MODULATIONAL INSTABILITY AND EVOLUTION OF NON UNIFORM LANGMUIR FIELDS IN PLASMAS

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MODULATIONAL INSTABILITY AND EVOLUTION OF NON UNIFORM LANGMUIR FIELDS IN

PLASMAS

T. Davydova, K. Shamrai

To cite this version:

T. Davydova, K. Shamrai. MODULATIONAL INSTABILITY AND EVOLUTION OF NON UNI- FORM LANGMUIR FIELDS IN PLASMAS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7- 561-C7-562. �10.1051/jphyscol:19797271�. �jpa-00219258�

JOURNAL DE PHYSIQUE CoZZoque C7, suppl6ment au n07, Tome 40, JuiZZet 1979, page C7- 561

MOOUATIONAI, INSTABILITY AND EVOLUTION OF N O N W O R M LANGMUIR FIELDS IN PLASMAS

T.A. Davydova and K.P. Shamrai.

I n s t i t u t e for NucZear Research, Kiev, U. S . S. R.

It has recently been observed in a set of experiments that the interaction of h.f.

fields with plaamse results in a formation of cavitons, Lee. density cavities with trapped field. To interpret these experi- ments one adopts as a rule a mbdulatioaal inotability of s uniform field, which is supposed to produce cavitons in the non- linear stage. But in a madority of experi- menta the pump fields are essentially non- uniform through inhomogeneity and finite size of a plasm, non-uniformity of exter- nal aources of field excitation ets. There- fore it seems necessary to develop a theory of modulartional instability of non-uniform f ields, which diff ers from that of unif o m field. Moreover, to investigate a possibi- lity of caviton formation and to determine a characteristic time of a process one must examine initial stage of plasma evolution under the action of non-uniform pump field.

lEhe evolution of some non-unif o m fields was a subject of a set of numerical simulations. !The present report deals with an analytic investigation of I-dimensional initial evolution of intensive non-uniform Langmuir field in two typical cases:

(i) localised pump field (isolated peak) and (ii) standing wave.

Let in the initial moment t19 Lanpuir

pump field lo= &o(x)cos(~ot) is switched on

"instantaneously" in a plasma. A pondero- motive force drives in plasma the coupled density and field perturbations. W e shall examine a temporal evolution of this pro- cess at a linear stage, i.e.neglecting the coupling of perturbations and their influ- ence on a pump. We use linearized eystem of equation6 by Zakharov / 1 /.

where 3 = 4 ~ e ~ n ~ / r n , P v $ = ~ ~ / m , r2.r2/a?, D T P -:0

T , &n is a density perturbation and

&

(x,t) is a real amplitude of field per-

turbation slightly depending on time.

'Eke abaolute instability exists if there are spatially bounded solutions of uniform system (1),(2), whioh depend on time as exp(-ia t) and

I ~ R

^>O. m e thresh- old intensity for a standing pump wave ( &o=&m~sin(kox) ) is determined f m m the

equations - ⁹

2 W ~ Q X &,ax

p n d n = i + z ~ ,

w"axP~-

at

a"

where

p,( T I ,

3,( q ) (n=2,3,.

.

^{) 81.e the}

ei-genvalues of Mathieu equation for func- tions sen and

ten.

The threshold is minimal

2 2

for eigenmodes se2(Q-6) and cep(ae9).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797271

Por a l o c a l i z e d pump f i e l d (&,+O as x.+-+m) t r a c t i o n and deepening with growth r a t e t h e i n s t a b i l i t y threshold i s of o d e r a2z1

ri& ^{& -}

^{3 exP m o r t}

2 no 4vc/mar ( C O S ~ ^-1 (csth))

/

/.

^Here

a 2 = ~ m x ~ 2 / r &

where A i s a s c a l e

' _a

_x2

of f i e l d l o c a l i z a t i o n region. There i s a ex^

(-

2

a

^fonh

y) w "Lcosh.Jd " ) .

f i n i t e d i s c r e t e spectrum of unstable modes The f i e l d amplitude i n t h e c e n t r e of cavi- both i n a standing wave f i e l d and i n a lo- t o n i n c r e a s e s with t h e same r a t e . A t l i n e a r c a l i e e d one. A number of them i s determin- s t a g e t h e d e n s i t y cavitiera s c a l e d o m up t o ed from i n e q u a l i t y 2n+lLCl (n i s a whole t h e width of t h e l o c a l i z a t i o n of t h e ground number). The i n s t a b i l i t y growth r a t e peaks mode, A / a

.

'Phe d u r a t i o n of t h e contrac- f o r a ground mode and decreases monotoni- t i o n is determined by t h e most i n e r t i a l c a l l y a s n increases. process, i.e. by sound i r r a d i a t i o n out of

To i n v e s t i g a t e a temporal evolution of caviton, t " A/c so

f i e l d and d e n s i t y perturbations we complete I f t h e i n s t a b i l i t y threshold i s weakly system (1),(2) by i n i t i a l conditions exceeded t h e r e a r e few unstable modes and

&(too)= 8nt(to0)=

&

( t = 0 ) = Gt(t=O)= 0. d e n s i t y p r o f i l e i s cusped strongly as t - Then i t i s e a s r t o o b t a i n t h e s o l u t i o n f o r

T&.

For a l o c a l i z e d f i e l d redundant den- small time

t<lmsx.

-1 Density perturbation s i t y i s squeezied out of t h e pump peak doma-

8 n / n o ~ ( 3 / 4 ) o ~ W n t 2 i s negative n e a r t h e i n and i s i r r a d i a t e d t o i n f i n i t y a s sound pump peaks and decreases i n time. A pon- compression waves with v e l o c i t y cS

deromotive influence of a pump f i e l d on a

a ,

³

L w , - , t

+

plasma g i v e s r i s e t o d e n s i t y c a v i t i e s .

T { Z [ (

S ! ~ ( x ~ c s ~ ~ - w ( x J } Their s c a l e i s i n i t i a l l y of order A ( x/ko ( [ x \ > A

1.

?or standing waye i t accumulates f o r a standing wave) and does not depend n e a r nodal points. Linear approximation on pump power. A t t h e same time t h e abso- breaks a t t - _,&1na

. _>> , -

¹

l u t e value of f i e l d perturbation grows a s I n conclusion, a l i n e a r s t a g e of t h e

\&I -

^t4. evolution of non-uniform Langmuir pump

To i n v e e t i g a t e evolution f o r t

> rm=

f i e l d r e e u l t s i n a formation near pmp we suppose t h e threshold t o be strongly peakrn of c o n t r a c t i n g and deepening d e n s i t y

exceeded, >>,W F$/A'> m/Y ( f o r a ~ t a n d - c a v i t i e s with trapped f i e l d ( c a r i t o w ) . i n g wave here and below one must s u b e t i t u -

Ref e r e m e s

.

t e h'ior ko). I n t h a t c a s e

~ m a x ~ q f ~ m a x m / ~ '

/2/ and d e n s i t y dynamics i s ' described by 1. Zakharov V.E. (1972) Zh.eksp. teor.Piz.

equation

/'/ -

62, 1745 (Sov.Phys.JFZP

22,

908).

2. Davydova T.A.,Shamrai K.P.(1977) P i s i k a Plazmy (Sov.J.Pla8ma Phys.)

2,

591.

with i n i t i a l conditions d;l(tr0)=8nt(tsO)=O.

3. Davydovcs T.A., Sham~ari K.P. ( 1978)PreprSnt It turned out t h a t for t

> r,,

t h e exci- 1nst.Theor.Pbys. 77-140P, Kiev ( i n Rue.).

t a t i o n of a modulational i n s t a b i l i t y near pump peake r e s u l t s i n d e n s i t y c a v i t i e s con-