Description of waves in a nonuniform magnetized plasma to all orders in electron and ion temperature

Volume 31A, number 1

DESCRIPTION TO ALL

PHYSICS LETTERS

12 January 1970

OF WAVES IN A NONUNIFORM MAGNETIZED PLASMA ORDERS IN ELECTRON AND ION TEMPERATURE

J . - L . MONFORT * and P. E . V A N D E N P L A S Laboratoire de Physique des Plasmas**, Ecole Royale Mililaire, Bruxelles 4, Belgium

Received 3 December 1969

A general description of waves (including the differential equation for the electromagnetic field} in a nonuniform magnetized plasma is given to all orders in the electron and ion temperature and permits the complete treatment of a bounded plasma.

V e r y r e c e n t l y , the e x i s t e n c e of new r e s o n a n c e s (and p a r t i c u l a r l y of m a i n r e s o n a n c e s ) in the m a g n e t o - i o n domain of a hot bounded p l a s m a has been p r e d i c t e d [1] and r e s o n a n c e s have a l r e a d y since been o b s e r v e d in this domain [2]. The t h e o r e t i c a l p r e d i c t i o n s w e r e b a s e d on the s t a n d a r d fluid model of a hot e l e c t r o n - i o n p l a s m a in which the equations a r e closed by a s s u m i n g p ~ = ~ K T ~ n ~ (~ = e or i) for the p e r t u r b e d exp (- iwt) p r e s s u r e s [3]. This c o n s t i t u t e s a s a t i s f a c t o r y f i r s t a p p r o x i m a t i o n but the n a t u r e and p o s s i b l e i m p o r t a n c e of these new r e s o n a n c e s r e n d e r s the d e r i v a t i o n of an exact d e s c r i p t i o n of the p l a s m a in the m a g n e t o - i o n domain n e c e s s a r y . On the other hand it has been shown that all the i n f o r m a t i o n contained in the Boltzmann equation to f i r s t o r d e r in the e l e c t r o n t e m p e r a t u r e T e can be obtained by an o p e r a t o r method [4,5]. In the p r e s e n t paper, we u s e such a method to give a g e n e r a l d e s c r i p t i o n of waves (including the c o r r e c t d i f f e r e n t i a l equation for the e l e c t r i c field) in a n o n - u n i f o r m m a g n e t i z e d p l a s m a to a l l o r d e r s in the e l e c t r o n and ion t e m p e r a t u r e and obtain the c o r r e s p o n d i n g d i s p e r s i o n r e l a t i o n . C o n s i d e r a n o n - u n i f o r m p l a s m a i m m e r s e d in a u n i f o r m m a g n e t i c induction B o = B o 1 z. Neglecting the static e l e c t r i c field Eo, the l i n e a r i z e d B o l t z m a n n equation for the exp (-iwt) d i s t r i b u t i o n function for (~ = e, i , . . . ) can be w r i t t e n as [ a / a ~ + P ~ (~b)]f~ = Qc~ (~b) in which P ~ (~b) is a l i n e a r o p e r a t o r containing a / a x , ~/~y and ~/~z; the v e l ocity v = [v± cos ¢ , v± sin ¢ , v, ] and ± m e a n s p e r p e n d i c u l a r to B o. This equation can be i n t e r p r e t e d and f u r n i s h e s f a u n d e r the f o r m of an o p e r a t o r r e p r e s e n t e d by a product of 10 infinite s e r i e s and acting on the p e r t u r b e d e l e c t r i c field E and on the n o n u n i f o r m e q u i l i b r i u m d i s t r i b u t i o n function i s Maxwellian and c e n t r e d on zero and if a / a z - o (the wave v e c t o r k is thus ± to Bo) , the equations for the c o m p o n e n t s of the e l e c t r i c field, to all o r d e r s in T e and Ti, are:

c2 Va2Ex

0)2 L ~y2

a--~J + Ex + ~ ~

(IV+1)(2N+ 1)'! N+I

N=O(2N+I) Iq 2 2

×

I=1

2

} / K T a ~N

2 -w-~= 0 .

* Chercheur agr4~ de l'Institut Interuniversitaire des Sciences Nucl4aires. ** Association Euratom - Etat belge. 11

Volume 31A, number 1

PHYSICS LETTERS

Eq. (2) is the s a m e as eq. (1) in which x--- y, y --, x and

/ Krs

12 January 1970 - ~ s ~ ~s •

2

2 --

(2) 0

(3)

l=O where T s : a b s o l u t e t e m p e r a t u r e , 00ps: p l a s m a f r e q u e n c y , ~s = Z s e B ^ / m ~ w , Zs: n u m b e r of positive c h a r g e s , m s : m a s s , K: B o l t z m a n n - c o n s t a n t , s = e or i for e l e c t r o n s ~md ions. We shaU see l a t e r that this leads to a c o n v e r g e n t d e s c r i p t i o n when the c y c l o t r o n f r e q u e n c i e s C0cs ¢ 0. T h i s method is p a r t i c u l a r l y i n t e r e s t i n g to u s e when one c o n s i d e r s a n o n u n i f o r m p l a s m a with different s p e c i e s of p a r t i c l e s ; the t e m p e r a t u r e can be n o n u n i f o r m . C o l l i s i o n a l damping is easy to introduce. If the quasi static a p p r o x i m a t i o n i s made (/~= - Vqb; note c a r e f u l l y that this i s m o r e g e n e r a l than c o n s i d e r i n g p u r e l y longitudinal waves, [e.g. 6, p. 8-10], eqs. (1) and (2) r e d u c e to:

N=o

17 (l

i)

l=l Eqs. (1), (2) and (3) lead to a r a t h e r c u m b e r s o m e e x p r e s s i o n for the g e n e r a l e l e c t r o m a g n e t i c d i s p e r s i o n equation of an u n i f o r m p l a s m a . In the g u a s i - s t a t i c a p p r o x i m a t i o n , the d i s p e r s i o n r e l a t i o n r e d u c e s to eq. (4) in which V2 i s r e p l a c e d by (- k2). T h i s i s a useful t e m p e r a t u r e - o r d e r e d f o r m of the d i s p e r s i o n r e l a t i o n , in which two i n f i n i t e s e r i e s (s = i and e) appear which a r e absolutely convergent when COce and ¢Oci ¢ 0. A sufficient condition for i n t e r r u p t i n g the s e r i e s by r e t a i n i n ~ the t e r m s up to o r d e r N in T s is that (N+ 1)2002s > w2 and 1(2N+3)k2KTct/m s I << I(N+2)2°°2s - 0J~. T h e s e conditions show that the power N of the t e r m s which m u s t be r e t a i n e d cannot be a p r i o r i decided but depends on the v a l u e s of Wcs , co, T s , m s , k 2 in the d o m a i n in which the wave i s being i n vestigated. F o r i n s t a n c e , when the m a g n e t i c field i s low ( W c s / w << 1), we m u s t u s e a r a t h e r high N and i n t r o duce the f i r s t (N+ 1) cyclotron h a r m o n i c s for a detailed d e s c r i p t i o n of the phenomena. T h i s does not n e c e s s a r i l y m e a n , however, that a s a t i s f a c t o r y d e s c r i p t i o n cannot be obtained when the above sufficient conditions a r e not met. It can be shown that eq. (4) with V2 = - k 2, which i s s i m i l a r to one r e c e n t l y obtained [7], i s e q u i v a lent to B e r n s t e i n ' s d i s p e r s i o n r e l a t i o n [8]. The l a t t e r e x p r e s s i o n is, however, m o r e unwieldly and did not p e r m i t the above c o n c l u s i o n s . It is also i m p o r t a n t to r e a l i z e that when "0cs ¢ 0, the u s e of the m o m e n t s method leads to exactly the s a m e d i s p e r s i o n r e l a t i o n as eq. (4) provided that all t e r m s up to T~N a r e included (see ref. 9 or ref. 6 page 154 for the d e t a i l s c o n c e r n i n g o r d e r one in Te). A s y s t e m a t i c c o m p a r i s o n between the k 2 given by the abov~ g e n e r a l d i s p e r s i o n r e l a t i o n s ( e l e c t r o m a g n e t i c and q u a s i - s t a t i c ) and the s t a n d a r d h y d r o d y n a m i c a l k ~ obtained with the c l o s u r e Ps = KTsnsTs is being c a r r i e d out and will be given, together with the d e t a i l s c o n c e r n i n g the d e r i v a t i o n of the above r e s u l t s , in a f u l l - l e n g t h paper. It is a p l e a s u r e to thank A. M. M e s s i a e n and R . W e y n a n t s for i n t e r e s t i n g d i s c u s s i o n s . 1. 2. 3. 4. 5. 6. 7. 8. 9.

12

P.E. Vandenplas, A.M. Messiaen, J.-L. Monfort and J. J. Papier, Phys. Rev. Letters 22 (1969) 1243. A.M. Messiaen, P.E. Vandenplas, J.-L. Montfort, Phys. Letters 29A (1969) 573. J . F . Denisse and J.L.Delcroix, Plasma waves (John Wiley, Interscience, London, New York, Sydney, 1963). S.J. Buchsbaum and A.Hasegawa, Phys. Rev. Letters 12 (1964) 685; A.Hasegawa, Phys. Fluids 8 (1965) 761. W.P.Allis and J.C.De Almeida Azevedo, Bull. Am. Phys. Soc. 11 (1966) 716; J.C.De Almeida Zevedo and M.L.Vianna, Phys. Rev. 177 (1969) 300. P.E.Vandenplas, Electron waves and resonances in bounded plasmas (John Wiley, Interscience, London, New York, Sydney, 1968). R.W.Fredricks, J. Plasma Phys. 2 (1968) 197. I.B. Bernstein, Phys. Rev. 109 (1958) 10. J. C. Nihoul and P. E. Vandenplas, Plasma Phys. (J. Nucl. Energy Part C) 7 (1965) 341.