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NONLINEAR WAVE INTERACTION AND CRITICAL FLUCTUATIONS IN PLASMAS
A. Sitenko
To cite this version:
A. Sitenko. NONLINEAR WAVE INTERACTION AND CRITICAL FLUCTUATIONS IN PLAS- MAS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-661-C7-662. �10.1051/jphyscol:19797321�.
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JOURNAL DE PHYSIQUE Colloque C7, suppldment au n07, ~ome 40, J u i l l e t 1979, page C7- 661
N O K I N E A R WAVE MERACTION A N D CRITICAL FLUCTUATIONS IN PLASMAS
A.G. Sitenko.
I n s t i t u t e for Theoretical Physics, Academy of Scidnces o f the Ukrainian SSR, Kiev, U. S . S. R.
The s p e c t r a l d i s t r i b u t i o n of plasma f l u c t u a t i o n s has a broad maximum i n the low frequency domain, which i s due t o the random motion of the i n d i v i d u a l p a r t i c l e s , and a s e t of sharp maxima a t plasma eigen- frequencies. These sharp maxima a r e ca- used by the c o l l e c t i v e f l u c t u a t i o n s , i .e.
the plasma random eigen o s c i l l a t i o n s . The equilibrium c o l l e c t i v e f l u c t u a t i o n l e v e l i s governed by the temperature; i t grows e s s e n t i a l l y i n nonequilibrium plasmas, e s p e c i a l l y i f the plasma s t a t e approaches the k i n e t i c i n s t a b i l i t y . This growth i s i n f i n i t e i n l i n e a r approximation, which i n d i c a t e s t h a t the l i n e a r approximation i s inadequate under such conditions. So one must take i n t o accound the nonlinear e f f e c t s
[I],
which l e a d t o the s a t u r a t i o n of the c r i t i c a l f l u c t u a t i o n s .When i n v e s t i g a t i n g the nonequilib- rium f l u c t u a t i o n s i t i s convenient t o u t i l i z e the nonlinear f i e l d equation, which follows d i r e c t l y from t h e Maxwell's
equations and the microscopic d e n s i t y equation t h a t describes the s t o c h a s t i c p a r t i c l e motion i n plasmas. The nonlinear wave i n t e r a c t i o n i s manifested most c l e a r l y when the frequencies and the wave v e c t o r s s a t i s t g the resonance conditions.
The s i m p l e s t example of the resonance wave i n t e r a c t i o n i s the three-wave one
t h a t causes the decay o r the explosive i n s t a b i l i t i e s . I n case the three-wave re- sonance conditions a r e n o t s a t i s f i e d , the resonance i n t e r a c t i o n s of f o u r o r more waves become important. Cne of the most s i g n i f i c a n t e f f e c t s t h a t a r i s e due t o the four-wave resonance i n t e r a c t i o n i s the frequency s h i f t s of the i n t e r a c t i n g waves.
I n p a r t i c u l a r , the resonance i n t e h a c t i o n i n plasmes causes the nonlinear shifts of the eigenfrequencies.
The nonlinear eigenfrequency shift A d t of a l o n g i t u d i n a l wave wieh the eigen-
frequency W k and the wave v e c t o r
k
i s a function of the wave i n t e n s i t yIt
and the plasma n o n l i n e a r s u s c e p t i b i l i t i e s :The p r o p o r t i o n a l i t y c o e f f i c i e n t 3 i s I
equal t o zero i n a cold plasma, i . e . the nonlinear wave i n t e r a c t i o n does n o t change the eigenfrequencies of cold plas- mas.
A s an example consider the Langmuir wave eigenfrequency s h i f t i n i s o t r o p i c plasmas [2,3]
.
The hydrodynamic t r e a t - ment of the e l e c t r o n motion i n n e g l e c tArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797321
of t h a t of i o n s l e a d s t o
However t h e hydrodynamic d e s c r i p t i o n i s v a l i d provided t h a t the p a r t i c l e thermal v e l o c i t y i s small n o t o n l y compared t o t h e wave phase v e l o c i t y , b u t a l s o i n compari- son w i t h the phase v e l o c i t i e s of t h e pul- s a t i o n s . A s t h e Langmuir waves do n o t sa- t i s f y t h e second c o n d i t i o n , then t h e hy- drodynamic approximation should n o t be a p p l i e d when c a l c u l a t i n g t h e n o n l i n e a r s h i f t . The k i n e t i c approach r e s u l t s i n
Note, t h a t t h e k i n e t i c
pk
t u r n s o u t t o be zero a tT
= 0 s i m i l a r l y t o t h e hydrodyna- mic one. However t h e a b s o l u t e values of t h e n o n l i n e a r eigenfrequency s h i f t s a r e d i f f e r e n t f o r t h e two approaches. The i o n motion may be n e g l e c t e d i fa ' / ?
> mM'
otherwise i t must be taken i n t o account.
The average i n t e n s i t y
Il.
of t h e equi- l i b r i u m plasma e i g e n o s c i l l a t i o n s i s go- verned by t h e t e m p e r a t u r e ' r .Ik
of nonequi- l i b r i u m plasma may d i f f e r e s s e n t i a l l y from the e q u i l i b r i u m v a l u e , thus making i t v e r y i m p o r t a n t t o t a k e i n t o account t h e n o n l i - n e a r eigenfrequency shifts. The s p e c t r a l d i s t r i b u t i o n of 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 i n nonequilibrium plasmas i sS i n c e t h e eigenfrequency
- Zk i s a f u n c t i o n
of l k ,
we can use ( 6 ) a s an equation f o r
I t ,
which i s t h e f l u c t u a t i o n i n t e n s i t y i n t h e i n s t a b i l i t y region. The e x i s t e n c e of a s o l u t i o n f o r t h i s equation i m p l i e s t h a t t h e wave i n t e r a c t i o n d r i v e s t h e u n s t a b l e plasma t o some s t a t i o n a r y s t a t e . The solu- t i o n i t s e l f determines t h e s t a t i o n a r y l e - v e l of t h e c r i t i c a l f l u c t u a t i o n e .Two examples have been considered t h a t i l l u s t r a t e t h e n o n l i n e a r s a t u r a t i o n of t h e c r i t i c a l f l u c t u a t i o n l e v e l i n non- e q u i l i b r i u m plasmas: f l u c t u a t i o n s i n a plasma w i t h a low-density compensated p a r t i c l e beam 2 , and those i n a magneto- a c t i v e plasma with an a n i s o t r o p i c p a r t i c l e d i s t r i b u t i o n [4].
I. A.G.Sitenko. Sov.J.Plasma Phys.,
1,
24 (1975)2. A.G.Sitenko. Fhysica S c r i p t a ,
2 ,
193 (1973).3. A.G.Sitenko, V.I.Zasenko. Ukrain.Piz.
Zh.,
3,
1277('l978)*4. A.G.Sitenko, V.I.Zasenko. Ukrain.Fiz.
Zh.,
22,
715 (1978).where
