for which a$/~% = as/at = 0, if a&/at = 0, 4 = 0, or ag/du = 0 let dg/du = 0 for u. < u < u1 where ua and u1 are determined,-and let + = 6, g(v) = F,(v) for all other u. For u. < u < = 0 and therefore

g = gp = g(uo ) = g(ul)

= F(uo)

ga - F and

= Flus)

thus (31)

“&G(u) =s,,(s-- F).

(32)

Now J?, g(u, t) - F,(u) = 0 from Eq. (29) since the total number of particles is considered, and therefore, since g(u, m) = Fe(u) for u < ~0, u > u1 J This, together with

gm = F(uo)

.,’ (gm - Fo) = 0. = F(uo)

f $(uo)

determines

= t II/

= 0.

gm , ug, ul and also yields

NONLINEAlt

PLASMA

487

OSCILLATIORiS

g,(v)

E*(v) l& FIG. 2. Initial F. and final, g, , electron and the final wave spectrum E as a function

Figure 2 illustrates

F”(u),

distribution of phase

gm(u) and q(u) IV.

MODE

in the velocity.

neighborhood

of the

bump

thus determined.

COUPLING

We now wish to consider the effect of the nonlinear mode coupling terms in Eq. (12) on the equilibrium spectrum. We first note that the equilibrium spectrum is relatively narrow and thus f, , and G--q are large only if 1Q 1, 1 k - Q 1 g E. , where k. denotes the center of the spectrum. It follows that 1~ must lie near 0, *2kc,. Thus to first order the mode coupling terms lead to waves near k = 0, f2ko, and to no change in the spectrum near lco. In second order the first order waves near lc = 0, f2ko will interact with the equilibrium spectrum. To find this we iterate Eq. (12) twice to obtain

fk = -; i’dt’Gk(t - t’)& 2 s(i) dt’G& x I” -

dt”G,(t

-

t”)E,(t”)

t -e3 dt’G& q41 =oi ‘m 4

x I” dt”G,(t’ 0

- t))E~+~(tl)

I# s

$ g(P) (33)

- t’)iLq

- P)Eq-*~(

X

;

t” ) g

dt”‘G,, ( t” -

0

;;

P’)E,t

5

g(P)

+

* - .

488

DRUMMOND

ASD

PINES

and

dt’G,&

- t’)Eh.&‘)

;

t’ X

dt”G,(t’

s

- t”)E,t(t”)

; g (34)

- i

s dv I’ dt’G&

(,fJ

-

&E&i)

2

t’ X

s0

dt”G,(t’

- t”)E,+

$

x 1’” dt”G,,(t”

- t”‘)E,Jt”‘)

0

We now attempt

a solution

; g . >

of the form Ek + E/?!

(35)

El” is of second order in the electric field, and we shall keep to terms in Eq. (34) only through third order in the electric field. To first order Ek is just the solution described in Section II. To second order we have: EL”

= / dv g

6’dt’Gk(t

-

(zi)

(ii)

+ T

t’)Ei2’(t’)

6’

$ g

dt’G&

- t’)E~-&‘)

5

(36)

I’

X Ei2’ will oscillate at two frequencies, the natural frequency (Jk . However, we at wk-* + wq since the terms with this way in the next order. For these terms instead of k) Ec2’(t) 4

=

c

Edt)Wt) Q

where

s0

dt”G,(t’

- t”)E,(t”)

the driving frequency are interested only in frequency will interact Eq. (36) yields (using

htq 4&)

s 7

Q,

q’>

;

.

W&.-q + wq and the oscillation in a resonant the variable q

(37)

NONLINEAR

PLASMA

489

OSCILLATIONS

and as before Ek has a time dependence of the form exp{l

dr}.

Sd7)

For q e 0, ( q’ 1 g ko S, =

-i3w,

pi q’ 1 [I

- 21

a2 =

-iqvo

[ al1 1-C

vo=31q’/

av<
and the integrand in h(q, S, , q’) has poles at v = v. and at v = wqI/q’. The pole at v = w,#/q’ leads to terms involving ag/av jv=wr,,pgand since ag/av is rapidly brought to zero by the nonlinear dispersion relation we neglect this pole. For q e 0 we thus obtain

sQ+”iqvaav s9’; iq,v; 2 (38) Eq* >zi=oo i Eqeq’ when P means take the principle part. For q g 2k0, l/(S, + iqv) has a pole near v = w~,/kOand we neglect this to obtain E(k,Sq-q’+Sq!)~l

-g=; *”

and ;

P

(q + q’]E,-,,(t)E,dt),

q E 2ko.

(39)

We are now in a position to find the third order (in E) corrections to Ek(t). From Eq. (34) we have

490 e&(t)

DRUMMOND

=

/

dv${

-;

lt

dt’Gdt

AND

-

tl)Ek(tl)

F(zz)” [6’

+

dtlGk(t - tl)Ekeq(tl)

+ it dt’Gk(t 0

- &E;‘(f)

2 s”

PINES

2

; lt’

(t’)

dtNGq(tl - t”)Ef’(t”)

dt”G+(t

- t”) E+(P)

2 (t”)

2

(40)

0

I -e t dt’G& - t’) E+(i) 99’ m 0 w t dt”‘G,,(t” - 1”‘) E,,(t”) X I0 -

; I’

dt”G,(t’

- t”) E&t”)

;

.

In the same way as in Section III this yields de aE, ask at -(

$

>

Sk0 Ek - $$ / dv

x s dv [6' dt’Q(t + l’dt’G,&

- t’) E&i)

xl? + 2 ’ (Sk + Ike) au 1 It’

dt”G,(t’

(!?)'

C' Q

- t”) E:‘(P)

F

0

- t’) E:‘(t’)

;; (41)

X

s0

t’ &“Gk-,(t

- t”) Ek-Jt”)

2 (t”)

x /- dv 1’ Gk(t - t’) Ekpq(t’)

(;)”

; /“dt”Gq(tf

0

x 6’” G,s(t”

- 2

- t”) E,-,!(P)

0

_ t,r,) ad av t’fl> E,,(P)

c q-2’ $

.

The first two terms on the right-hand side of Eq. (41) are just those obtained in Section III, and we see that the mode coupling terms introduce an additional time dependence to Ek( t) . We consider first the case of p E 0. The term involving Ek-qE(2) yields c h(h w’

4, q’)Ek-,E,,-%

where h 1 =!6

e

1

J~(lc, q, q’) = wi (i)/dv Sk-q

+

sq-qf

(42) +

sql

+

ikv

av

NONLINEAR

PLASMA

491

OSCILLATIOiYS

It is convenient to expressJ1 in terms of the dielectric constant A& + A%-,~ + S,l by Sk and integrating by parts we obtain

t( y, S,). Denoting

1

1 ag ___--1 8, + iyv n an

where 316%

Y,q’> = --idme (Sk + 1ikv0)2 ’

2 e -

fwp

m

S dv [(Sk

:ikv)2

-

1 (Sk + ikzkJ2

1

1 lag (S, + iqv) k av

has no pole at v = v. and thus is purely imaginary. We thus obtain

2 1 Y')= [WI0f (Sk +&.vo)2 +g 01 ;5 1 las+c!. { ‘.l”(s, T iyv) t (s,, + $8) i av 1y 1 a21[ (S,,

hl(h!L ’

The term in Eq. (41) involving

Ep’Ek-,

c Mk, QQ’

1i

(44)

1-- ag l + iq’v) n aa Il=VO*

yields

q, q’)Ek-,E,-,&

where h2 = 5 h e Jz(lc,y,q’)

1 ag ’ “[p t--] (Sk + ikv) au Skeq + z(k - q)v n au

= g(;)/dv

(45) 2

2 WP

e -

dv (Sk :

0sm

ikv)2 (Sk-q + i(,k - y)v) i ii

which has no pole at v = v. and hence is purely imaginary. The term involving Ek-,E,+E,t yields c where

0s

Mk,

y, y’b%qEq-q&y

2

h3=--wp2

2

m

1 (Sk + ikv)’

1 a (& + iqv) %

-1

1 1 ag (Sq’ + iq’v) n av

(46)

492

DRUMMOND

AND

PINES

2 j;/aaU S,, + 1 0iz(Sk +likvo)z

the imaginary

part of which is

1 1 at7 _iq’v ii aV - cd,” 1 and we note that this just cancels the imaginary part of the term in hl which does not involve E. Thus H(lc, q, q’) = hl + hz + h, -

H(k,Q,n’>= w,”(zJ+[(sk:ikvo)~

- (SkAkV)J

1

($1+ Jd

+

and we note that the imaginary part of e. Making use of the identity

E

part of H comes entirely

(47)

h(k, Q,!I’>

from the imaginary

Sk + ikvfl = Sk + ikv - i ( - iqvo + iqv ) = Sk + ikv - i (S, + iqv )

we obtain 1

1 II (Sk :

ikv)2 -

(Sk

+

1

(S,

+

1

= -- k q

ikv$

(Sk

+

H(k,q,a’)= ; w;($ + (Sk + ikvf&k

1

ikv#

and thus, after integrating

iqv)

(Sk

+

+

ikv)

1

1

(Sk + ikvo)

(Sk

ikv)2

+

1

(48)

by parts P/dv[ (Sk

-I- ikv)3

+

ik ikvo)2(Sk + ikv)2

1

(f&s : iq’v) t 2

+ f ($1 + Jdhk, 9, q’>.

(49)

The combination H(k, q, k) + H(k, q, q - k) will be of particular interest and we note that vo(q’) is the same for q1 = k and q’ = q - k. Thus J1 , Jt , and c which depend on q’ only through ~(4’) will be the same for q’ = k, and q’ = q - k.

We note that, 1

+ (Sk -I- ikv)

1 sq-k

i-

i(q - k)v

(f4 + iqv) =

(Sk

+

ikV)(s,k

+

i(q

-

k)v) (50)

=-

___(Sk

iq(v

+ ikv)(X,-k

-

f-41)

+ i(q -

0)

NONLINEAR

PLASMA

493

OSCILLATIONS

and thus

h(k, q, ICI + h(k, q, q - k) = -JI

+ Jz = --iq

0

csk +‘ikv

;

I2

(51)

-(,$I

2

H(k,

0

+ Jd

q, k) + H(k, q, q - k) = G’cw,” e 0m

ilc

xN[ '

(Sk

Xl!! ndv

(Sk

-iq

+

+

ikvo)2(&

ikv)(&-k

1 (Sk

iku)2+(8k

+

v- 210

e 0 ?%

+

+

ikvO)2

+

i(q

-

($I+

J2)

k)u)

ih$~fik

+

ikv)3

1

(52)

I

-;

Cdl+

J2,>'.

Again making use of the identity, Eq. (48), we can eliminate the (l/S, + ip) from the integrand in $1 . Thus we can express the parts of H(k, q, k) + H( k, p, q - Ic ) not involving t in terms of integrals which have poles only in the neighborhood of v = ukO/ko and since we are neglecting these poles the integrals may be evaluated by expanding the denominators to obtain, after integrating, an expansion in powers of (ka)2. Neglecting (k~)~ << 1, we obtain HOc,q,k)

+H(kq,q

-k)

=

5 23 0m

(6k2 - 3kq) +yJ$L~

-D’.

(53)

To obtain the dependenceof c(q, S) on a” we introduce the variables v/e = x, vo/o = x0. Defining

g(v) =;gm, [J

1

dvg(v) = 1

we obtain

Further Re

s

dxL% N 1 for (x - x0) ax -

FG3koa

<< 1

while Im

s

dx--=

(x

1 -

ail x0)

ia (1

ax

ds 0: ia Q

I 4 I ax

thus Re e a l/(ya)’ and Im e a [(koa)/(ya)“]y/l Re t by a factor of koa << 1. The quantity 1&,k-,

1’ = 1H(k

Y, k) + H(h

x0 E 3ir

I’ll

ka

<< 1

9 1 and Im E is smaller than

Y, Q - k) + H(k, 2~2-

+ H(k, 2 - Y, k - a) + H(k,

Q IYI

q, k)

0, Y - k) + H(k,

0, k -

(55) q> 1’

will be of interest. After some tedious algebra we obtain for Hrsk-, , neglecting (hi4 << 1

(56)

We note that the imaginary part of H comes entirely from Im 6, and since Im e S (koa) Re e, the contribution to j Hk,keV I2from Im E is smaller than the other contributions by a factor of (Ic~u)~ << 1 and can be neglected. Thus the leading terms in the mode coupling yield a contribution to dEk/at of --ae aEk = c H(k q, n’> Edt) ask dt > % qq’

-f&At)

-G(t),

q=O

(57)

We consider first the terms in Eq. (57) which are in phase with Eke . These are (since g # 0) q’ = k and q - q’ = k. (The term with y = 0 leads to the nonlinear dispersion relation and is included there. The terms q’ = k and Y - y’ = k might a1so have been considered as part of the nonlinear dispersion relation, but their effect is small compared to the q = 0 term and it does not matter.) We consider only the imaginary part of H for these terms since only this leads to a change in amplitude. From Eq. (53)

Im W(h a k) + Hk 4, Y - k)l

NONLINEAR

PLASMA

495

OSCILLATIONS

and = EkiL)I;9;k4

2 1 [I - tq/k)12 __1 1 ag --,c 1 E p [ q / au v. ’ Ek-q ’ WP q cl3

2 0m

(59)

Im l/c is of the order of (qa)*ka and the right hand side of Eq. (59) hence contains a factor (!~a)~ << 1, and thus this contribution to aE/& is very small. This term arises from the fact that in second order (in E) the waves interact with the distribution function at v = VO. For k >, ko the interaction is such as to cause the waves to grow. In terms of energy al Ek 1’ at > nl

(60)

= ] Ek lzwk 9ak4

f 0

2 -$ c

” - ;f’k’lz

&

h

P

;) v

IE

k-q

j2.

80

It is easy to show that zk dl Ek 12/dt = 0 and thus there is no net transfer of energy between the particles and the waves, and the interaction simply distorts the equilibrium spectrum towards the lower k values. The in-phase terms also lead to a change in g(v, t), but since they are small and since there is no net transfer of energy, we shall neglect this. We now consider the terms from Eq. (44) which are out of phase with Eke . These will lead to transitions and can be treated in just the same way as transitions in quantum mechanics. To calculate the transitions to the kth mode we have

ae dEk ask

at

>m

=

2’

H(k,

q, q’)E~-,(t)E,-,,(t>Eq,(~).

(61)

out of phase

Ek has the natural freqUenCy Ok and if we Write Ek(t) = &k(t)e+@ for these terms we obtain, after integrating over a time, T, which is long compared to l/w and short compared to the time in which the amplitudes of the driving terms change T$

k

&k(t

+

x

7)

exp

=

-$&k(t)

[+i(Wk i(Ok

Multiplying

+

k -

c

YP’

Wk-q Wk-q

-

H(k,

4, q’) (62)

WqFqr

-

Wq’)7]

w*-*I

-

W>

-

1 &k-q(t)&,-q~(t)&,‘(t>.

this equation by its complex conjugate

yields

496

DRUMMOKD

AND

PINES

=5’ H(k,q,qyR(k, p,p’) PP’

X

exp [i(wk (

i(Wk

- wk-p - wp-p’ - Wp’)Tl - 1 &k-q &,-qf &,I &T-p &-pf t&’ WkLp ‘dppr b&T) >

+

-

(63)

l >

i

x &-p( t)8,-qt(t)&(t)

+ C.C. >

where the bar and C.C. mean the complex conjugate. For large 7 the first term on the right hand side of Eq. (63) will be proportional to 7 whereas the terms in the curley brackets will be independent of r and can be neglected. The result will be time proportional transitions to the lath mode. Since the phases of the initial data were assumed to be random and since no phase correlation is introduced by the nonlinear dispersion relation, the only terms which survive c are those for which the phases cancel. The result is w’ PP’

ei(wk-Yk-*-w~-nf-wu')r

x

_

1

2

(wk _ wk-q _ wq-q, _ wq,>

+ a(k,

H(k

k + q’ - q, q’) + B(k, + mk,

In addition also survive, have already qf E k. is of

Q, q’){~k

Pt q’) + a(% 9, q - q’)

(64)

k + q’ - q, k - a)

ii - 4, q - q’) + R(k,

k - q’, k - q)).

there are terms for which Wk - w&q - wqmqr + wql = 0 which but these are just those terms which are in phase with Ek(O), and been treated. The quantity wk - W&-q - wq-qt - wq’ , for g 2% 0, the form Uk

-

Wk-q

-

Wqmq’

-

Wq’

= -3wpa2q(q’

- k)

!l/gk

0

and for q G 0, q’ G --k. ‘dk - ‘dk--qJ - ‘+ql

- Wq’ = 3Wf12q( q’ + k - q),

q’ Lx ko .

‘d&q - Wq-q* -

= -wp3a2(k

q z 2lco.

For q = 2k0 Wb

-

Wq’

- q’)[q’ -

(q - k)],

NONLINEAR i(w~--W~~*-W*-q

Thus the term

PLASAMA F-w*

2

‘)T

(in _ wlc--9 _ wq--9, -w,s)

has resonance at Q’ = k, p - k for

q G 0,2k and at q = 0 for 9’ g &ko. Converting we obtain [4(q, q’) is any smooth function] (w, - co&* - c&--q -

&

1q _’ 2k 1

+ ,&

s&$2

the sum over qq’ to an integral,

UP’)

Mq, k) + dq, q -

+ &,

497

OSCILLATIONS

k)l

Mq, k> + dq, q -

1q’ : k 1 do, n’) + ,gk,

&

where the quantities $&,a2, l/j q 1, etc., are simply After some manipulation this yields I Ek(t)

I2 -

I Ek(O)

= ‘2

k)l 1q’ : k 1 ‘+(‘, n’)

the densities

of states.

I2

&$

I J-L-9 I41El, 1’1a(k,

+ l?(k, 2k - q, k) + a(k,

q, k) + R(k,

q, q -

k)

(65)

226 - q, k - q)

+ B(k, 0, q - k> + fl(k, 0, k - n>l” (66) where Hksk-, is given in Eq. (56). As discussed Im Hk,k--q for q Z 0 is very small compared and can be neglected and

al Ek(t) 1’ at

> X-K

=

g2

I Ek

1’ c,

L”

I

I I Hk,h

to Re H(k,

I*

X z6 k.

q, q’)

(67)

Thii simply gives the transition probability of energy from al1 other modes to the kth mode. Since we take only the largest interaction term and this is hermitian, the transitions from the kth mode to all the other modes are given by

81&s(t)jz at

Lw, IEk14q k-.X = - 12U2

!f;i~~~!’

Xgk

(68)

and the net change in the energy in the kth mode is given by

=2

- I Ek 1”/Ex 1’1Hx~I”>. (6g) 1Ek 1’F (1Ex I41Hkx1’lk--XI

498

DRUMMOND

It should be noted that c

AND

PINE

al Ek 1’ = 0, i.e., electric energy is conserved dt

as

k

would be expected. We note that if k is such that 1Ek /’ is near the peak of the spectrum d 1EI, /“/at),, < 0 whereas if Ic is such that Ek is near the edge of the spectrum and hence smaller than the average 8 1Ek /“/at)m, > 0. The net effect of the mode coupling terms is thus to introduce a distortion towards lower k, (Eq. 47), and flattening of the spectrum, (Eq. 56). In terms of the phase velocity we can thus write for u g VP ,

(71)

LWp2 c Cr3=m

(#x”

1 Ha

x

j2 -

‘bk

Ik--XI

J/A

1 Hxk

1”) .

As discussed in Section III, pl$ plays the role of a diffusion coefficient in Eq. (71). When the mode coupling terms were neglected aj/au on the outer edges of the distribution was negative and thus rl, damped towards zero, thus reducing the diffusion. The mode coupling terms however tend to slowly distort the equilibrium spectrum such as to build up the outer edges, especially at the lower edge, and this increases the diffusion at the edges. This in turn causes af/at to become more negative thus increasing the absorption of the waves at the edges. Thus there is a sink for plasma waves at each edge of the spectrum and as the mode coupling terms slowly feed energy to the edges, it is absorbed. The result is that the equilibrium spectrum slowly decays to zero as t + 00, leaving g(u) a monotonically decreasing function of u. The energy which had been fed into the part of the velocity distribution with u << vP is removed in the same way when aflat becomes negative and as t + 00 all this energy is fed into the distribution function at the edges of the spectrum. V.

DISCUSSION

The essential feature which allows one to take only the lowest order nonlinear terms and to neglect mode coupling is that the diffusion of the background distribution function, go(v), reduces the growth rate of the unstable waves.

NONLINEaR

PLASMA

OSCILLATIONS

499

This leads to a wave spectrum which has such a small amplitude that the higher order mode coupling terms are unimportant except for very long times. This general behavior can be expected for a wide class of the so called “velocity space” instabilities for which the growth rate, Tk , is proportional to the velocity gradient of go , and (yk/wk) << 1 and in addition mode coupling is not resonant, i.e.,

ukl

RECEIVED:

+

Wkz

#

January

Wkl+k2

.

13, 1964