MODULATION INSTABILITY AND PLASMA ELECTRODYNAMIC CHARACTERISTICS

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MODULATION INSTABILITY AND PLASMA ELECTRODYNAMIC CHARACTERISTICS

A. Litvak, V. Mironov, A. Feigin

To cite this version:

A. Litvak, V. Mironov, A. Feigin. MODULATION INSTABILITY AND PLASMA ELECTRODY- NAMIC CHARACTERISTICS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-635-C7-636.

�10.1051/jphyscol:19797308�. �jpa-00219298�

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CoZZoque C7, suppZIrnent au n07, Tome 40, JuilZet 1979, page C7- 635

MODULATION INSTABILITY AND PLASMA ELECTRODYNAMIC CHARACTERlSTlCS

A.G. Litvak, V.A. Mironov and A.M. Feigin.

Applied Physics I n s t i t u t e , Academy of Sciences of the U.S.S. R., Gorky/U.S. S. R.

1, It is well known that the featmes of interaction between strong electromag- netic waves and a high density collieion- less plasma is associated with modulatio- nal instability of Langmuir oscillations and further formation of Langmuir soli- tons. In moat of papers the dynamics of plasma modulation is inveatigated in or- der to determine the anoroaloue absorpti- on characterized b the effective colli- sional frequency

f .

The other sepuen- ce is a change of t8ifrefractive index

(or the real part of the permittivity) of en electromagnetic wave produced by such a small-scale structure. meere effects give rise to an essential change of the field distribution a d apperently alter the energy absorption in plasma.

'Phe purpose of the present report is to consider the electromergnetic wave penelm- tion into a eupercritial plagnns for the case of the nonlinear changing of the refractive index,

The paper deals with a stationary selfaction of a T?EIU-wave in the initidly homogeneous layer of a eupercritial plas-

~ i a . The field distribution in a layer and

the dependence of the transmission c o r n - cient on the amplitude of an incident wa- ve have been found.

2. Let us consider the normal incidence of a plane monochromatic wave upon a plama layer with permittivity bo(Z)

.

It has frequency O close to plasma frequency

o

In the linear approximation a wave is gflected from the region

6, (Z)

< 0 .

In the nonlinear case the

field can penetrate deep into the super- critical. p

ya_.a.

Then-the complete elect- ric field

E (x,.z,~)

xo is a super- position of the fields of an electromag- netic (vortex) and plasma (potent~al) waves. They have appreciably different space scales. 'Phie permite to make use of the averaging approach and solve the prob- lem by two stages. Piret we shall seek the stationary distribution of plasma as- vee of the croas eection emconst excited

(in the quasi-static approxhation) by the given electric displacement

2 azE

D=-k

- = C O ~ S ~ . .This distribution is

0 at

described by theequation for a slowly va- rying amplitude of the electric field

Then, averaging (1) over the potential wave scale we have the equation which dm- cribes eelf-action of the mean (small-

-space average) fieldcbapmp wave in a medium characterized by the efficacious

( 2 )

~ P = C - M ~ S ~

^A

.

^{( 3 )}

The-line maans the epecial averagihg of stationary field distribution in the crag- csection oconst. To find an exact value

of & eff

= d /c ,

one should eolve the non-

stationary problem. We should turn here to numerical calculations, Though the neces-

numerical that the re-

lopment is formation of a chain of soli- tons. We may naturally expect that in the stationary caee the distancc between eoli- tons coincides with a scale of the moat rapidly growing perturbation in the linea- rised problem, So, in

b]

the stratifica- tion scale pas assumed to be the same all over the instability region and is deter- mined by the maximum value of the field at the initial moment of time (teO). Estima- tions of the transparence parameters (pe- netration time and the threshold field va-

lue) obtained in this case describe rather well the data [4,5] and eome

other approxima- in thie p a p the local relation between the stratifi- cation scale and the electric displacement is assumed. Thie approach in contraat to that used in

n]

permits to solve a self- consistent problem of penetration of an electromagnetic wave into a high density plasma in the general case taking account of reflection. The efficacious permittivi- ty in such a model is defined as

tctf(dadmd ⁼ Vrd

⁽⁴

At d

<

dmod (d~lodol&1~'~/fi i the threshold value of displacement for modnlational in- stability development) stratification d a s not occar and Eeff is equal to the unper- turbed linear value &ef f (d d dmod )= 6 It follows from (4) that in the given ap- proxiatation the transparence of the initi-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797308

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ally homogeneous supercritical plasm ta- kes place when

For d C d:$h the stratification of plae ma is inaxgnificant for its transpasence,

and the plasm remains supercritical.

With the following decrease of d

grows in modulus and achieves the unper- turbed value at d=dmod.

3.

Determination of Eeff reduces the problem at the second stage to a usual electrodynamic one. k t us coneider the simplest problem of euch a type:viz. the normal incidence of a plane wave from va- cuum to the initially homogeneous layer of a supercritical plasma with a thick- ness of 2e,-e<z<e, Inside the layer, the field amplitude and phase distributi- on is described by a ~lystem of equations

(2), (3),(4). Here the parameter C cor- responds to the power flux of a wave pas- sing through the layer. 'Phis system should be added with the boundary conditions.

That- is a continuity of thtnean field

a

and its derivative

aa/a&

^{both at}

the boundaries of a layer 2 s 2

e

and at tho- se point8 within the layer where the me- an field is compared to the threshold va- lue for the modulational instability Zmod=dmod/(&l

.

^Aisolution of a system of equations (2)-(4) was investigated qualitatively by the example of integral curve ba&vi_aur on a phase plane of the system (a,

aa /aLa

and determine the transmission fac or T. Fi e l e h o m the dependence of T on

eo= ^a.

^fmpb(

^a

^{ie the}

amplitude of an incident wa e) for a lay- er of le th L35bo (

ho

is wavelength in with the unperturbed permitti- vity

Eo

ⁿ^-0.1. The dependence is of hysteric form, a complete transparency

(Pol) be'ng realized for the certain va- lues of b o

.

^{In Fig.1} these resoneb sta- tes are characterized by the number defi- ned as a ratio of the layer length to the period of a nonlinear wave. Depending on the power flux C, there are two tyges of eolut'ions of Eqs.(2)-(4). At C C C =8,{

s layer ie divided into sublayers with supercritical (unperturbed) an$ t r a n s p d

(stratified) plasmas, At C

>

^C ^{the layer}

becomes transparent (stratified) all over, Perssing from o3e type of solution to the other at C

=

C explains the existence of two resonant etates with the same number (see Fig.1). Note that the described de- pendence T(bo) in the resonant region8 is smoother than in the case of cubic nonli- nearity [7-91 B u s , even the case of purely ohmic

.

loerses, absorption of energy of the inci- dent radiation increases appreciably due to the essential lengthening of the regi- on occupied by the electromagnetic field.

References

1, A.G.Litvak, V,A.lironov, G.M.Etrsiman, ZhETF Letters,

22,

368-371 (1975) 2. A.G.Litvak, V.Yu,Trakhtengerts, T.X.

Fedoseeva, G,X.likaiman, ZhETF Letters, 20, 544 (1974)

3. B.A.Allterkop,

-

A.S.Volokitin, V.P.Ta- rakanov, ZhTF Letters, f, Ho 11,

(1975)

4. Yu,Ya.Brodsky, B.G,Eremin, A.G.Litvak, Yu,A.Sskhonchik, ZhETF Letters,

2,

136 (1971)

5. G.IYI,Batanov, V.A,Silin, ZhEW Letters,

x,

445 (1971)

6. Yu.Ya.Brodaky, V.L.Goll ternan, V.A.Pi-

ronov S.I.Xechuev, ZhETF Letters,

a,

1636 t1978)

9. L.H.Gorbunov, K.Zauer, Fizika Plasmy,

2,

Ho 5 (1977)

MODULATION INSTABILITY AND PLASMA ELECTRODYNAMIC CHARACTERISTICS

HAL Id: jpa-00219298

https://hal.archives-ouvertes.fr/jpa-00219298

MODULATION INSTABILITY AND PLASMA ELECTRODYNAMIC CHARACTERISTICS

A. Litvak, V. Mironov, A. Feigin

To cite this version:

f .

.

o

< 0 .

ya_.a.

E (x,.z,~)

D=-k

~ P = C - M ~ S ~

.

= d /c ,

b]

n]

tctf(dadmd = Vrd

<

3.

a

aa/a&

e

.

aa /aLa

eo= a.

a

ho

Eo

.

>

=

.

22,

-

2,

x,

a,

2,

tctf(dadmd ⁼ Vrd

eo= ^a.

^a