Turbulence and Self-Consistent Fields in Plasmas

Nonlinear Problems: Present and Future A.R. Bishop, D.K. Campbell, B. Nicolaenko (eds.) 0 North.Holland Publishing Company, 1982

435

Turbulence and Self-consistent F i e l d s i n Plasmas

D. Pesme and D. DuBois Theoretical D i v i s i o n University o f California Los Alamos National Laboratory Los Alamos, NM 87545 USA This paper i s concerned w i t h the r o l e o f self-consistency o f the e l e c t r i c f i e l d i n 1-0 plasma turbulence. We f i r s t show t h a t i n the non-self consistent e l e c t r i c f i e l d problem excell e n t agreement i s found between numerical experiments and quasi1 inear theory whenever t h e imposed e l e c t r i c f i e l d Fourier components have random phase. A discrepancy i s e x h i b i t e d between q u a s i l i n e a r p r e d i c t i o n and numerical simulations i n t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d case. This discrepancy i s explained by t h e c r e a t i o n o f a long c o r r e l a t i o n time o f the e l e c t r i c f i e l d r e s u l t i n g from a strong wave-particle i n t e r a c t i o n . A comparison i s made between q u a s i l i n e a r and renorma1i z e d propagator theories, and the Dupree "Clump" theory. These three theories are found t o be s e l f - c o n t r a d i c t o r y i n the regime o f strong wave-particle i n t e r a c t i o n because they make an a p r i o r i quasi-gaussian assumption f o r t h e e l e c t r i c f i e l d . The D i r e c t I n t e r a c t i o n Approximation i s then shown t o be a closure t a k i n g i n t o account i n a s e l f - c o n s i s t e n t way nongaussian e f f e c t s . INTRODUCTION

This paper i s concerned w i t h the r o l e o f s e l f - c o n s i s t e n t electromagnetic f i e l d s i n the theory o f t u r b u l e n t plasmas. I n recent years dramatic progress has occurred i n our understanding o f the motion o f charged p a r t i c l e s i n prescribed electromagnetic f i e l d s : We have learned t h a t t h e r e are thresholds f o r the onset of chaotic motion which are governed by conditions such as resonance ( o r i s l a n d ) overlap; the chaotic motion i s characterized by d i f f u s i o n o f p a r t i c l e s i n veloci t y and by t h e exponential divergence o f i n i t i a l l y nearby t r a j e c t o r i e s . Above the stochastic threshold p a r t i c l e s can wander a l l over and indeed t h e i r t r a j e c t o r i e s appear t o completely f i l l c e r t a i n regions o f phase space. The d e t a i l e d studies o f i n t r i n s i c s t o c h a s t i c i t y have concentrated on t h e prescribed f i e l d problem. I n t h i s paper we w i l l t r y t o make a p r e l i m i n a r y assessment o f t h e importance o f the self-consistency o f the e l e c t r i c f i e l d s i n the t u r b u l e n t o r c h a o t i c behavior o f plasmas. We w i l l f i r s t review i n Section 2 some numerical studies o f presc r i b e d f i e l d models i n parameter regimes w e l l above t h e stochastic threshold. I n c e r t a i n cases i t i s known t h a t , even f o r a s i n g l e r e a l i z a t i o n o f t h e imposed f i e l d , a s t a t i s t i c a l theory on the l e v e l o f t h e f a m i l i a r q u a s i l i n e a r theory provides an accurate d e s c r i p t i o n o f the d i f f u s i o n i n v e l o c i t y space when compared t o numerical solutions. I t i s l e s s well-known t h a t t h e exponential separation o f o r b i t s i s l i k e w i s e accurately p r e d i c t e d by a s t a t i s t i c a l theory o f t h e twop o i n t c o r r e l a t i o n function. I n Section 3 we review t h e r e s u l t s o f s t a t i s t i c a l theories which involve the ensemble average over many r e a l i z a t i o n s o f t h e imposed f i e l d s ; i n p a r t i c u l a r the imposed e l e c t r i c a l f i e l d s a r e considered t o be quasi-

D. PESME. D. F. DUEOlS

436

gaussian random v a r i a b l e s , i n a sense which w i l l be d e f i n e d i n Section 2. Usua l l y t h i s p r o p e r t y i s r e a l i z e d by assuming random phases f o r each mode. We exami n e the c o n d i t i o n s under which t h e p r e d i c t i o n s o f such a s t a t i s t i c a l t h e o r y are a b l e t o describe t h e p r o p e r t i e s o f a s i n g l e r e a l i z a t i o n such as a numerical simul a t i o n run. I n Section 4 we w i l l review some aspects o f t h e s e l f - c o n s i s t e n t f i e l d case. S e l f - c o n s i s t e n t numerical s i m u l a t i o n s i n t h e q u a s i l i n e a r parameter regime a r e n o t i n agreement w i t h t h e imposed f i e l d s t a t i s t i c a l t h e o r i e s . The reason i s t h e f o l l o w i n g : t h e s e l f - c o n s i s t e n t Maxwell equations a r e n o n l i n e a r and impose phase c o r r e l a t i o n s between modes. T h i s i n v a l i d a t e s t h e random phase o r quasigaussian f l u c t u a t i o n assumption and introduces new c o r r e l a t i o n times i n t o t h e problem; t h i s i s shown i n an i t e r a t i v e c a l c u l a t i o n i n Section 5. The imposed f i e l d s t a t i s t i c a l t h e o r i e s cannot be made s e l f - c o n s i s t e n t a p o s t e r i o r i because t h e y n e g l e c t important nongaussian c o r r e l a t i o n s . I n a sense ( t o be discussed) t h e assumption t h a t E i s a gaussian random v a r i a b l e i s an assumption o f maximum chaos. Since s e l f - c o n s i s t e n c y i n v a l i d a t e s t h i s assumption then a s e l f - c o n s i s t e n t system must have l e s s than maximum chaos. Self-consistency i m p l i e s a k i n d o f s e l f - o r g a n i z a t i o n o f t h e chaos. The f o r m u l a t i o n o f a s e l f - c o n s i s t e n t , s t a t i s t i c a l d e s c r i p t i o n o f a t u r b u l e n t plasma i s being pursued w i t h renewed v i g o r a t t h e p r e s e n t time. An approach t h a t shows considerable promise i n v o l v e s t h e a p p l i c a t i o n o f t h e s t r o n g turbulence approximation known as t h e d i r e c t i n t e r a c t i o n approximation ( D I A ) t o Vlasov turbulence. T h i s approximation i n c o r p o r a t e s c e r t a i n nongaussian c o r r e l a t i o n s i n t o a s t a t i s t i c a l t h e o r y t h a t r e t a i n s t h e b a s i c conservation laws and o t h e r r e a l i z a b i l i t y p r o p e r t i e s . I n Section 5 we w i l l examine c e r t a i n f e a t u r e s o f t h i s theory and make some comparisons w i t h t h e imposed f i e l d t h e o r i e s .

REVIEW OF INTRINSIC STOCHASTICITY STUDIES I N PRESCRIBED FIELDS The imposed f i e l d s t u d i e s seek t o e s t a b l i s h t h e p r o p e r t i e s o f s i n g l e part i c l e t r a j e c t o r i e s i n p r e s c r i b e d e l e c t r i c and magnetic f i e l d s . Here we w i l l r e s t r i c t our c o n s i d e r a t i o n s t o one-dimensional e l e c t r o s t a t i c models w i t h e l e c t r i c f i e l d s o f t h e form E(x,t) =

IEklexp(iek) expi[kx-wn(k)t] n=l

(2.1)

k

We consider a f i n i t e system o f l e n g t h L so t h a t t h e k modes a r e d i s c r e t e and we a l l o w t h e p o s s i b i l i t y o f a s e t o f frequencies wn(k) associated w i t h each mode, where

1 5 n 5 Nm.

For s i m p l i c i t y we w i l l w r i t e most o f our equations e x p l i -

c i t l y f o r t h e case Nm = 1 and wl(k)

= w(k).

Newton's equations f o r t h e motion

o f a p a r t i c l e o f charge q and mass m i n t h i s f i e l d a r e o f course: x = v

=

E(x,t)

=

q

(2.2)

I Ekl

expi [ k ~ - w ( k ) t + 8 ~ ]

and a r e h i g h l y n o n l i n e a r . We f i r s t focus on a s i n g l e resonance associated w i t h a g i v e n mode ko; Newton's equation f o r a p a r t i c l e i n t e r a c t i n g w i t h t h i s s i n g l e mode can be w r i t t e n

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

x = v

431

(2.3)

= 9 E

ko

sin[kox-u(ko)t]

I t i s convenient t o transform t o t h e coordinates X,V

i n t h e wave frame:

x = v (ko)t + X

4

(2.4)

v = v (k ) + V $

0

where v ( k ) = w(ko)/ko i s the wave phase-velocity. $

0

o f motion are x = v

\j = 9 E

ko

I n t h i s frame the equations

* s i n koX

(2.5)

These are the well-known equations f o r t h e nonlinear pendulum whose s o l u t i o n i s e x a c t l y known. The phase space i s d i v i d e d i n t o t r a p p i n g and passing regions as shown i n Fig. 1 by t h e s e p a r a t r i x whose equation i s

v=*

[

2 q+]I::'

cos

y2

"T

Fig. 1

Fig. 1. S e p a r a t r i x i n phase-space f o r a s i n g l e resonance.

The " t r a p p i n g h a l f width" o f the i s l a n d s o f t h e s e p a r a t r i x i s

(2.7) I n the case o f many modes we can l a b e l each resonance by i t s C..x.e v e l o c i t y and t r a p p i n g h a l f - w i d t h as i f i t were an i s o l a t e d resonance. The d i s t a n c e i n v between t h e phase v e l o c i t i e s o f the i s o l a t e d resonances i s 6v = Vg(kn+l) =

& ve(k)

-v ( k 1 $ n x 2n

D. PESME, D. F. DUBOIS

430

where k = 2 d L . For very small amplitudes t h e resonances, a t a f i r s t approximation,"can be considered as i s o l a t e d and we have the p i c t u r e o f superimposed resonances shown i n Fig. 2.

VA

Fig. 2.

Approximate resonance domains i n phase space f o r several resonances such t h a t kS << 1.

x

0

The work o f Chirikov parameter kS:

1 showed t h a t i t was useful t o d e f i n e t h e s t o c h a s t i c i t y

He showed t h a t f o r k << 1 the motion i s r e g u l a r (except i n small disconnected regions o f phase spa&e) b u t f o r ks > 1 i t i s c h a o t i c i n the phase space domain f o r which the v e l o c i t y belongs t o the i n t e r v a l o f phase v e l o c i t i e s o f t h e resonances. As an example we d i s p l a y the r e s u l t s o f Escande and Dovei12 f o r the case o f two we1 1-separated resonances w i t h t h e equations o f motion

= M1sin x + M p i n k(x-t) where v (k) = 1 and v (0) = 0.

4

4

They computed t h e o r b i t s x ( t ) , v ( t ) as a func-

t i o n o f t from some i n i t i a l conditions xo, vo a t t = 0.

The r e s u l t s are summar-

i z e d i n Fig. 3 f o r the case kS << 1. The primary resonances are d i s t o r t e d as evidenced by t h e d e v i a t i o n o f t h e separatices from the sinusoidal shape. In addl t i o n , new secondary resonances appear. The t r a j e c t o r i e s near t h e separat i c e s become fuzzy forming a s t o c h a s t i c layer. The important r e s u l t i s t h a t there s t i l l e x i s t s d e f i n i t e passing t r a j e c t o r i e s and these i s o l a t e t h e phase space i n the sense t h a t a p a r t i c l e cannot t r a v e l from one s e t o f t r a p p i n g i s l a n d s v I i s bounded so a p a r t i c l e t o another. Thus t h e motion i s s t a b l e and I v ( t ) As ks i s increased c a n ' t wander f a r i n v e l o c i t y space; thus t h e r e i s no'diffusion.

-

(but ks

2 1) t h e width o f the s t o c h a s t i c l a y e r grows as does t h e s i z e o f the

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

secondary resonances.

439

There e x i s t s a value o f kS (near u n i t y ) where a l l t h e

s t o c h a s t i c l a y e r s j o i n together a t which p o i n t t h e system undergoes a gross s t o c h a s t i c i n s t a b i l i t y i n which p a r t i c l e s can wander over t h e phase space between the resonances. C h i r i k o v o r i g i n a l l y p o s t u l a t e d t h a t t h e s t o c h a s t i c t h r e s h o l d occurred a t ks = 1. More r e c e n t work3”

has shown t h a t t h e p r e c i s e

value o f t h e t h r e s h o l d i s somewhat l e s s t h e u n i t y . I n t h e f o l l o w i n g we w i l l assume kS >> 1, w e l l above the c h a o t i c threshold. We w i l l n o t attempt a p r e c i s e d e f i n i t i o n o f chaos here; s u f f i c e i t t o say f o r our purposes, t h e c h a o t i c regime i s c h a r a c t e r i z e d by d i f f u s i o n o f p a r t i c l e s i n v e l o c i t y and exponential d i v e r gence o f nearby o r b i t s . The former i s a very p r a c t i c a l consequence o f chaos t h e l a t t e r i s a more t h e o r e t i c a l consequence r e l a t e d t o t h e r a t e a t which informat i o n i s l o s t by t h e system as c h a r a c t e r i z e d by the s o - c a l l e d Kolmogorov entropy.

Fig. 3.

D i s t o r t i o n o f primary resonances and appearance o f secondary resonances f o r kS = 0.4 (from (Escande e t a l . , Ref. 2).

Fig. 3 There have been many numerical studies o f these q u a n t i t i e s f o r v a r i o u s models o f t h e imposed f i e l d o f t h e form i n equation (2.2) which appear t o f a l l i n t o t h r e e classes:

i) The f i r s t c l a s s considers o n l y a s i n g l e wavenumber b u t associates an i n f i n i t e numer o f frequencies (harmonics) w i t h t h i s wave number ( i . e . , wn(k) = nwo and N, = =. This c l a s s includes t h e famous C h i r i k o v - T a y l o r 4 5 mapping and has been i n v e s t i g a t e d by Rechester and White and others. ii) A second c l a s s o f models has an i n f i n i t e ( o r l a r g e ) number o f modes k each w i t h an i n f i n i t e number o f harmonics o f a fundamental frequency ( i . e . , wn(k) = nw(k) and Nm = =). Such models have been i n v e s t i g a t e d by Rechester, Rosenbluth and 8 e t al.

Molvig

iii) The class o f models c l o s e s t t o our i n t e r e s t s have an i n f i n i t e ( o r l a r g e ) spectrum o f k values each w i t h a d e f i n i t e frequence w(k) ( i . e . , N, =

D. PESME, D.F. DUEOIS

440

= w(k)).

1 and wl(k)

Flynn9 and D o v e i l . l 0

Simulations o f t h i s case have been c a r r i e d o u t by I n these numerical c a l c u l a t i o n s values o f l E k l f o r

Eq. 2.2 were assigned and t h e phases Ok were randomly chosen between 0 and 2n w i t h no c o r r e l a t i o n between d i f f e r e n t values o f k. T h i s i s u s u a l l y This procedure generates a c a l l e d t h e Random Phase Approximation (RPA). s t o c h a s t i c process which i s a s y m p t o t i c a l l y e q u i v a l e n t t o a gaussian process 12 S. P. Gary and D. Montgomery, i n t h e l i m i t o f a l a r g e number o f modes."

K. N. Graham and J. A. Fejer13 a l s o d i d s i m i l a r c a l c u l a t i o n s , though they used somewhat d i f f e r e n t processes than random phases i n order t o generate a quasigaussian e l e c t r i c f i e l d . I n t h e f o l l o w i n g , f o r s i m p l i c i t y , we w i l l consider i n t h e imposed e l e c t r i c f i e l d case o n l y t h e models w i t h random phase F o u r i e r components. The s i n g l e p a r t i c l e t r a j e c t o r i e s x(xo, vo, t) and v(xo, vo, t) were computed n u m e r i c a l l y from i n i t i a l c o n d i t i o n s xo, vo a t t = 0. V e l o c i t y d i f f u s i o n was s t u d i e d by computing t h e s p a t i a l average over t h e i n i t i a l p o s i t i o n s o f t h e squared v e l o c i t y d e v i a t i o n 2 Av (t)

(V(XO,VO,t)

X

-

vo)

2O

(2.10)

where we d e f i n e the s p a t i a l average o f any f u n c t i o n +(x) by

Results f r o m these numerical c a l c u l a t i o n s showed t h a t : i) 3 was p r o p o r t i o n a l t o t (except f o r very s h o r t times). The observed v e l o c i t y d i f f u s i o n c o e f f i c i e n t i n t h e numerical experiments can thus be d e f i n e d as Dexp ( d i f ) h 2ti

qTj

(2.11)

ii) The observed d i f f u s i o n c o e f f i c i e n t was i n good agreement w i t h t h e pred i c t i o n o f quasilinear theory (2.12) where

DQaL = m

'c k

n

IEkI2

(t

dr exp i [wn(k)

-

kv]r

(2.13)

0

The q u a s i l i n e a r theory i s a s t a t i s t i c a l theory, averaged over many r e a l i z a t i o n s o f the phases, 0 , which are assumed t o be random. We w i l l discuss s t a t i s t i c a l t h e o r i e s i n t h e k e x t section. S i m i l a r good agreement between t h e observed d i f f u s i o n c o e f f i c i e n t and t h e pred i c t i o n s o f q u a s i l i n e a r theory was found i n t h e work o f Rechester, Rosenbluth and White

7 as i t can be seen i n Fig. 4.

For t h e i r model t h e q u a s i l i n e a r pre-

d i c t i o n i s DQL = c2M/4 where c i s t h e amplitude o f t h e F o u r i e r components and M i s t h e number o f harmonics (Nm i n o u r n o t a t i o n ) .

44 1

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

f

Fig. 4.

Comparison between numerical experiments (dots 0) and the quasi 1i n e a r p r e d i c t i o n ( s o l i d l i n e ) f o r the value o f t h e d i f f u s i o n c o e f f i c i e n t (from Rechester e t a l . , Ref. 2).

I

,

,

,

,

,

,

Nuatrirol Rinla

D

Id'

,

* O I

ai

.01

a

10

1.0

Fig. 4 Rechester, Rosenbl u t h and White a1 so computed t h e s o - c a l l e d "Kolmogorov entropy" d e f i n e d as h =

lim

t-

1 t

l i m In do+O

- . .

11

(2.14)

do

where d i s t h e "distance" between two nearby o r b i t s .

We can take t h i s t o be

(2.15)

where x1 = x ( t , xo,l

vo),l

v1 = v ( t , x,ol

vo)l

and x2 = x ( t , xO2, vO2), v2 =

x ( t , xo2, vo2) corresponding t o d i f f e r e n t b u t nearby i n i t i a l c o n d i t i o n s a t t =

0; do = d ( t = 0).

The n o r m a l i z a t i o n constants i n t h i s expression a r e v+ = %(vl

v ) and k+ = k(v+); k(v) i s t h e s o l u t i o n o f t h e i m p l i c i t equation w(k(v)) = 2 k(v)v. P o s i t i v e values o f h are i n t e r p r e t e d t o be a s i g n a t u r e o f c h a o t i c behav i o r . Roughly speaking, i n p h y s i c a l terms, t h i s d e f i n i t i o n i m p l i e s t h a t f o r s h o r t times d ( t ) 2 d exp h t and t h i s i m p l i e s a loss o f memory o f i n i t i a l condit i o n s and a s u f f i c i e g t c o n d i t i o n f o r m i x i n g and e r g o d i c i t y . +

The r e s u l t s o f the numerical experiments o f Rechester e t a l . ,6'7 a r e shown i n e x i s t s and i s independent o f i n i t i a l condif i g u r e 5. They showed t h a t h exP t i o n s i f do i s s u f f i c i e n t l y small and obtaiped e x c e l l e n t agreement w i t h t h e p r e d i c t i o n h Q e Lobtained from a s t a t i s t i c a l theory assuming a quasigaussian e l e c t r i c f i e l d . This theory, which i s discussed i n t h e n e x t s e c t i o n provides an e x p l i c i t r e l a t i o n o f h Q a L *t o t h e q u a s i l i n e a r d i f f u s i o n c o e f f i c i e n t h Q e L *= 0.36

(

Q*L3

k+D

13'

(2.16)

442

D. PESME, D. F. DUBOlS

h

F i g . 5.

Comparison between numeric a l experiments (dots 4) and t h e q u a s i l i n e a r p r e d i c t i o n ( s o l i d l i n e ) f o r the value o f the Kolmogorov entropy ( f r o m Rechester e t a l . , Ref. 6, 7).

Numoical hints

Fig. 5 The numerical experiments showed t h a t t o e x c e l l e n t accuracy hexp =

2 Q. L.) 13' hQ' L' = 0.36 (k+D

(2.17)

STATISTICAL TURBULENCE THEORIES FOR PRESCRIBED RANDOM FIELDS We next g e n e r a l i z e our considerations t o a system o f many-charged p a r t i c l e s I f t h e f i e l d s a r e p r e s c r i b e d , we have a system moving i n e l e c t r o s t a t i c f i e l d s . o f independent p a r t i c l e s whose t r a j e c t o r i e s have t h e p r o p e r t i e s discussed i n Section 2. The many p a r t i c l e s i t u a t i o n i s c o n v e n i e n t l y described by t h e one p a r t i c l e phase space d i s t r i b u t i o n f u n c t i o n f ( x , v , t ) which evolves a f t e r a t i m e t from some i n i t i a l d i s t r i b u t i o n fo(x,v) a t t = 0 according t o t h e formula f(x,v,t)

[

= I d x o I d v o 6 x-x(t,xo,vo)]

where x(t,xo,vo) Section 2.

[at +

and v(t,xo,vo)

6 [v-v(tlxolvo)

1

f o (x,v)

(3.1)

are j u s t t h e p o s i t i o n and v e l o c i t y discussed i n

The phase-space d e n s i t y evolves according t o t h e L i o u v i l l e equation: v

L

ax

+

$ E(x,t)a,,lf(x,v,t)

= 0

(3.2)

J

which expresses t h e conservation o f d e n s i t y along phase-space t r a j e c t o r i e s . I n t h e s e l f - c o n s i s t e n t case we can s t i l l use Eq. 3.2 i n which E(x,t) i s t h e s e l f 14 c o n s i s t e n t f i e l d determined by Poisson's equation 4.1. I n t h e p r e s c r i b e d f i e l d problem we assume E(x,t) i s a random v a r i a b l e over an ensemble each r e a l i z a t i o n o f which i s given by an expression o f t h e form (2.1). This i s t h e s o - c a l l e d s t o c h a s t i c a c c e l e r a t i o n problem based on t h e l i n e a r stoThis problem has r e c e i v e d much attention.'' We g i v e c h a s t i c equation (3.2). here t h e r e s u l t s o f t h i s a n a l y s i s i n t h e q u a s i l i n e a r regime which i n v o l v e s t h e f o l l o w i n g assumptions

i) kS >> 1

TURBULENCE AND SELF-CONSISTENTFIELDS IN PLASMAS

443

iii) quasigaussian approximation f o r e l e c t r i c f i e l d f l u c t u a t i o n s , e.g., phase Bk random and uncorrelated. Condition i) simply assumes t h a t t h e system i s w e l l above t h e s t o c h a s t i c t h r e s hold. I n c o n d i t i o n iii),t h e quasigaussian approximation i s d e f i n e d as t h e case i n which t h e c o r r e l a t i o n f u n c t i o n s o f any order a r e c h a r a c t e r i z e d by a unique c o r r e l a t i o n length. More p r e c i s e l y , throughout a l l t h i s a r t i c l e , we name quasigaussian any random process f o r which an i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n o f any order i s n o t zero o n l y i f a l l the arguments l i e i n s i d e a same i n t e r v a l o f l e n g t h Q , where Q i s t h e c o r r e l a t i o n l e n g t h o f t h e two p o i n t c o r r e l a t i o n function: This c o f i d i t i o n can be r e a l i z e d by t h e random phase approximation. I n c o n d i t i o n ii)we have d e f i n e d a c h a r a c t e r i s t i c time o f the Kolmogorov entropy;

th

o f i n i t i a l conditions i s l o s t .

T~

as t h e i n v e r s e

i s a measure o f t h e t i m e over which i n f o r m a t i o n This time, rh, must be much l e s s than t h e

time o f e v o l u t i o n o f t h e mean d i s t r i b u t i o n f u n c t i o n , T.<~>. l a t t e r as

We can estimate t h e

(3.4) where Avf i s a measure o f the w i d t h i n v e l o c i t y space f o r which waves i n t e r a c t resonantly w i t h p a r t i c l e s and DQ’L’ 2.13.

Note t h a t DQ’L’

(q/m)‘

t h e q u a s i l i n e a r d i f f u s i o n c o e f f i c i e n t o f Eq.

tCwhere r c = [Ak(v-V,)]-l

i s the correlation

time o f the e l e c t r i c f i e l d f l u c t u a t i o n s as seen by a t y p i c a l resonant p a r t i c l e ; We can then w r i t e V i s the group v e l o c i t y o f the wave w i t h wave number k(v). cl % ~ < ~ >=/ (t I ~J J ~ ~where ~ ) - ~wb = i s o f t e n c a l l e d t h e e f f e c t i v e bounce frequency; i . e . ,

t h e frequency o f a p a r t i -

c l e trapped i n a monochromatic wave o f amplitude % and wave number i . these d e f i n i t i o n s i t i s found t h a t c o n d i t i o n ii)can a l s o be w r i t t e n as

With (3.5)

Under these c o n d i t i o n s i t i s known16 t h a t t h e equation o f e v o l u t i o n o f t h e ensemble averaged o r mean d i s t r i b u t i o n < f > i s

a t c f > = a V D Q - ~ . ( V ) av

cf>

(3.6)

where DQeL. is given i n 2.13. The ensemble average i s over many r e a l i z a t i o n s o f t h e e l e c t r i c f i e l d s . We assume s t a t i s t i c a l homogeneity, i . e . = 0 and fo(X,V) = fo(v). An equation f o r t h e two-phase p o i n t , one time c o r r e l a t i o n f u n c t i o n
bf(x:v:t’)>

, Gf(x,v,t)

can a l s o be d e r i v e d i n t h i s l i m i t (Dupree

= f 17

).

-

,

,

I n t h e l i m i t o f small separation

i n v e l o c i t y space I k ( v + ) v - ) << sup [ ( k DQ’L’)1/3, [(k2(v+)DQ‘L’ )-1/3,.y,1]



y,],

and Ix-/v+l

t h i s equation can be w r i t t e n as:

<< i n f

0.PESME, 0.F. DUBOIS

444

(3.7) where

D12

i s the r e l a t i v e d i f f u s i o n c o e f f i c i e n t

These equations are expressed i n r e l a t i v e coordinates: v- = v - v ' , x- = x - x ' and - v+v' c e n t r i x coordinates: v+ - - assuming s p a t i a l homogeneity. The propagator G k , r o ( ~ ) t o be used i n 3.7 i s t h e usual renormalized propagator given by: (3.9) Dupree17 was t h e f i r s t t o show t h a t equation (3.7) p r e d i c t s t h a t i n i t i a l l y nearby t r a j e c t o r i e s w i l l diverge e x p o n e n t i a l l y i n time. He considered t h e source f r e e s o l u t i o n t o eqn. (3.7),

G2(x-,v-;v+,t),

and generated a closed s e t

JdxJdv x_? vlk;,. The s o l u t i o n o f 2 2t h e r e s u l t i n g equations showed t h a t , increased from t h e i r

o f equations f o r the moments

t ) . p r e c i s e c o n t a c t between t h e i n i t i a l values as exp ([4k+ DQ * L * ] 1 /A3more

dynamical d e f i n i t i o n has been made by Rechester, Rosenbluth and White.6 Since t h e q u a n t i t y h as d e f i n e d by Eq. 2.14 i s n u m e r i c a l l y observed t o be independent o f i n i t i a l c o n d i t i o n s , these authors argue t h a t i t i s a s t a t i s t i c a l q u a n t i t y . More e x p l i c i t l y they invoke ergodic p r o p e r t i e s which p e r m i t them t o r e p l a c e t h e t i m e average In 0

]

d(t')

[do

by the s t a t i s t i c a l average

dt' =

a <

1 I n d(t) t

In

do

>.

By w r i t i n g t h e q u a s i l i n e a r equation

corresponding t o t h e two-point, one-time c o r r e l a t i o n f u n c t i o n they f i n d :

(3.

lo)

These authors observed an e x c e l l e n t agreement between t h e s t a t i s t i c a l p r e d i c t i o n hQ'L' and t h e experimental value h obtained from numerical i n t e g r a t i o n o f t h e o r b i t s (Fig. 5). exP Thus t h e r e i s e x c e l l e n t agreement between t h e p r e d i c t i o n s o f t h e q u a s i l i n e a r s t a t i s t i c a l theories f o r velocity diffusion (i.e.,

DexpY D Q ' L * ) and t h e corres-

N hQ'L.) i n t h e case h ex? o f p r e s c r i b e d quasigaussian random f i e l d s when t h e c o n d i t i o n s (3.3) f o r t h e I n Section 4 we w i l l show t h e r e v a l i d i t y o f q u a s i l i n e a r theory are s a t i s f i e d . i s no corresponding agreement between standard s t a t i s t i c a l t h e o r y and t h e r e s u l t s o f c e r t a i n s e l f - c o n s i s t e n t computer s i m u l a t i o n s .

ponding p r e d i c t i o n s f o r t h e Kolmogorov entropy ( i . e . ,

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

445

We complete this section by pointing out the following question: why the results of an ensemble averaged statistical theory can accurately describe the results of a single numerical experiment; i.e., a single realization of the ensemble. The answer is really not obvious as it is sometimes considered to be; the statistical quantities DQ'L* and hQ'L* clearly do not depend on the phases, but the and h of one-numerical experiment with a given assignment of results D exP exP the phases could in principle depend on the phases. The simulations described above are essentially spatially homogeneous and explicitly calculated only spatial-averages. In particular, the spatially averaged phase space distribution f(v,t) is defined as

Pesme and Brisset18 have considered the following situation: for each realization the mode amplitudes lEkl form a smooth function of k and the phases Ok are randomly distributed and-independent of each other. For such a given realization they _have shown that f obeys the quasilinear equation (3.5) with replaced by f the conditions ( 3 . 3 ) are satisfied. More precisely they have shown that the corrections to this equation are statistically small under these conditions. The condition k >> 1 is easily shown to be equivalent to the following condition on the lsngth of the system:

L/v >> h-l or L >> !Lh

E

v

th

(3.11)

In this case during the transit time of a particle o f velocity v through the system it sees many independent partitions of the system which are equivalent to independent realizations. This is because on one hand, after a time h-l the particle has lost all memory of its initial conditions. On the other hand, the correlation length of the electric field fluctuations is ac- l/A k and when the conditions 3 . 3 are satisfied, we have Ilc << a, so that the electric field in each partition is independent. Therefore spatial averaging is then equivalent to ensemble averaging. COMPARISON OF SELF-CONSISTENT NUMERICAL SIMULATIONS WITH STANDARD QUASILINEAR THEORY Here we will review the simulations of the bump on tail instability in a onedimensional, one-component plasma which were carried out and analyzed by Adam, Lava1 and Pesme." The conditions of these simulations were those commonly thought to be in the regime of validity of quasilinear theory.*' In fact, the simulations did not follow the quasilinear predictions and this failure is now attributed to the breakdown of the random phase assumption in a self-consistent plasma. The fact that self-consistency results in qualitatively and quantitatively different behavior than in the imposed field case has also been verified in a rather different physical situation by the simulations of Bezzerides, Gitomer and Forslund." These authors investigated the production of hot electrons in resonance absorption of light. Detailed consideration of this case i s not possible here and we refer the reader to this work. In the self-consistent problem the electric field is, of course, determined by the charge density of all the particles in the system. Poisson's equation is written as:

0. PESME. 0.F. DUBOIS

446

V

*

E(x,t) = 4nq Jdvf(x,v,t)

-

4nqN

(4.1)

where f i s a smooth function14 and N i s t h e i o n d e n s i t y . The Vlasov-Poisson system o f equations thus becomes a n o n l i n e a r f i e l d theory. I n t h e s e l f - c o n s i s t e n t case t h e f i e l d E(x,t) cannot be s p e c i f i e d , b u t evolves dynamically from t h e i n i t i a l c o n d i t i o n s (fo(xlv) = f(x,v,t=O)). The bump on t a i l i n s t a b i l i t y develops from t h e i n i t i a l c o n d i t i o n s shown i n Fig. 6a. Here i s shown t h e i n i t i a l s p a t i a l averaged d i s t r i b u t i o n f(v,t=O) = The phases o f fk(v,t=O) a r e randomly chosen from one mode t o t h e fk=O(v,t=O). next. The f o u r i e r amplitudes Ek(t=O) f o r k # 0 are computed from Poisson's equation, i . e . ,

Ek(t=O)

amplitudes I E k ( t =

Fig. 6.

= (4nq/ik) Jdvfk(v,t=O)

b u t a r e n o t shown.

The i n i t i a l

0)l are shown s c h e m a t i c a l l y i n Fig. 6.a.

Time e v o l u t i o n o f t h e spaceaveraged d i s t r i b u t i o n f u n c t i o n f ( v . t ) and o f t h e e l e c t r i c f i e l d made amplitude IEkI(t), a. i n i t i a l c o n d i t i o n s , b. e a r l i e r stage: growth o f lEkl and no m o d i f i c a t i o n o f f , d. stationary f i n a l state.

4% /ii

6.b

6.L

0

Fig. 6

I t i s well-known t h a t t h e l i n e a r d i s p e r s i o n r e l a t i o n d e r i v e d from EO(k,w) = 0

where E ~ ( J U U ) i s t h e l i n e a r d i e l e c t r i c f u n c t i o n p r e d i c t s t h e f a m i l i a r Landau growth r a t e f o r Langmuir waves p r o v i d e d y:'L'/wp

<< 1:

P a r t i c l e s w i t h v e l o c i t i e s v near t h e phase v e l o c i t y v (k) = w(k)/k o f t h e wave

4

s e e - e s s e n t i a l l y a s t a t i c f i e l d and i n t e r a c t s t r o n g l y w i t h t h e wave. I f (avf)v=v > 0 then t h e waves grow because t h e r e i s energy t r a n s f e r from 4J

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

p a r t i c l e s t o the wave.

For the i n i t i a l c o n d i t i o n on

447

chosen by Adam e t a l . ,

t h e r e i s a band o f wave numbers whose growth r a t e s ( Y Q ~ . >~ '0) a r e very c l o s e i n value and f o r s h o r t times a broad spectrum o f waves grows up as shown i n Fig. 6b. A t l a t e r times (Fig. 6c) t h e spectrum becomes n o i s y and uneven b u t s t i l l broad and-begins t o cause d i f f u s i o n o f p a r t i c l e s i n v e l o c i t y which tends t o f l a t t e n o u t f i n t h e unstable r e g i o n o f v e l o c i t i e s . A t longer times (Fig. 6d) f becomes e s s e n t i a l l y f l a t i n t h i s region, t h e waves cease t o grow and a broad b u t noisy q u a s i s t a t i o n a r y spectrum o f waves r e s u l t s . Except f o r v e r y s h o r t times the spectrum has grown t o a l e v e l f o r which kS >> 1 and t h e c o n d i t i o n f o r s t o c h a s t i c i t y expected f o r an imposed f i e l d i s exceeded. s i m u l a t i o n t h e c o n d i t i o n f o r q u a s i l i n e a r theory h - l = s a t is f ied.

Also, throughout t h e

<<

T~

17 i s

well

The comparison between t h e numerical r e s u l t s and t h e q u a s i l i n e a r p r e d i c t i o n f o r t h e growth r a t e f o r a t y p i c a l unstable mode i s shown i n Fig. 7. The numeric a l r e s u l t s f o r the ensemble average over t e n s i m u l a t i o n runs ( s o l i d l i n e ) l i e s i g n i f i c a n t l y above t h e p r e d i c t e d q u a s i l i n e a r growth r a t e (dashed l i n e ) .

Fig. 7.

Growth r a t e s versus phase v e l o c i t y : t h e dashed l i n e i s t h e quasi 1i n e a r p r e d i c t i o n , t h e d o t t e d curve i s obtained by numerical time d e r i v a t i o n o f t h e ensemble average o f t h e mode energy and t h e s o l i d curve i s a l e a s t square f i t o f t h e preceding one (from Adam e t a l . , Ref. 19).

3

4

6

6

7

8

9

10

v/vc.

Fig. 7 Now r e c a l l t h e usual q u a s i l i n e a r equation f o r t h e t i m e development o f t h e mean i n t e n s i t y i n mode k: (4.3) I n t h e usual q u a s i l i n e a r theory t h e source term Sk which a r i s e s from mode-mode c o u p l i n g e f f e c t s i s argued t o be n e g l i g i b l e when t h e c o n d i t i o n s (3.3) a r e f u l f i l l e d . The coupled equations (4.3) w i t h Sk = 0 and (3.5) c o n s t i t u t e t h e usual q u a s i l i n e a r theory. Together they conserve p a r t i c l e number, momentum, and energy. The simulations appear t o show t h a t t h e growth r a t e which should appear i n (4.3) i s g r e a t e r than yQkSL-;i n t h i s case equation (3.5)

must a l s o be a l t e r e d

i n order t o m a i n t a i n conservation o f energy. !e.g., D # i n f e r from t h e simulations t h a t

Therefore, we

448

D. PESME, 0.F. DUBOIS

Since o f t h e c o n d i t i o n s (3.3) f o r t h e v a l i d i t y o f q u a s i l i n e a r theory b o t h

i)( k s >> 1) and i f ) ( h - l << T < ~ > )are s a t i s f i e d d u r i n g t h e s i m u l a t i o n we must conclude t h a t iii) t h e quasigaussian f l u c t u a t i o n assumption i s n o t v a l i d i n t h e sel f - c o n s i s t e n t f i e l d case. This statement i s n o t s u r p r i s i n g i n i t s e l f since we know t h a t t h e s e l f - c o n s i s t e n t Poisson equation i s nonlinear. As a r e s u l t nongaussian e l e c t r i c f i e l d f l u c t u a t i o n s w i l l evolve from i n i t i a l l y gaussian e l e c t r i c f i e l d f l u c t u a t i o n s . SELF-CONSISTENT STATISTICAL THEORIES OF PLASMA TURBULENCE Introduction The understanding o f t h e s e l f - c o n s i s t e n t s t a t i s t i c a l d e s c r i p t i o n o f Vlasov t u r bulence i s i n a new stage o f development today and t h e study o f t h i s problem w i l l c e r t a i n l y be t h e s u b j e c t o f f u t u r e research i n t h e n o n l i n e a r physics o f plasmas. I n t h e p r e s c r i b e d e l e c t r i c f i e l d problem we were concerned w i t h t h e motion o f p a r t i c l e s i n random forces. The p r o p e r t y f o r t h e e l e c t r i c f i e l d o f b e i n g p r e s c r i b e d d i d n o t appear e x p l i c i t l y i n t h e d e r i v a t i o n o f t h e equations o f e v o l u t i o n f o r < f > and <€if €if’>. These equations (eq. 3.6 and 3.7) a c t u a l l y reduce t o two Fokker-Planck equations and t h e y can be e a s i l y d e r i v e d by computing t h e f r i c t i o n and d i f f u s i o n c o e f f i c i e n t s corresponding t o t h e motion o f a p a r t i c l e i n t h e random f i e l d . T h i s i s t h e reason why i t has been thought f o r a long t i m e t h a t t h e s t a t i s t i c a l d e s c r i p t i o n o f t h e p a r t i c l e motion i n s e l f - c o n s i s t e n t Vlasov t u r b u lence was t h e same as i n the p r e s c r i b e d e l e c t r i c f i e l d case. We have, however, t o keep i n mind t h a t a basic v a l i d i t y c o n d i t i o n f o r q u a s i l i n e a r t h e o r y as w e l l as f o r t h e Fokker-Planck equation i s t h e quasigaussian p r o p e r t y o f t h e e l e c t r i c f i e l d . I n t h e s t o c h a s t i c a c c e l e r a t i o n problem t h i s p r o p e r t y was always imposed, e.g., through random phases, b u t i n t h e Vlasov-Poisson case i t i s necessary t o check t h e v a l i d i t y o f t h i s hypothesis.

-

-

We consider e x p l i c i t l y a s e l f - c o n s i s t e n t Vlasov t u r b u l e n c e s i t u a t i o n f o r which t h e c o n d i t i o n s 3.31 and 3 . 3 i i a r e s a t i s f i e d . We assume a l s o t h e usual weak turbulence hypothesis t h a t assigns N, we1 1-defined frequencies wn(k) t o a wave number k. These frequencies have t o be computed s e l f - c o n s i s t e n t l y through a n o n l i n e a r d i s p e r s i o n r e l a t i o n . For s i m p l i c i t y we w i l l t a k e N, = 1. A l l t h e q u a n t i t i e s such as k,

wb, t c are now computed through t h e instantaneous values

o f t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d . I n e q u a l i t y 3.3i) i s e a s i l y s a t i s f i e d i n unstable s i t u a t i o n s . I n e q u a l i t y 3 . 3 i i ) i s u s u a l l y considered as t h e v a l i d i t y c o n d i t i o n o f q u a s i l i n e a r t h e o r y and i s f u l f i l l e d , f o r instance, i n t h e warm beam-plasma i n s t a b i l i t y . We d e f i n e as quasigaussian t h e o r i e s t h e s t a t i s t i c a l d e s c r i p t i o n s o f Vlasov t u r bulence which assume a p r i o r i t h a t t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d has quasigaussian p r o p e r t i e s . I n t h i s s e c t i o n we w i l l f i r s t p r e s e n t t h e usual argument given f o r j u s t i f y i n g t h i s assumption. We w i l l then show t h a t t h e quasigaussian t h e o r i e s are s e l f - c o n t r a d i c t o r y i n t h e regimes where t h e n o n l i n e a r i t y p l a y s a r o l e . The reason f o r discrepancy w i l l be e x h i b i t e d through a p e r t u r b a t i v e c a l culation: the s t a t i s t i c a l properties o f the self-consistent e l e c t r i c f i e l d

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

are not characterized by a unique c o r r e l a t i o n l e n g t h stochastic d e c o r r e l a t i o n l e n g t h Rh = vth.

449

= l / A k b u t a l s o by the

We close t h i s Section by describing some o f the features o f the d i r e c t i n t e r a c t i o n approximation ( D I A ) which i s a renormalized theory which takes i n t o account c e r t a i n nongaussian e f f e c t s . The quasi gaussi an hypothesis I n the e a r l i e r d e r i v a t i o n s o f quasi1 i n e a r theory the random-phase assumption was usually i m p l i c i t l y made. z z 9 z 3 The underlying reason has been made e x p l i c i t by OlNei124 as follows: Let us assume on physical grounds t h a t w e l l above t h e stochastic threshold the e l e c t r i c f i e l d i s i t s e l f t u r b u l e n t and i s characterized by an a u t o c o r r e l a t i o n I n a f i n i t e box o f length L, the F o u r i e r components o f the e l e c t r i c length Qeff. f i e l d ar& defined as:

1 L

Ek(t) =

dx E(x,t) exp-ikx

I f we now d i v i d e t h e i n t e r v a l [O,L]

n2n with k =

eff i n t o segments o f l e n g t h go where gc

<< go << L, t h e e l e c t r i c f i e l d i n each segment i s independent o f t h e f i e l d i n other segments. Ek =

Therefore Ek can a l s o be w r i t t e n as

dx E(x,t) gcN j=l (j-l)Qo

exp-ikx

w i t h N = L/Qo

.

Thus Ek appears t o be the sum o f many independent v a r i a b l e s and by t h e c e n t r a l

l i m i t theorem Ek behaves as a Gaussian v a r i a b l e i n t h e l i m i t L/go very large. Let us emphasize t h a t the o n l y hypothesis i n t h i s demonstration i s t h e existence o f an a u t o c o r r e l a t i o n l e n g t h gFff and t h a t seems q u i t e reasonable i n a turbulent situation. However, O'Neil observedz4 t h a t the argument o u t l i n e d above cannot be used t o j u s t i f y t h e quasigaussian assumption (which he c a l l s the random-phase approximation).

This assumption would say, f o r example, t h a t the t l2>. The neglected terms are24 i n d i v i d u a l l y can be replaced by 2
D. PESME, D. F. DUBOlS

450

On t h e b a s i s o f t h e previous s e c t i o n , t h e quasigaussian t h e o r i e s assume a i.e., the e l e c t r i c p r i o r i t h a t t h e l a s t c o n d i t i o n iii)o f 3.3 i s s a t i s f i e d f i e l d behaves as a quasigaussian process.

-

The scheme o f these t h e o r i e s i s t h e f o l l o w i n g :

1) Two Fokker-Planck equations describe t h e e v o l u t i o n o f < f > and < 6 f 6 f ' > , namely t h e q u a s i l i n e a r equation 3.6 and t h e Dupree equation 3.7. The f r i c t i o n and d i f f u s i o n c o e f f i c i e n t s i n v o l v e t h e s p e c t r a l d e n s i t y < l E k l 2>. 2 2 ) The spectrum < I E k l > i s then computed by w r i t i n g Poisson's equation as: k2 = (471)2q2 Jdvdv' c 6 f 6 f ' > k

(5.1)

We thus o b t a i n a closed s e t o f t h r e e equations f o r t h e t h r e e unknowns < f > , < 6 f

2

6 f ' > , and .

However, i t i s necessary t o go beyond t h i s l e v e l i n o r d e r t o

compute t h e i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s o f t h e e l e c t r i c f i e l d o f h i g h e r order than 2. This scheme would be c o n s i s t e n t o n l y i f these c o r r e l a t i o n funct i o n s s a t i s f y t h e d e f i n i t i o n o f a quasigaussian process given i n 52. An i n d i r e c t demonstration o f t h e i n c o n s i s t e n c y o f t h i s scheme i n t h e regime ~ ~ solved t h e (k2D)1/3 >> y f a L * has been given by Adam, Lava1 and P e ~ m e . They Dupree equation 3.7 and by i n s e r t i n g i t s s o l u t i o n i n t o Poisson's equation 5.1; they found a growth r a t e d i f f e r e n t from t h e q u a s i l i n e a r p r e d i c t i o n . This r e s u l t , combined w i t h q u a s i l i n e a r equation 3.6 does n o t a l l o w energy conservation. These authors conclude t h a t t h e q u a s i l i n e a r theory i s i n c o n s i s t e n t i n one dimension and i n t h e regime ( k DQ'L')1/3

>> y f e L * and showed t h a t t h e c o n t r i b u t i o n o f t h e 4-

point irreducible correlation function i s not negligible, i n contradiction t o t h e quasigaussian hypothesis. Even though t h e quasigaussian scheme appears t o be i n v a l i d , t h e s o l u t i o n o f t h e Dupree equation 3.7 e x h i b i t s c e r t a i n f e a t u r e s worth b e i n g mentioned. I n t h e

l i m i t o f i n e q u a l i t i e s 3 . 3 i ) and ii).and f o r Ik(v+) v-I (k2DQ*L.)1/3),

and Ix-/v+I

<< i n f [(k(v+) DQ.L)-1/3,

y;]',

<< sup (Y!'~., equation 3.7

becomes:

[at + v-ax - - 2 D Q ' L * ( ~ + ) 2 DQ'L*(v+) cos (k(v+)x-)

< 6f 6fl> kv;f>]

=

2

The l a s t term o f t h e R.H.S. o f Eq. 5.2 i s c a l l e d t h e mode-coupling term s i n c e i t couples t o g e t h e r t h e k and k-k(v+), k + k(v+) F o u r i e r components o f < 6 f 6 f ' > . The r a t i o R = (k2DQ*L')1/3/yf*L* t u r n s o u t t o be a measure o f t h e n o n l i n e a r i t y and can be considered as a Reynolds number. For R << 1 t h e mode-coupling term i s n e g l i g i b l e and t h e q u a s i l i n e a r r e s u l t s a r e recovered. For R >> 1 t h e mode c o u p l i n g terms can be shown t o be o f t h e same order o f magnitude as t h e o t h e r and generate terms. The mode c o u p l i n g terms g i v e a c o n t r i b u t i o n f o r v - v ' = w k / k

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

451

harmonics coupled together through t h e resonant p a r t i c l e s . As a r e s u l t , t h e c o r r e l a t i o n f u n c t i o n c 6 f 6 f ' > i s found t o have t h e f o l l o w i n g convenient harmonic expansion <6f(x,v,t)

Bf

(XI , V ' , t ' ) >

= cgn(v-)

exp

n k(v+)(x-x')

] [-inw exp

Ik(v+)l (t-t')]

(5.3)

I n t h e regime R >> 1, where t h e n o n l i n e a r i t y p l a y s a r o l e , t h e c o r r e l a t i o n funct i o n i s thus the sum o f an i n f i n i t e number o f harmonics coupled together. When i n s e r t e d i n t o Poisson equations t h e c o n t r i b u t i o n coming from t h i s harmonic coupl i n g appears as a source term r a d i a t i n g coherently. We can, t h e r e f o r e , say t h a t t h e n o n l i n e a r i t y r e i n f o r c e s t h e coherence o f t h e s o l u t i o n . As a consequence, Adam, Laval, and Pesme indeed have found an increase o f t h e growth r a t e compared t o the q u a s i l i n e a r r e s u l t . This f e a t u r e i s generalized i n a s e l f - c o n s i s t e n t way i n the D I A . The Dupree "clump theory" l i k e w i s e p r e d i c t s an enhanced r a d i a t i o n due t o macroscopic g r a n u l a t i o n o f t h e phase space d e n s i t y a s o r t o f enhanced discreteness noise due o n l y t o n o n l i n e a r processes i n t h e continous Vlasov approximation. The Dupree t h e o r y a p p l i e s t o a "strong" turbulence l i m i t i n which no d e f i n i t e normal mode frequency e x i s t s f o r a given k. Therefore k(v) = wk(,,)/v i s not defined i n t h i s l i m i t .

-

The r e s u l t s o f Adam, Laval and Pesme apply i n t h e l i m i t o f q u a s i l i n e a r o r d e r i n g (eq. 3 . 3 i and 3 . 3 i i ) when yk/wk << 1 so t h a t a d e f i n i t e frequency i s associated w i t h a d e f i n i t e k. P a r t i c l e s o f v e l o c i t y v i n t e r a c t s t r o n g l y w i t h waves i n t h e spectrum o f wave number k(v) such t h a t p a r t i c l e s tend t o be p e r i o d i c a l l y c o r r e l a t e d w i t h t h e separation 2n/k(v). P h y s i c a l l y , t h e p a r t i c l e s are more l i k e l y t o be found near t h e troughs o f these resonant waves. The quasigaussian hypothesis r e v i s i t e d I n t h i s subsection we w i l l show i n a p e r t u r b a t i v e way how t h e mode-coupling terms generate a nongaussian e l e c t r i c f i e l d . We w i l l f o l l o w e s s e n t i a l l y t h e 26 arguments o f Laval and Pesme. Let io(v,t)

t h e space-averaged d i s t r i b u t i o n f u n c t i o n and d e f i n e 6 f as 6 f =

Following Kadomtsev, t h e equation f o r 6 f can be w r i t t e n i n f(x,v,t)-fo(v). F o u r i e r space as:

i n which

01

provides a contour p r e s c r i p t i o n f o r v - i n t e g r a t i o n .

Equation 5.5 can be solved i t e r a t i v e l y beginning w i t h i n a f u n c t i o n a l s e r i e s i n powers o f E. equation ikEkw = 4nq J6fb(v)dv

= g k [Ek fOlw T h i s s e r i e s i s then i n s e r t e d i n t o Poisson's

and y i e l d s a n o n l i n e a r equation f o r Ekw o f t h e

f o l l o w i n g form: &(k,w)Eb

= u2(k,w)E

E + p3 (k,w)EEE +

...

(5.6)

D. PESME, D.F. DUBOlS

452

(5.7)

(5.8a)

with Vku,,k'w' =

Ig-

h0

0

0

ku, g k - k ' ,

w-w'

(5.9)

f~dv

Equation 5.7 can be solved i t e r a t i v e l y by s e t t i n g

E = E(l)

+

E(2)

+

E(3)

...

(5.10)

The lowest order gives z E ( l ) = 0. T h i s i s t h e approximation o f q u a s i l i n e a r theory i n which the imaginary p a r t o f t h e r o o t o f E(k,w) determines t h e quasif o r s i m p l i c i t y i n t h e f o l l o w i n g we s e t y f a L ' = yk.

l i n e a r growth r a t e

= Ek(0) exp[-iwkt + y k t ] .

We have E(kl)(t) field.

We assume Ek(0) as a random phased

The mode-coupling terms w i l l generate nongaussian c o r r e l a t i o n s :

E(2) = I-r2 E ( l ) E ( l )

(5. l l a )

We now consider t h e 4 - p o i n t c o r r e l a t i o n f u n c t i o n 1 1 2 3 On s u b s t i t u t i o n o f t h e expansion (5.10) f o r each Ek we f i n d f i r s t o f a l l t h e usual terms
E-$ ( l ) E-k3 ( l ) > which, because o f t h e gaussian assumption

f o r E ( l ) can be c a s t i n t o a sum o f < E ( ~ ) E ( ~ ) .> L e t us now compute t h e nongaussian c o n t r i b u t i o n s .

A t t h e lowest order we o b t a i n t h e sum o f f o u r

E ( l ) > w i t h t k m = 0. The terms E(2) a r e indeed s m a l l e r terms l i k e
ki

453

TURBULENCE AND SELF-CONSISTENT FIELDS IN PLASMAS

where Qk = wk + i y k and 2Q = R k l + Rkll + Rk-kl-kll expression i n t o

.

After substitution o f t h i s

and use o f t h e gaussian assumption f o r E (1) ,

we o b t a i n t h e f o l l o w i n g expression:

exp- i w

( tl- t2) exp- i w k 2 ( tl- t 3) exp- iwk3( tl- t4) kl

(2kl+k2+k3)

3 -2iu

W

W

+ permutations (kl,k2,

k3)

, with u = y

kl

(5.13)

The Here t h e m a t r i x element V has been e x p l i c i t l y evaluated u s i n g 5.8b. d i e l e c t r i c f u n c t i o n has been taken t o be on resonance, i , e . , wkl + wk2 + wk3 which i m p l i e s t h a t frequencies and wave numbers have n o t a l l t h e same +k +k kl2 3 sign. We have a l s o assumed t h a t t h e d i s p e r s i o n o f t h e r o o t s wk i s weak so

2:w

t h a t e(m)

N

(E)

.

2 i ~ t . ~ .

The expression 5.13 enables us t o compute t h e

contribution o f the irreducible 4-point correlation function i n the evaluation o f the mode amplitudes. A t the lowest order we have

which leads t o t h e r e s u l t : (5.15)

D. PESME, D.F. DUBOlS

454

The m o d i f i c a t i o n o f t h e spectrum as g i v e n by t h e expression i n s i d e t h e b r a c k e t of Eq. 5.15 shows again t h a t t h e mode c o u p l i n g terms l e a d t o an increase o f t h e energy t r a n s f e r from resonant p a r t i c l e s towards waves. T h i s r e s u l t i s s t r i c t l y v a l i d o n l y f o r y!'L'>>

( k 2 D ) , l l 3 b u t i t c l e a r l y shows t h a t t h e i r r e d u c i b l e

4 - p o i n t c o r r e l a t i o n f u n c t i o n g i v e s a n o n - n e g l i g i b l e c o n t r i b u t i o n as soon as (k2D)1'3

2

y i ' L. which i n v a l i d a t e s t h e quasigaussian assumption.

By F o u r i e r transform o f Eq. 5.13 we o b t a i n t h e expression i n r e a l space o f t h e 4-point correlation function
E(l)(x2,t2)

expi kl(xl-x2)

E(')(x3,t3)

expi k2(x1-x3)

E(')(x4,t4)>

=

(5.16)

expi k3(x1-x4)

I f the expression i n t h e c u r l y brackets o f t h e R.H.S. o f Eq. 5.13 was a smooth f u n c t i o n o f kl, kz and k3, then t h i s c o r r e l a t i o n f u n c t i o n would be e s s e n t i a l l y a product o f t h r e e two-point c o r r e l a t i o n f u n c t i o n s o f t h e form

$([,r) = 2 exp-i[wkr-k[l For c o r r e l a t i o n s along t h e unperturbed o r b i t s l a t i o n time

T~

5

= v t , t h i s has t h e usual c o r r e -

= I A k ( v - V g ) l - l discussed p r e v i o u s l y .

On t h e c o n t r a r y , however, the expression i n c u r l y brackets o f t h e R.H.S. o f Eq. 5.13 i s a peaked f u n c t i o n f o r vlkl+kzl

5 u.

2

I n t h e regime R = ( k 0)

1/3

/yk

>> 1, Eq. 5.15 shows t h a t t h g s e r i e s expansion i n power o f E ( l ) i s n o t convergFnt. 9

The f r e e propagator g

may then be replaced by t h e renormalized propagator

m

gw,k

= (-q/m$

drexp

i(w-kv)r.

exp

-

[(k21f*L')~]3

3 av

(5.17)

0

I n the e s t i m a t i o n o f order o f magnitudes i t i s s u f f i c i e n t t o r e p l a c e u = y i m L . by v = ( k DQ'L*)1/3.

The nongaussian c o r r e l a t i o n f u n c t i o n i s , t h e r e f o r e , charac-

t e r i z e d by t h e time (k2DQ*L*)-1'3

which i s o f t h e o r d e r o f t h e i n v e r s e o f Kolmo-

gorov entropy hQ. L' = 0.36( k2DQ*L'

)ll3 encountered e a r l i e r .

Since ( k2D)-1'3

>>

tC

we see t h a t s e l f - c o n s i stency i n t r o d u c e s a tendency toward s e l f - o r g a n i z a t i o n o f t h e t u r b u l e n t sys tem.

A d e t a i l e d diagrammatic expansion leads t o t h e conclusion t h a t i n t h e regime (k2D)1'3

>> y k a l l t h e i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s c o n t r i b u t e a t t h e same

order o f magnitude as the q u a s i l i n e a r term. T h i s r e s u l t i n v a l i d a t e s t h e t h e o r i e s which c o n s i s t o f a simple r e n o r m a l i z a t i o n o f t h e propagator such as developed by

TURBULENCE AND SELF-CONSISTENT FIELDS IN PLASMAS

Rudakov and Tsytovich, Choi and H o r t ~ n . ~In~ the regime defined by ii) they lead to the quasilinear result since they don't contain the ling term of Eq. 5.2. It can be shownz8 that a partial summation of irreducible correlation functions leads to the Dupree equation 3.7. we may summarize the different levels of renormalization: 1.

455

3.3i) and mode coupthe neglected Therefore,

WEAK TURBULENCE THEORY - This makes the usual quasilinear prediction but suffers from the defect of secular divergences arising in the neglected irreducible correlation functions. These secularities are removed by renormalization of the propagator which leads to:

2.

SIMPLY RENORMALIZED THEORIES - These again make the usual quasilinear prediction but still suffer from non-negligible contributions from neglected irreducible correlation functions. A partial summation of the irreducible correlation functions leads to:

3.

DUPREE'S THEORY - This theory predicts a zero order modification of the quasilinear result coming from mode coupling terms. However, this theory considers only part of t h e irreducible correlation functions and is not properly self-consistent.

SELF-CONSISTENT STATISTICAL DESCRIPTIONS AND THE DIA It appears from the considerations above that most familiar statistical theories of Vlasov turbulence until recently, have been quasigaussian theories in the sense that they explicitly or implicitly make the assumption of quasigaussian electric field correlations. A self-consistent statistical description of Vlasov turbulence must take into account the relationship between perturbations in the averaged phase space distribution = and perturbations in the mean electric field = as well as the relationship between the These fluctuations in these quantities: 6fl = fl- and 6E1 = EI-. relationships are, of course, imposed by Poisson's eqn. (4.1). For the discussion here we follow closely the results of reference 29. From the Vlasov eqn. (32) the well-known equations for the mean distribution function is easily found: r

(5.18)

This couples to the fluctuation correlation function <6E 6f>. The averaged Poisson equation is (5.19) V1 * = 4nq J dvl - 4nqN We have added an artificial phase space source density q1 = q(xlxltl) which we can consider as a probe of the system. Suppose for r) = 0 the system of equations has a well-defined solution , ; we then add a local phase space source perturbation 6r) which changes to + 6 and to + &. The increments 6 are functional differentials such as those used in the calculus of variations of continuous fields. A relationship between the increments 6, 6 and 6
0. PESME, 0.F. DUBOIS

456

G;:

6

a

+

r126 = 6ql v1

(5.20)

By t a k i n g t h e f u n c t i o n a l d i f f e r e n t i a l o f eqn. (5.18) c o e f f i c i e n t s as

i t i s easy t o i d e n t i f y t h e

(5.21) (5.22) To s i m p l i f y w r i t i n g t h e equations we a r e u s i n g a summation convention where, f o r example, :;G

6

Jdv2Jdx2Jdt2 G;:

6, etc. and 6(1-2)

6(vl-v2)

6(x1-x2)6 (tl-t2). The l a s t terms on t h e r i g h t hand s i d e o f eqns. (5.21) and (5.22) c o n t a i n f u n c t i o n a l d e r i v a t i v e s o f t h e c o r r e l a t i o n f u n c t i o n <6E16fl> w i t h respect t o changes i n t h e mean values and , r e s p e c t i v e l y . These funct i o n a l d e r i v a t i v e s completely describe t h e renormal i z a t i o n o f t h e response funct i o n s G12 and rI2. The response f u n c t i o n G12

i s c a l l e d t h e " q u a s i p a r t i c l e propagator"; w i t h o u t

r e n o r m a l i z a t i o n i t i s c l e a r from eqn. (5.21)

t h a t G12

simply describes p a r t i c l e

propagation along unperturbed o r b i t s ( i n t h e mean f i e l d

which i s zero i n

a homogeneous system.) The r e n o r m a l i z a t i o n term takes i n t o account t h e s e l f c o n s i s t e n t response o f the f l u c t u a t i n g t u r b u l e n t "background" through which t h e p a r t i c l e propagates. I n an imposed f l u c t u a t i o n l i m i t t h i s propagator reduces t o t h e f a m i l i a r Dupree renormalized propagator g i v e n i n eqn. 3.9. The response f u n c t i o n

aVlrl2

i s c a l l e d t h e " q u a s i p a r t i c l e p o t e n t i a l source."

I t takes i n t o account t h e e f f e c t i v e source d e n s i t y 6q which must a r i s e when t h e r e i s a change i n t h e mean e l e c t r o s t a t i c f i e l d 6 i n o r d e r t o keep < f > constant. Without r e n o r m a l i z a t i o n , t h i s term i s simply p r o p o r t i o n a l t o a . v1 The r e n o r m a l i z a t i o n o f t h i s q u a n t i t y i s n o t f a m i l i a r b u t was a c t u a l l y considered

by K a d o m ~ t e vi ~n ~Section I11 o f h i s 1965 book (equation 21a). (Kadomstevls renormalized theory has some elements i n common w i t h t h e D I A , b u t d i f f e r s i n several important respects r e l a t e d t o s e l f - c o n s i s t e n c y . ) We next consider t h e f l u c t u a t i o n s . By s u b t r a c t i n g eq. (5.18) from eqn. (3.2) and n o t i n g t h e d e f i n i t i o n s (5.21) and (5.22) we can d e r i v e t h e exact equation f o r bf,

6fl

= 6f;nCA

+ ;q/m)

G12 8,

[6f26E2-<6f26E2>

2

-

6<6f26E2> 6f3

6

-

(5.23) 6<6f26E2> <6E3>

D. PESME. D.F. DUBOIS

We have i n d i c a t e d t h a t 6fl

457

can be separated i n t o a coherent p a r t p r o p o r t i o n a l t o

t h e e l e c t r o s t a t i c f l u c t u a t i o n 6E and an incoherent p a r t .

I n a s p a t i a l l y homo-

geneous plasma i t i s easy t o see t h a t t h e F o u r i e r component 6 f p h i s p r o p o r t i o n a l t o 6Ek.

For t h e incoherent p a r t terms l i k e 6f16E1

such as

i n v o l v e F o u r i e r convolutions

6fk-k16Ekl ( w i t h 6Ekz0=0) and thus i n v o l v e s t h e c o u p l i n g o f d i f f e r e n t

F o u r i e r modes. I f t h e expression (5.23) i s i n s e r t e d i n t o Poisson's equation Vl* GE1=4nqJdvl

[ 6 f i o h + 6 f i n c ] one can rearrange t h e coherent c o n t r i b u t i o n t o w r i t e

V1 where

-

E~~

cI2

6L2 = 4nqJdv16fl

inc

(5.24)

i s t h e d i e l e c t r i c operator 29

The coherent p a r t has produced t h e s u s c e p t i b i l i t y f u n c t i o n i n t h e d i e l e c t r i c operator. For a homogeneous s t a t i o n a r y system, t h e F o u r i e r c o e f f i c i e n t o f E can be w r i t t e n

2

2

E(k,w) = 1 + (4nq /mk ) Jdv3 Jdv

4

Gv

3 4

i k * a r (k,w) v4 v4

(k,w)

We see from eqn. (5.24) t h a t t h e charge d e n s i t y associated w i t h 6finC a c t s as an incoherent source f o r t h e e l e c t r o s t a t i c f l u c t u a t i o n . For example, t h e twop o i n t e l e c t r o s t a t i c c o r r e l a t i o n f u n c t i o n can be w r i t t e n as

i n terms o f t h e inverse d i e l e c t r i c operator.

Or

n

(5.26b) f o r t h e s p e c t r a l i n t e n s i t y i n a homogeneous, s t a t i o n a r y case. The two-point c o r r e l a t i o n f u n c t i o n f o r t h e phase space d i s t r i b u t i o n can l i k e w i s e be w r i t t e n as c6f16fll

> = (q 2 /m 2 1 G 1 2 ~ v 2 ~ 2 3 G 1 1 2 1 ~ v ; ~ 2 ~ 3 ~ ~ ~ E 3 ~ E 3 ~ >

where P12

6(1-2)P1,2,-P126~l'-21)

i s a " p o l a r i z a t i o n " operator

1

2 a v3r 34(v2& 1-l '12 = (4nq lm) '13 4 42

<6f2i n c6f2' inc,

(5.27)

(5.28)

The f i r s t term on t h e r i g h t side i s t h e c o r r e l a t i o n <6fyh6fi:h> while the f i r s t inchf inc, term i n square brackets a r i s e s from <6fl 1l The second and t h i r d term i n

.

458

D. PESME, D.F. DUBOlS

square brackets a r i s e r e s p e c t i v e l y from <6finc6f::h>

coh inc, and <6fl 6fll

.

In

w r i t i n g these terms i n terms o f the operator P we have used Poisson's equation i n the form o f eq. (5.24) t o e l i m i n a t e 6E i n terms o f 6finc.

By doing t h i s we

have expressed c o r r e l a t i o n f u n c t i o n s o f t h e form <6E6E6f> i n terms o f < 6 f i n c 6finc 1l > which can be seen f r o m t h e d e f i n i t i o n o f 6finc, eq. 5.23, t o i n v o l v e 4 - p o i n t and higher c o r r e l a t i o n f u n c t i o n s . The vanishing o f t h e cross c o r r e l a t i o n terms i s made e x p l i c i t l y i n t h e theory o f Dupree ( r e f . 2 eqn. (21)). The equations w r i t t e n i n t h i s s e c t i o n up t o t h i s p o i n t a r e f o r m a l l y exact. We inc6f inc, see t h a t the system i s n o t closed since t h e c o r r e l a t i o n f u n c t i o n <6fl i n v o l v e s t h r e e - p o i n t and f o u r - p o i n t c o r r e l a t i o n f u n c t i o n s . An approximation which closes these equations c a l l e d t h e " D i r e c t I n t e r a c t i o n Approximation," D I A , has a t t r a c t e d renewed i n t e r e s t . This t y p e o f theory, f i r s t a p p l i e d by Kraichnan t o Navier-Stokes t u r b ~ l e n c e ,t r~e~a t s c e r t a i n nongaussian c o r r e l a t i o n s i n a n o n p e r t u r b a t i v e way. This approach was reasonably successful f o r moderate Reynold's numbers b u t f a i l e d t o a c c u r a t e l y d e s c r i b e t h e i n e r t i a l range f o r h i g h Reynold's numbers. The f a i l u r e was due t o t h e i n a c c u r a t e treatment o f t h e convection o f s h o r t scale turbulence by t h e l o n g s c a l e motion. 32 The D I A was w r i t t e n down f o r t h e Vlasov-Poisson system by Orzsag and Kraichnan i n 1967. This work r e c e i v e d very l i t t l e a t t e n t i o n u n t i l r e c e n t l y when DuBois and E ~ p e d a showed l~~ how t o " f a c t o r " t h e t h e o r y i n t o s i n g l e (quasi) p a r t i c l e and c o l l e c t i v e responses. This exposed t h e existence i n t h e D I A o f new s e l f - c o n s i s t e n t f l u c t u a t i o n ( o r n o n l i n e a r p o l a r i z a t i o n ) c o n t r i b u t i o n s which were n o t found i n t h e o l d e r quasigaussian imposed f i e l d t h e o r i e s . The new terms a r e proport i o n a l t o t h e d i e l e c t r i c response E - ~such as t h e terms i n t h e p o l a r i z a t i o n operator i n eqn. (5.27); various estimates have shown these terms t o be o f t h e same order as t h e imposed f i e l d terms. The D I A based theory was a l s o i n v e s t i gated by Krommes and coworkers34 who a l s o considered a p p l i c a t i o n t o t h e g u i d i n g 35

center model and t o d r i f t wave turbulence.

Here we w i l l l i s t the approximations t o t h e s e t o f exact equations l i s t e d above which a r e e q u i v a l e n t t o t h e D I A :

1.

y

The incoherent c o r r e l a t i o n f u n c t i o n <6fl inchf i > i s computed t o terms o f

q u a d r a t i c order i n t h e c o r r e l a t i o n s u s i n g a quasigaussian approximation f o r t h e mixed f o u r - p o i n t c o r r e l a t i o n f u n c t i o n

q2/m 2 G12av2G1121av21[<6E26E21><6f26f21>+ <6E26f2'> <6f26E21>]

(5.29)

Note t h a t t h i s i s n o t a t a l l j u s t t h e quasigaussian approximation f o r t h e c o r r e l a t i o n f u n c t i o n o f f o u r e l e c t r i c f i e l d s . The f l u c t u a t i o n 6 f c o n t a i n s , as a consequence o f t h e incoherent c o n t r i b u t i o n , 6finc orders i n t h e e l e c t r i c f i e l d f l u c t u a t i o n 6E.

, i n eqn. (5.23), terms o f a l l (These terms can be obtained from

eqn. (5.23) by i t e r a t i o n s t a r t i n g w i t h 6 f y ) = 6 f E o h . )

Thus t h e D I A c o n t a i n s

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

459

e l e c t r c f i e l d c o r r e l a t i o n s o f a l l orders; i n p a r t i c u l a r i t produces i n t h i r d 2 order n (6E ) t h e nongaussian c o r r e l a t i o n s c a l c u l a t e d by Lava1 and Pesme. The inverse propagator G - l and t h e response f u n c t i o n lowest (linear)order i n the c o r r e l a t i o n f u n c t i o n s .

n

L.

r

a r e computed t o

To c a r r y o u t the l a s t step t h e f u n c t i o n a l d e r i v a t i v e s o f <6E6f> i n eqs. (5.21) and (5.22) must be evaluated t o t h i s order. This can be accomplished by u s i n g an exact r e l a t i o n connecting the f l u c t u a t i o n s which i s e a s i l y d e r i v e d from eq. (5.23): <6f16ElI>

=

{ G12av;23GE36E11>

+

4nq a l(V~lcl121 )-1,6finc1 6f2inc,

(5.3ij

x1

This equation can a l s o be used f o r t h e r i g h t - h a n d s i d e o f eq. (5.18) f o r t h e mean d i s t r i b u t i o n . I n computing t h e r e q u i r e d f u n c t i o n a l d e r i v a t i v e s o f <6f6E> f o r s e l f - c o n s i s t e n t f l u c t u a t i o n s i t i s necessary t o consider t h e f u n c t i o n a l dependence on < f > o r o f t h e d i e l e c t r i c response, c-', which e n t e r s e x p l i c i t l y i n the l a s t t e r m o f eq. (5.31) and a l s o i n <6E6E> as shown e x p l i c i t l y i n eq. (5.26). We can g i v e here o n l y the r e s u l t o f t h i s c a l c u l a t i o n f o r :G; homogeneous, s t a t i o n a r y system:

-1 Gvvl (k,u) = -i(w-k.v)b(v-v')

+

42m2 av. Jdk'Jdw'

I

for a

<6E6E>klul

~ v v I ( k - k ' , u - w ' ) + Jdy P v ( k - k ' , w - w ' ) G ~ v l ( k - k ' , w - w l )

1

(5.32)

The l a s t two terms i n c u r l y brackets a r e s e l f - c o n s i s t e n t f l u c t u a t i o n terms which a r i s e from t h e f u n c t i o n a l dependence discussed above and do n o t occur i n t h e quasigaussian f i e l d t h e o r i e s . These s e l f - c o n s i s t e n t f l u c t u a t i o n c o n t r i b u t i o n s have been estimated i n various cases t o be as important as t h e remaining term which appears i n t h e quasigaussian f i e l d t h e o r i e s and produces t h e renormal i z e d 1f o r I n the q u a s i l i n e a r o r d e r i n g o f ~ ~ < < t and < ~ kS>> > propagator i n eq. 5.1. a one-dimensional one-component plasma we have been a b l e t o show t h a t t h e s e l f c o n s i s t e n t f l u c t u a t i o n terms cannot be neglected. A s i m i l a r equation r e s u l t s f o r the renormalized Tv(k,u) which we w i l l n o t discuss here. F u r t h e r r e s u l t s a r e given i n reference 29. I f the D I A approximation o f eqn. (5.30) f o r <6finc6finc> i s i n s e r t e d i n t o eq. (5.27) we o b t a i n t h e f o l l o w i n g equation f o r t h e two-point c o r r e l a t i o n f u n c t i o n

460

D. PESME, D.F. DUBOIS

r

s . c. A

= (q2/m2) G12~v;23G1121~vh~2131<6E36E31>

(5.34)

When the terms labelled S.C. are dropped in this expression and only the imposed field renormalization terms in G are retained, this equation can be shown to reduce to the Dupree equation for <6f(x,v,t)6f(x'v1t)> given in equation (3.7). The latter approximation is valid only for quasigaussian imposed electric field fluctuations; we can see explicitly in this case that the remaining contribution from <6finc 6fivc> in eqn. (5.34); i.e., (q 2/m 2)G12G11216 v 2 6 v 2 1 ~ 6 f 2 6 f 2 1 ~ ~ 6 E 2 6 E 2 1 ~ reduces to the mutual diffusion (or mode coupling) term of equations (3.7) or (5.2). It is this term which produces the finite Kolmogorov entropy, h, discussed in Section 3. Thus there is a direct connection in this case between a finite level for the incoherent noise function <6f~"6f~vc> (sometimes called the nonlinear noise) and the intrinsic stochastic behavior of the system. The reduction of these DIA equations to a calculable form is very difficult even in the limit of quasilinear ordering. The harmonic expansion of eq. (5.3) still appears to be useful in the DIA. Using this technique we have been able to show that n ~ n eof the self-consistent terms can be dropped even in quasilinear ordering when R >> 1. Only the equations for the harmonic components n = 0, ? 1 and 2 are significantly changed by the self-consistent terms. Work is still in progress in deducing the consequences of the DIA in the quasilinear limit. The DIA is the natural first approximation in the renormalized perturbation theories of Kraichnan, Martin, Siggia and Rose (MSR)36 and others. The general conditions for the convergence of the MSR expansion are completely unknown; probably only convergence in the sense of an asymptotic expansion is expected at best. It appears to be a possible, but extremely tedious, task to examine the size o f the leading vertex corrections (in the sense of MSR) to the DIA. There is some Q that in quasilinear ordering these vertex corrections may be negligible. This hope is based on the following points: i) In the imposed field case the DIA theory becomes exactly quasilinear theory (QLT) in quasilinear ordering; the corrections in this case are o f order (T~/T<~,) << 1. ii) In the self-consistent field case the DIA differs from QLT in quasilinear ordering; the differences due to nonlinear polarization effects are the same order as the QLT terms. iii) If self-consistency also does not change the order of the vertex corrections, the corrections to the DIA will also be of order (T~/T<~,) as in i). 37 To make a point of contact between this formalism and the results of Adam, Lava1 and Pesme, we write an equation for the time evolution of ClbEl 2k>:

where (5.36)

TURBULENCEAND SELF-CONSISTENTFIELDS IN PLASMAS

This can be shown t o be an exact consequence o f eq. (5.26a)

’ 9 , i s t h e zero it,) = 0. I n t h e found t h a t Qk and 9 ,

where Qk + E(k,lk

+

provided

461

pk<
o f t h e d i e l e c t r i c f u n c t i o n i n t h e complex w plane:

l i m i t of q u a s i l i n e a r o r d e r i n g ( r h < < t c f > , kS>>l)

d i = w2+3k2Ve2, P = y i e L * . I n conventional q u a s i l i n e a r theory i t i s assumed t h a t Sk a r i s e s 2 2 from mode-mode c o u p l i n g e f f e c t s o f o r d e r <16Ekl > and i s thus n e g l i g i b l e . it i s

are given by t h e i r l i n e a r values:

9,

However, t o account f o r t h e r e s u l t s o f Adam, Laval and Pesme discussed i n Section 4 i t i s necessary t h a t t o l e a d i n g order

(5.37) where y y p i s t h e observed growth r a t e .

We see t h a t Sk i s r e l a t e d t o t h e r i g h t

hand side o f eq. (5.14). Comparison can be made w i t h t h e i t e r a t i v e c a l c u l a t i o n o f Laval and Pesme (see Section 5.3) by i t e r a t i n g Eq. (5.34) s t a r t i n g w i t h <6f16fl,> = [ r i g h t hand s i d e o f eq. (5.34)] and i n s e r t i n g t h i s formal expansion i n powers o f <6E6E> i n t o eqs. (5.31)

and (5.29).

The combination o f t h e l a s t

t w o equations then leads t o a formal expansion o f < 6 f ~ n c 6 f ~ : c > and a l s o o f

2 3 t h i s expansion agrees w i t h t h e r e s u l t s o f Laval and Pesme which was o u t l i n e d i n Section 5.3; because Sk i n powers o f .

To terms f o r m a l l y o f order <6E >

o f wave-particle resonances f o r yk<(k2DQL)1’3

2

t h e terms i n Sk o f t h i s formal

order appear t o be a c t u a l l y o f order <16Ekl >, a r e s u l t which i s c o n s i s t e n t w i t h eq. (5.37).

A complete s o l u t i o n o f t h e D I A equations f o r < 6 f ~ n c 6 f v l n C > k w

remains t o be c a r r i e d out.

The s e l f - c o n s i s t e n t ( n o n l i n e a r p o l a r i z a t i o n ) terms

2

o f the D I A do n o t appear t o change t h e order i n <16Ekl > o f q u a n t i t i e s such as Sk b u t they are n u m e r i c a l l y important and must be included, f o r example, t o pre-

serve energy conservation. Conc 1 us ion I n t h e f i r s t p a r t o f t h i s paper we have reviewed t h e r e s u l t s o f i n t r i n s i c s t o c h a s t i c i t y t h e o r i e s i n the case where t h e imposed e l e c t r i c f i e l d has random phase F o u r i e r components. Whenever t h e mode amplitudes a r e w e l l above t h e t h r e s h o l d f o r i n t r i n s i c s t o c h a s t i c i t y , an e x c e l l e n t agreement has been demons t r a t e d between t h e numerical experiments and t h e quasi1 i n e a r p r e d i c t i o n s f o r t h e d i f f u s i o n c o e f f i c i e n t and the Kolmogorov entropy. I n t h e second p a r t we have considered t h e Vlasov-Poisson turbulence i n which t h e e l e c t r i c f i e l d i s s e l f - c o n s i s t e n t l y r e l a t e d t o t h e p a r t i c l e s motion through Poisson’s equation. We have f i r s t observed a s t r i k i n g d i f f e r e n c e compared w i t h t h e imposed e l e c t r i c f i e l d case: t h e numerical s i m u l a t i o n s e x h i b i t a s i g n i f i cant discrepancy w i t h t h e q u a s i l i n e a r p r e d i c t i o n . The o r i g i n o f t h i s d i s c r e pancy has then been shown t o be a consequence o f a s t r o n g enhancement o f t h e wave-particle i n t e r a c t i o n . As a r e s u l t , the s t a t i s t i c a l p r o p e r t i e s o f t h e s e l f - c o n s i s t e n t e l e c t r i c f i e l d are c h a r a c t e r i z e d by a second c o r r e l a t i o n time Self-consistency (k2D)-1’3 much longer than the usual c o r r e l a t i o n time t leads thus t o a s o r t of s e l f - o r g a n i z a t i o n o f t h e turbulgnce. Dupree’s theory now appears t o be a p a r t i a l summation o f the i r r e d u c i b l e c o r r e l a t i o n f u n c t i o n s o f t h e e l e c t r i c f i e l d which are neglected i n t h e simply renormalized t h e o r i e s . I t has, however;a remaining l a c k of consistency i n t r i n s i c t o a l l quasigaussian

.

462

0. PESME, D.F. DUBOlS

theories. We have then displayed the c h a r a c t e r i s t i c features o f D I A a p p l i e d t o Vlasov turbulence and we have shown how t h i s c l o s u r e takes i n t o account i n a s e l f - c o n s i s t e n t way the strong wave-particle i n t e r a c t i o n and t h e r e s u l t i n g nongaussian p r o p e r t i e s o f the e l e c t r i c f i e l d . ACKNOWLEDGEMENTS We are pleased t o acknowledge h e l p f u l comments from D r . J . C. Adam, D r . G. Laval, D r . H. Rose and O r . T. O'Neil. Conversations w i t h D r . F. Doveil, D r . 0. Escande and D r . A. Rechester are a l s o acknowledged. REFERENCES

52,

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3

nhD>>lwhere n i s the plasma d e n s i t y and AD t h e Debye length.

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464

0. PESME. D.F. DUBOIS

appears to be only a restriction on the level 37. The condition of electric field fluctuations whereas the condition for smallness of the vertex corrections to the D I A must also involve some restriction on <6f2>. The fluctuation level <6f2> is asymptotically in time proportional to <6E2> and both are bounded. Convergence would seem possible for sufficiently small and smooth initial fluctuations < 6 f2>.