WKB MODEL OF THE COLLAPSE OF LANGMUIR WAVES

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WKB MODEL OF THE COLLAPSE OF LANGMUIR WAVES

B. Breizman

To cite this version:

B. Breizman. WKB MODEL OF THE COLLAPSE OF LANGMUIR WAVES. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-563-C7-564. �10.1051/jphyscol:19797272�. �jpa-00219259�

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JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuilZet 1979, page C7- 563

WKB M O D U Of THE COLLAPSE OF LANGMUlR WAVES

B.N. Breizman.

I n s t i t u t e o f NueZear Physics, Novosibirsk, U.S.S.R.

The h y p o t h e s i s on t h e c o l l a p s e of Langmuir waves h a s been suggested by V,E,Zakharov i n 1972 /I/. Soon a f t e r w a r d s , a number of a t t e m p t s have been made t o f o r m u l a t e t h e c o n d i t i o n s under which t h e c o l l a p s e t a k e s p l a c e and t o d e s c r i b e qua- l i t a t i v e l y i t s dynamics /2-5/. The b a s i c e q u a t i o n s i n papers /1-5/ were t h e equa- t i o n s f o r t h e e l e c t r i c f i e l d amplitude and f o r t h e low frequency p e r t u r b a t i o n of plasma d e n s i t y . With such an approach t h e l e n g t h of Langmuir wave is u s u a l l y asswn- ed t.o be comparable w i t h t h e s i z e of a po- t e n t i a l w e l l wherein t h e waves a r e t r a p p - ed, i.e, i t is implied t h a t t h e number of eigenmodes i n a w e l l is small. This s i t u a - t i o n i s v e r y complicated f o r a n a l y t i c a l s t u d y s i n c e one h a s t o f i n d t h e e x a c t "wave f u n c t i o n s t t of Langmuir o s c i l i t a t i o n s . I n o r d e r t o avoid t h i s d i f f i c u l t y , we s h a l l c o n s i d e r below a w e l l which i s wide enough and c o n t a i n s many eigenmodes s o t h a t W E approximation i s v a l i d and t h e o s c i l l a - t i o n s can be d e s c r i b e d by L i o u v i l l e equa- t i o n /6,7/:

Here, t h e i n v e r s e plasma frequency i s cho- s e n , a s a u n i t of time, t h e l e n g t h and t h e wavevector a r e normalized t o r, ^andr,-'

,

r e s p e c t i v e l y , fl is t h e p e r t u r b a t i o n of plasma d e n s i t y normalized t o t h e u n p e r t u r - bed v a l u e ,

I n a d d i t i o n t o eq.(l ), we make use of l i n e a r i e e d e q u a t i o n s of hydrodynamics, which i n c l u d e t h e p r e s s u r e of plasmons:

The i o n v e l o c i t y 7 ^ismeasured h e r e 2n t h e u n i t s of e l e c t r o n thermal v e l o c f t y , and t h e i o n mass

-

i n t h e u n i t s of e l e c t - r o n mass.

From t h e p h y s i c a l p o i n t of view t h e c o n s i d e r a t i o n o f t h e c o l l a p s e problem i n t h e framework of eqs. ( I ) - ( 3 ) seems t o be q u i t e n a t u r a l s i n c e t h e q u a l i t a t i v e a r - guments f o r t h e two- o r three-dimensional c o l l a p s e and a g a i n s t t h e one-dimensional one /8/ a r e completely a p p l i c a b l e t o t h e V@33 c a s e , It should be emphasized t h a t t h e s e arguments a r e n o t a s s o c i a t e d with whatever assumptions w i t h r e s p e c t t o pha- s e c o r r e l a t i o n . T h e r e f o r e , they a r e v a l i d , i n p a r t i c u l a r , i n t h e c a s e of random pha- s e s .

The system (1 ) - ( 3 ) h a s t h r e e integrals:

t h e number of plaslnons

,

t o t a l momentum and energy a r e conserved. T h i s system g i v e s a l s o t h e f o l l o w i n g r e l a t i o n s h i p

( v i r i a l theorem):

We s h a l l apply i t t o t h e slow ( s u b s o n i c ) c o l l a p s e f o r which t h e d e n s i t y p e r t u r b a - t i o n "f ollowsT1 th e p r e s s u r e of plasmons, i.e. n -

S

/j/d< ^and?-- 0

.

Yith t h i s s i m p l i f i c a t i o n s we g e t

The right-hand s i d e h e r e is c o n s t a n t be- cause o f t h e c o n s e r v a t i o n o f energy. In- e q u a l i t y ( 5 ) shows t h a t any d i s t i b u t i o n of plasmons w i t h n e g a t i v e v a l u e of Q be- comes s i n g u l a r w i t h i n a f i n i t e time. This r e s u l t i s s i m i l a r t o Talanovts theorem f o r t h e s e l f - f o c u s i n g /9/, For t h e spherical-symmetric problem i t has been prov-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797272

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ed previously without WRB approximation /I/. I n WKB case i t remains t r u e even i n t h e absence of symmetry,

Let u s now consider t h e supersonic c o l l a p s e , It is p o s s i b l e t o g e t t h e ana- l y t i c a l s o l u t i o n of t h e system (1)-(3) f o r t h i s case. We s h a l l demonstrate i t by t h e example of two-dimensional problem with axisyrnmetric d i s t i b u t i o n of plasma d e n s i t y *

.

In a d d i t i o n , we assume t h a t t h e c h a r a c t e r i s t i c time of c o l l a p s e is long a s compared t o t h e period of plasmon motion i n t h e p o t e n t i a l well.

Then due t o f a s t phase mixing; t h e spect-

rum of plasmons t u r n s out -to be only de- pendent on t h e angular momentum

111

= K,+,r

and t h e r a d i a l a d i a b a t i c i n v a r i a n t

( E ^{i s}t h e energy of plasmon). A fonn of t h e f u n c t i o n ~ ( ? n ;

I

) is determined by i n i t i a l conditions.

It w i l l be seen from what follows t h a t f o r some i n i t i a l s t a t e s t h e behavior of d e n s i t y n (I-;+) is s e l f - s i m i l a r :

This s e l f - s i m i l a r i t y e x i s t s not only i n IImB problem of collapse but a l s o i n t h e exact one ( s e e , e.g., /4/). It is impor- t a n t t o note t h a t i f t h e r e l a t i o n (6) holds, then t h e boundary which s e p a r a t e s t h e trapped plasmons from t h e untrapped ones on t h e plane of v a r i a b l e s

l"fi

and

I ,

does not depend on time. Therefore, one can choose t h e f u n c t i o n N('?n; I ) being equal t o some constant /lJo f o r trapped plasmons and t o zero f o r untrapped ones.

It is now easy t o c a l c u l a t e t h e p r e s s u r e of plasmons

and t o d e r i v e t h e following equation f o r

f

( 9 )

from t h e system (2)-(3):

The second s o l u t i o n of eq.(7) is omitted s i n c e i t g i v e s t h e divergent space integ- r a l of 17

In t h e case of high enough i n t e n s i t y of Langmuir waves ( N, > 3/T ) formula ( 8 ) d e s c r i b e s t h e c o l l a p s e of i n i t i a l d i s t r i - bution. It i s easy t o s e e t h a t t h e a p p l i - c a b i l i t y c o n d i t i o n s f o r 1KKB approximation and f o r t h e conservation of a d i a b a t i c in- v a r i a n t do hold during t h e whole process of c o l l a p s e provided they a r e f u l f i l l e d a t t h e i n i t i a l moment.

References:

/ I / V.E.Zakharov. Zh. Eksp. Teor. F i t . 6 2 , 1745, 1972.

/2/ L.Pd.Degtyarev, V.E.Zakharov. P r e p r i n t N0106 I n s t , of Appl. Math., Moscow,

1974.

/3/ A D A , Galeev, P, Z. Sagdeev

,

^{Yu. S}

.

^Sigov

^,

V.D.Shapiro, V.I.Shevchenko. Piz.

Plasmy 1, 10, 1975.

/4/ V,E.Zakharov, A.P.Mastryukov, V.S.Sy- nakh. Fiz. Plasmy -l, 614, 1975.

/5/ L.M.Degtyarev, V.E.Zakharov, L.I.Ru- dakov. Zh. Eksp. Teor. Piz. 68, 115, 1975.

/6/ A.A.Vedenov, L.I.Rudakov. Dokl. Akad.

Nauk SSSR 159, 767, 1963.

/7/ A.A.Vedenov, A.V.Gordeev, L.I.Rudakov.

Plasma Phys. 2, 719, 1967.

/8/ A.A,Galeev, P.Z,Sagdeev, V,D.Shapiro, V.I.Shevchenko. Zh. Eksp. Teor. P i s ,

a,

1352, 1977*

/9/ S .N.Vlasov, V.A. Petrishchev, V. I.Tala- nov, Izv. Vutov Radiofiz.

s,

^1353,

1971

* S i m i l a r s o l u t i o n s can be found f o r t h e three-dimensional spherical-symmetric ca- Be.