DISPERSION CHARACTERISTICS OF PLASMA WAVE-PACKETS

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DISPERSION CHARACTERISTICS OF PLASMA WAVE-PACKETS

R. Vidmar, F. Crawford

To cite this version:

R. Vidmar, F. Crawford. DISPERSION CHARACTERISTICS OF PLASMA WAVE-PACKETS.

Journal de Physique Colloques, 1979, 40 (C7), pp.C7-559-C7-560. �10.1051/jphyscol:19797270�. �jpa- 00219257�

JOURNAL DE PHYSIQUE CoZZoque C7, supptlment au n07, Tome 4 0 , J u i l l e t 1979, page C7- 559

DISPERSW CHARACTERISTICS OF PLASMA WAVE-PACKETS

R.J. Vidmar and F.W. Crawford.

I n s t i t u t e for Plasma Research, Stanford University, Stanford, California 94305 U.S.A.

We s h a l l c o n s i d e r propagation of a wave-packet through an i n f i n i t e homogeneous plasma which may b e l o s s l e s s , a b s o r p t i v e o r u n s t a b l e , s o t h a t t h e ener- gy of t h e wave-packet may b e conserved, d i s s i p a t e d , o r i n c r e a s e d a s i t propagates.

I n 1914, B r i l l o u i n and Sommerfeld [l] d e t e r - mined t h e evolving shape of a wave-packet, and p r e - d i c t e d t h e occurrence of p r e c u r s o r s t h a t . o u t r u n t h e body of t h e packet ( s e e Fig. 1). The s a d d l e - p o i n t methods t h e y introduced have s i n c e been r e f i n e d and widely used [2]. Concepts such a s phase v e l o c i t y , group v e l o c i t y and energy v e l o c i t y emerge, and a r e u s u a l l y understood by r e f e r e n c e t o d i s p e r s i o n ( ~ r i l l o u i n ) diagrams showing t h e frequency v a r i a - t i o n (w, r e a l ) with wavenumber (k, r e a l ) . D i f f i c u l - t i e s a r i s e f o r a b s o r p t i v e o r u n s t a b l e media, f o r which group v e l o c i t y may b e complex o r i n f i n i t e . We suggest t h a t i n such c a s e s an o b s e r v a t i o n a l d i s - p e r s i o n diagram i s u s e f u l , e f f e c t i v e l y d e s c r i b i n g t h e r e s u l t s of complex w,k measurements on wave- packets. We s h a l l u s e a c o n s i s t e n t d e f i n i t i o n of group v e l o c i t y t o e s t a b l i s h t h e form of t h i s d i s - p e r s i o n r e l a t i o n , and i t s dependence on t h e s o u r c e e x c i t i n g t h e wave-packets.

We c o n s i d e r two sources, b o t h d e l t a - f u n c t i o n s i n t h e d i r e c t i o n of propagation, z : a d e l t a - f u n c t i o n s o u r c e in time, ~ & ( t , z ) = 6 ( 2 ) 6 ( t ) > and t h e switch-on of a continuous wave, S c ( t , z ) = b ( z ) ~ ( t ) s i n ( m O t ) . Here, 8 i s a d e l t a - f u n c t i o n , H i s a Heaviside step-function, and m0 i s a c o n s t a n t frequency. These i d e a l i z e d s o u r c e s d i f f e r by t h e presence (5,) o r absence ( S ) of a p o l e i n t h e t r a n s f o r m of S. They a r e consequently r e p r e s e n t a - A t i v e of s o u r c e s which a r e e i t h e r switched on and maintained i n d e f i n i t e l y , o r a r e p u l s e d ,

THEORY

Although we a r e i n t e r e s t e d i n plasmas, t h e wave e q u a t i o n may be l e f t i n g e n e r a l form t o accom- modate any medium and v a r i a b l e , , t h a t e x h i b i t s d i s p e r s i v e propagation. For l i n e a r propagation i n one dimension we have [3],

where t h e c o e f f i c i e n t s p , a . , d e s c r i b e t h e p r o p e r t i e s of t h e medium i t s l d c i t a t i o n by t h e source, S.

We d e f i n e a Laplace t r a n s f o r m i n time, and a F o u r i e r t r a n s f o r m i n space,

m-iu m

By assuming an i n i t i a l l y q u i e s c e n t medium, and r e s t r i c t i n g ai t o c o n s t a n t c o e f f i c i e n t s , (1) may b e transformed t o

D(W, k ) ~ ( o , k ) = A(W, k) ~ ( w k) , ( 3 )

where D(m,k) i s t h e d i s p e r s i o n r e l a t i o n

To i n v e r t t(m, k), t h e k - i n t e g r a t i o n i s evalu- a t e d by u s e of t h e r e s i d u e theorem, t a k i n g account of t h e p o l e s d e f i n e d by ~ ( w , k ) = 0 . P o l e s i n t h e upper h a l f - p l a n e [ s e e Fig. 2 ( a ) ] a r e included i n t h e c o n t o u r i f t h e y o r i g i n a t e i n t h e lower h a l f - p l a n e f o r v a l u e s of m c o n s i s t e n t with t h e m - i n t e g r a t i o n along a Bromwich contour m = tor - ^{i m} . ^For

f u r t h e r discussion, s e e 14) .

The remaining i n v e r s e transform i s of t h e form a - i a

where k(m) i s determined from D ( ~ , k) = 0 , ^{and F}

i s t h e sum of k - i n t e g r a l r e s i d u e s m u l t i p l i e d by t h e m-transform of t h e source, which may c o n t a i n a pole.

Branch-points of f u n a t i o n F must b e avoided i n t h e

3.Ml-

-

( a ) k - i n t e g r a t i o n ( z I > 0) ( b ) m - i n t e g r a t i o n ( t > 0 ) FIG. 2 . INTEGRATION COWOURS

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797270

k r

( b ) D e l t a - f u n c t i o n modes >

I

( c ) Group v e l o c i t y FIG. 3. BEAM-PLASMA DISPERSION DIAGRAMS AND GROUP VELOCITY

m-plane [ s e e Fig. 2 ( b ) ] . They r e s u l t from f a c t o r s between t h e s e two r e s u l t s i s t h a t wi=O f o r normal i n t h e d i p e r s i o n r e l a t i o n such a s k2 = aw o r modes, b u t t h a t we may have # 0 f o r d e l t a -

= d2($ - w 2 ) , f o r whioh branch-points occur a t f u n c t i o n e x c i t a t i o n . For s t a b l e media, t h e two

~u = 0 and cu = * ^{W ,} - , ^- r e s p e c t i v e l y . modes a r e i d e n t i c a l . I n s t r o n g l y absorbing, o r SADDLE-POINT METHOD h i g h l y u n s t a b l e media, t h e d i f f e r e n c e between k(cuo)

and k(w-) may become s i g n i f i c a n t . We may approximate ( 4 ) by expanding t h e i n t e - .

A s an example, c o n s i d e r an u n s t a b l e system grand about t h e extrema of t h e e x p o n e n t i a l term.

c o n s i s t i n g of a monoenergetic e l e c t r o n beam and a The s a d d l e - p o i n t s of ( 4 ) a r e t h e n r e q u i r e d t o

s a t i s f y s i m u l t a n e o u s l y plasma, f o r which

where dk/& must be r e a l and p o s i t i v e s i n c e t h e c o n t o u r s i n Fig. 2 a r e d e f i n e d f o r t and z r e a l and p o s i t i v e . The saddle-point, us , i s a f u n c t i o n of t and z , and corresponds t o t h e v e l o c i t y , v,,,.

- The approximation of (4) has two forms which depend on t h e p r e s e n c e o r absence of a p o l e i n F, i . e . on t h e n a t u r e of t h e source. I f F i s analy- t i c , t h e i s o l a t e d s a d d l e - p o i n t method can b e used f o r l a r g e z [ 2 ] . For S,, , we t h e n have

q w s y k(as))

~ ( t ~ ~ ) ~~ X P xi wst z)], ( 6 )

U) [ ⁽

where G i s a smooth f u n c t i o n . This approximation f o r 5 i s u s u a l l y v a l i d when z exceeds about t e n wavelengths corresponding t o k(ms).

The p o l e from gc w i t h s i n wot = I m exp(io, t ) r e q u i r e s t h e simple p o l e s i n g u l a r i t y method [ 2 f ,

1 / 2 where b = i i l ( m s u n ) t - (*(mB) -*(mo)) z j / , ^{U and}

T a r e smoot unctions, e r f c 1s t h e comp ementary e r r o r f u n c t i o n , and t h e =k s i g n i s chosen s o t h a t e r f c " 0 f o r non-causal v a l u e s of t and z. Pre- c u r s o r s ( s e e Fig. 1 ) a r e d e s c r i b e d by t h e ~ ( m , ) terms. A s waves of lower v,, a r r i v e from t h e source, a ~ ( m s ) t e r m i n ( 7 ) dominates t h e r e s p o n s e , a f t e r t h e phase of b changes sign. This phase change occurs a s t h e r e a l p a r t of %(= wl" + b i ) approaches UC); e r f c changes i n magnitude from w 0 t o m 2 .

The main wave t r a i n of a s o u r c e l i k e Sc p r o p a g a t e s with t h e v e l o c i t y , corresponding t o ~ e ( t u , ) =wo.

B r i l l o u i n termed t h k :'signal v e l o c i t y " . DISCUSSION

The d i s p e r s i o n c h a r a c t e r i s t i c s corresponding t o ( 6 ) and ( 7 ) a r e b o t h of t h e form ~ ( c u , k ) = 0 ,

b u t with m = u s f o r s o u r c e s without p o l e s ( d e l t a - f u n c t i o n modes) and w i t h w r e a l f o r s o u r c e s w i t h p o l e s (normal modes). The p r i n c i p a l d i s t i n c t i o n

Here, , ^{i s} t h e plasma frequency, i s t h e plasma fgequency of t h e beam, and vb%is t h e beam v e l o c i t y . Tlle d i s p e r s i o n diagrams of Figs. 3 ( a ) and ( b ) , f o r normal modes and d e l t a - f u n c t i o n e x c i t a - t i o n , a r e s i m i l a r f o r t h e s t a b l e branch from A t o By b u t d i f f e r s u b s t a n t i a l l y f o r t h e branch from C t o E : i n t h e normal mode diagram, t h e group velo- c i t y may a p p a r e n t l y b e n e g a t i v e o r even i n f i n i t e . The d e l t a - f u n c t i o n modes a l s o d i f f e r s u b s t a n t i a l l y below t h e plasma frequency, w i t h t h e branch from F t o G i n t h e normal mode diagram d i s a p p e a r i n g . Values of v, corresponding t o t h e two modes of e x c i - t a t i o n a r e p l o t t e d i n F i g . 3 ( c ) f o r comparison.

An o b s e r v a t i o n a l d i s p e r s i o n diagram such a s Fig. 3(b), f o r a p o l e - f r e e source, has t h e advant- age over a normal mode diagram such a s Fig. 3 ( a ) t h a t it i l l u s t r a t e s t h e c a u s a l group v e l o c i t i e s , f r e q u e n c i e s and wavenumbers t h a t would a c t u a l l y b e observed. S i n c e t h e v e l o c i t y % of a s p e c t r a l com- ponent a t frequency w has been d e f i n e d a s r e a l and p o s i t i v e , it corresponds t o t h e u s u a l d e f i n i t i o n of group v e l o c i t y i n a s t a b l e medium, and e x t e n d s t h a t concept c o n s i s t e n t l y t o a b s o r p t i v e and u n s t a b l e media. I t i s a n t i c i p a t e d t h a t t h e o b s e r v a t i o n a l

approach and d i s p e r s i o n diagram w i l l b e u s e f u l i n t h e s t u d y of i n s t a b i l i t i e s i n inhomogeneous plasmas, and i n i o n o s p h e r i c r a y t r a c i n g .

This work was supported by t h e NASA and t h e NSF. The a u t h o r s have b e n e f i t e d from d i s c u s s i o n s w i t h Dr. K. J. Harker.

[11 B r i l l o u i n , L., Wave Propagation and Group

Velocity, ~ c a d e v

[ 2 ] Felsen, L. B. and Marcuvitz, N., R a d i a t i o n and S c a t t e r i n g of Waves, P r e n t i c e - H a l l (1973).

[ 31 ^{Whitham, G.} B., Linear and Nonlinear Waves Wiley (1974) .

[ k ] Briggs, R, J., Electron-Stream I n t e r a c t i o n s with Plasmas, MIT P r e s s (1964).