THE INSTABILITY AND DAMPING OF THE LANGMUIR WAVES WITH DIFFERENT AMPLITUDES AND PHASE VELOCITIES

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THE INSTABILITY AND DAMPING OF THE LANGMUIR WAVES WITH DIFFERENT AMPLITUDES AND PHASE VELOCITIES

N. Buchelnikova, E. Matochkin

To cite this version:

N. Buchelnikova, E. Matochkin. THE INSTABILITY AND DAMPING OF THE LANGMUIR WAVES WITH DIFFERENT AMPLITUDES AND PHASE VELOCITIES. Journal de Physique Col- loques, 1979, 40 (C7), pp.C7-633-C7-634. �10.1051/jphyscol:19797307�. �jpa-00219297�

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JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tom 40, JuiZZet 1979, page C7- 633

THE INSTABILITY AND DAMPING OF THE LANGMUIR WAVES WITH DIFFERENT AMPLrrClDES AND PHASE VELM=ITIES

N.S. Buchelnikova and E.P. Matochkin.

I n s t i t u t e of NucZear Physics, Novosibirsk, U.S.S.R.

The n v e r i c a l experiment was made t o WO 0,3 1 , 6 11 18 36 115 i n v e s t i g a t e t h e i n s t a b i l i t y of onedimen

-

^b#%

- - -

^{0 , l} ^4,6 ²⁰

s i o n a l running Langmuir waves with high

€ 2 ²

amplitudes W, = -9 LKord) and phase 8fihT

v e l o c i t i e s Vph/ 7;. 1 : WO '0,3-10 2 'fr

( V P L / ~ ~ = 16); VpL/v, ^c3-48 ^(WO = 1.6).

The increase of w,, and decrease of Vph i n such range 1ead.s t o the increase of plasma e l e c t r o n n o n l i n e a r i t y from the li- n e a r case up t o the case of e l e c t r o n t r a p - ping, The l i n e a r case and the case of

W ~ Q 2,5 6,2 18,4 22,7

- -

Wmaxhe 8 3,9 1 , 7 112

- -

trn~,h-,, 37 1 7 4.5 4,5

- -

In cases 1-1

-

^1-4 i n agreement w i t h /I/ t h e modulational i n s t a b i l i t y with t h e l e n g t h of p e r t u r b a t i o n hC A O develops and l e a d s t o t h e formation and contrac

-

t i o n of t h e d e n s i t y c a v i t y and t o t h e COI+

c e n t r a t i o n of t h e e l e c t r i c f i e l d i n i t ,

E ^'marc

s m a l l n o n l i n e a r i t y ~ e r e investigated ear- so t h a t e n e r g y d e n s i t y W r , = i n c r e a - 8 ~ i m f

l i e r /I/. To c h a r a c t e r i z e the nonlineari- sea (Pig. I ) . I n a p e c t r a l d e s c r i p t i o n 2 P % z '

t y t h e parameters o\ s:

-

^and t h i s process shows i t s e l f i n the e x c i t a

-

2 Vph

-

^VT

d'=wo h d ) were used. Electron nonli- t i o n of more and. more s h o r t wavelength n e a r i t y shows i t s e l f i n t h e increase of

t h e perturbed e l e c t r o n v e l o c i t y , i n the sharpening of e l e c t r o n d e n s i t y perturba- t i o n and i n t h e steepening of Eo(). It was shown t h a t e l e c t r o n n o n l i n e a r i t y be

-

I

comes e s s e n t i a l ifd ^{0 , 4}^{( d}

+

^10'~).

The l i m i t i n g case of high e l e c t r o n nonli- n e a r i t y is the trapping of plasma e l e c t - rons by t h e wave. The trapping becomes e a s e n t i a l i f 4 ^),1 , 3 ( d l + 5-10'~).

The information about numerical experiments with d i f f e r e n t amplitude waves is given i n t h e t a b l e 1 and Fig. 1.

1-1 1-2 1-3 1 1-5 1-6

o( 0&6 0,71 I s 1 7 1933 1957 2910

modes with decreasing phase v e l o c i t i e s . When the d e n s i t y p e r t u r b a t i o n beco

-

.v

mes high ( %o 3 1 0 - ~ - 1 0 - ~ ) t h e conver

-

s i o n process ke tpk;+ ⁼1,2...

,

k e , k,

-

wave v e c t o r s ) becomes e s s e n t i a l and l e a d s t o the e x c i t a t i o n of more and more s h o r t wavelength d i r e c t and backwardmodes

The damping of the e l e c t r i c f i e l d begins when t h e modes with phase veloci- t i e s low enough t o t r a p plasma e l e c t r o n 8 a r e excited. The trapping and accelera- t i o n of e l e c t r o n s l e a d s t o t h e f u l l ab

-

s o r p t i o n of e l e c t r i c f i e l d energy. The i n c r e a s e o$ i n i t i a l wave amplitude l e a d s t o t h e i n c r e a s e of perturbed e l e c t r o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797307

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v e l o c i t y and of t h e a d d i t i o n t o i t due t o t h e e l e c t r o n n o n l i n e a r i t y s o t h a t modes w i t h h i g h e r Vph b e g i n t o t r a p e l e c t r o n s and t h e damping s t a r t s e a r l i e r (Fig. 1,

kmax i n t h e t a b l e ) . Due t o t h e c r o s s i n g of t h e modest t r a p p i n g r e g i o n s t h e e l e c - t r o n s a r e a c c e l e r a t e d up t o h i g h v e l o c i - t i e s ( i n t h e c a s e s 1-3, 1-4 up t o V .v

v p k +"tr of t h e i n i t i a l wave, Vtv = 2 e

and forme t h e t a i l on t h e v e l o c i t y d i s t - r i b u t i o n f u n c t i o n .

I f i n i t i a l wave amplitude i s h i g h enough, s o t h a t d. & 1,3, t h e t r a p p i n g of e l e c t r o n s by t h e i n i t i a l wave becomes es- s e n t i a l . I f

-

^{1 , 3}a n d t h e p a r t of e l e c - t r o n s

R / ~ N ~

t r a p p e d d u r i n g f i r s t plasma p e r i o d TOP i s s m a l l ( c a s e 1-4) t h e de

-

c r e a s e of \/Vm due t o t h e t r a p p i n g and i n c r e a s e of i t due t o t h e i n s t a b i l i t y a r e of t h e same o r d e r (Fig. 1 ). If d and PN _N

i n c r e a s e t h e t r a p p i n g becomes t h e main p r o c e s s ( c a s e s 1-5, 1-6). The damping r a t e i n t h e s e c a s e s i s many o r d e r s high- e r t h a n Landau damping r a t e . In a l l t h e s e c a s e e t h e c o n v e r s i o n a l a o p l a y s r o l e l e a - d i n g t o t h e f u l l damping of t h e e l e c t r i c f i e l d .

The i n f o r m a t i o n about numerical ex- p e r i m e n t s w i t h d i f f e r e n t phase v e l o c i t i e s i s g i v e n i n t h e t a b l e 2

n/ 2-1 2-2 2-3 2-4 2-5 2-6

w i t h d i f f e r e n t W o o r Vph shows t h a t t h e main p r o c e s s e s a r e t h e same. The c h a r a c t e r of phenomena depends on t h e p a r a m e t e r @

-

i n t h e c a s e s of d i f f e r e n t Wo and Vph but e q u a l d t h e main f e a t u r e s of t h e i n s t a - b i l i t y and damping ( i n c l u d i n g t h e c u r v e s

& ( t ) ⁾a r e s i m i l a r . The i n c r e a s e of wo

parameter d l e a d s t o t h e e a r l i e r damping of t h e e l e c t r i c f i e l d (Fig. 1 ) . So t h e main p r o c e s s e s of t h e Langmuir wave insta- b i l i t y and damping a r e modulational i n s t a - b i l i t y , conversion on d e n s i t y p e r t u r b a

-

t i o n s and t r a p p i n g of e l e c t r o n s by t h e i n i t i a l wave o r i t s modes. The e l e c t r o n n o n l i n e a r i t y d o e s t n t i n f l u e n c e t h e i n s t a -

b i l i t y development ( i n agreement w i t h t h e o r e t i c a l r e s u l t s /2/) u n t i l t h e t r a p - ping by t h e i n i t i a l wave becomes e s s e n t i - a l . The e l e c t r o n n o n l i n e a r i t y p l a y s r o l e f o r damping because i t l e a d s t o t h e per- t u r b e d e l e c t r o n v e l o c i t y i n c r e a s e . 1. I?.S.Buchelnikova, E.P.Matochkin "Phe- nomena i n I o n i z e d Gases", v. 11, p.831, 1977, B e r l i n . "Waves and I n s t a b i l i t i e s i n Plasmas", p. 71, 1977, P a l a i s e a u .

2. E,A,Kuznetsov Sov. J. Plasma Phys. 2,

327, 1976.

The comparison of n u m e r i c a l experiments

Fig. 1.