MODULATIONAL INSTABILITY OF LANGMUIR OSCILLATIONS IN THE FIELD OF AN ELECTROMAGNETIC WAVE

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MODULATIONAL INSTABILITY OF LANGMUIR OSCILLATIONS IN THE FIELD OF AN

ELECTROMAGNETIC WAVE

I. Kol’Chugina, A. Litvak, T. Fedoseeva, G.M. Fraiman

To cite this version:

I. Kol’Chugina, A. Litvak, T. Fedoseeva, G.M. Fraiman. MODULATIONAL INSTABILITY OF

LANGMUIR OSCILLATIONS IN THE FIELD OF AN ELECTROMAGNETIC WAVE. Journal de

Physique Colloques, 1979, 40 (C7), pp.C7-629-C7-630. �10.1051/jphyscol:19797305�. �jpa-00219295�

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CoZZoque C7, suppZdment a u n 0 7 , Tome 40, J u i Z Z e t 1979, page C7- 629

MODLlLATIONAL INSTABILITY OF LANGMUIR OSCILLATIONS IN THE FIELD ff AN ELECTROMAGNETIC WAVE

I.A. Kollchugina, A.G. Litvak, T.N. Fedoseeva and G.M. Fraiman.

I n s t i t u t e o f A p p l i e d P h y s i c s U. S. S . R., Academy o f S c i e n c e s Gorky, U. S . S . R.

The paper in devoted to the analysis of modulational instability of Langmuir oscillations exited near plasma resonan- ce region by the field of an electromag- netic wave. A detailed substantiation of qugsi-static approximation equations used for numerical study of a Langmuir turbulen- ce is presented.

Using these equations the one-dimen- sional Langmuir turbulence in the elect- ric field with the given solenoidal com- ponent is numerically calculated.

1. To describe the interaction between an electromagnetic wave and plasma oscil- lations in an nonisothermal plaama(T >>T

)

near plasma resonance IW -up(<< o we aka11 use a set of equations (averagd$ over the field p e ~ i o d ) fsr slow pplitude of the electric field E E & ~ + ) e " and small slect- ron density perturbution n = 6 n / ~ ,

^,

Here E s =

(

- ^u;/u2-' is the linear plasma permitt~vity, No[s thpunperturbed eleb tron density, U8 - ^{s the ion}

sound velocity,

E;:~'yj

XM )& ie the cha- racteristic field of nonline& effects.

In order the equations to be aimpli- field we shall uee the existence of two easentialy different spatial scales in the problem: the electromagnetic wave lenght Xt and plasma oscillation ecaleXp

X, >> A,. (3

Since small-scale plasma oscillatians are almost potential and the electromagne- tic wave is aLmost of solenoidal charac- ter, it is possible to represent the full field as a sum of solenoidal and potential components:

-C

E =G+G,div5=0, rot 5 ^-0. ^{( 4 )}

S bstituting (4) &

^we

obtain the following set bf equa- in Eq.(l) for and tions

:

It is seen from (5) that solenoidal and potential oscillations are coupled due to inhomogeneous density perturbations.

Therefore a solenoidal part of the fi- eld should also have a small-scale com- ponent. The amplitude of this component

Esy is not difficult to be evaluated

using

( 5 )

f i = ( T ~ p ) L n < < 4

W

is the basic parameter which permits to use the so-called qua- si-static approximation. According to this approximation the Langmuir oscil- lations distribution may be found. if the solenoidal (electromagnetic) field com- ponent il~grersumed to be known. The equa- tion for Es may be derived by subse- quent averaging of the obtained eolution over small scale oscillatione.

If it is assumed that just the one- -dimensional plasma oscillations are ex- cited the problem is aimplifed moat es- sentially. Under thia assumption for plasma oscillatione

we

have an equation

2i aE, 2 d ~ ^c2

⁺

--

^(&)

K+kn)E, +3r

d o + ?=

--,(rot a

^h

rot E).(?)

where the displacement r= ³

C'

^rotrot

is given.

^% 0

Averaging of the obtained solution over a small scale enables ue to intro- duce the effective permittivity (see [1] ,

PI

which defines the mean solenoidal field by the equation

In a number of papers (see, for example, [3-51

)

the problem of exitation of one- -dimensional Langmuir oscillations was solved within the given full mean elect- ric field (rather than its solenoidal component) approximation. In our opini- on such a formulation is more adequate to the case of exicitation of Langmuir oscillatione due to stream instabiliti- es but not to the roblem of Langmuir

turbulence in the Bield of an electro- magnetic wave,

2. Let us analyae in brief the reqults of numerical study of an one-dimensional

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797305

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Langmuir turbulence

i n

the em wave field.

The corresponding equation may be written in dimensionless form as

. - -

wheri - ³ ⁼ ^const.

We use periodic boundary conditions and initial conditions under which the modu- lational instability should be realized.

Humsrical studies show th8t at the initi- al sta e the uniform electric field E,

(Pigelf increase. primarily up to the va- lue exce ding the steady-state one

E o > D& In thia case the system goes out of resonance with the pump due to the nonlinear frequency shift - ^E, . ^{The next}

stage of thia process without the diesi- pation is similar to the caae of modula- tional instability of free Langmuir oscil- lations [6 , ⁷¹

^:

the saall scale compo- nent of the electric field increases and a soliton structure forms without the change in the full energy of the electric field (Pig.2). As in [6, 71 the electric field distribution is repreeented as a succession of equidistant eolitons with essentially different amplitudes, the dis- tanse between solitons corresponding to the optimal scale of modulational ineta- bility. Despite the soliton amplitude pul- sations the full energy of plasma oscilla- tions is almoat constant. The time-avera- ged spectra of Lanmuir turbulence rea- l i z h g in our calculations appear to be exponential (~ig.3

)

* The dependence of the relaxation time versus the diaplace- ment D shown in Fig.4 is well approximated by

2-'

Note that the above considered pro- .

cess of modulational inetability differs from that resulted from calculations with the given mean electric field 13-51 where the stationary state did not exist if the dissipation was neglected: the syertem ca- me to the quaei-stationary etate only when the dissipation in small ecalea is consi-

derable. References

1. A.G.Litvak, **V.ABBironov, G.M.Praiman, ZhE'PF Letters, 22, 368 (1975) 2 . A.G.Litvak, A.M.Feigin, V.AMironov,**

Report of this Conference

3. G.J.Morales, Y.C.Lee, R.B.White, Phys.

**RevoLett*, 32, 457 (1974)**

4 . A.A.Galeev, R.Z.Sagdeev, V.D.Sha iro, V.I.Shevchenko, ZhETF, u, 1352 PI9771 5. B.A.Al'terko A.S,Volokitin, V,P,Ta-

rakanov, ~iz!ia Plasmy, 2, 59 (1977) 6. A.G.Litvak, V.Yu.Trakhtengerta, T.I.

Fedoaeeva, G.I.Fraiman,

ZhETF