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Transverse instabilities in streaming collisional plasmas
Volume 32A, number 7
PHYSICS LETTERS
(a'cE/n'o 2) 6t> ,/(",Xo) 2 + (5) 5t/ZXXo) 2 - zXxo
(3)
w h e r e z~xo is the i n i t i a l width of the wave packet. T h i s condition should be c o m p a r e d with (1) and it will be noted that it is s i m p l y that the w a v e packet is l e s s steep than the 'light shock'. The 'light shock' will f o r m in a time.
~" ~ no2 LXXo/COL'E
(4)
The s u b s e q u e n t s t r u c t u r e of the w a v e - p a c k e t will be s i m i l a r to a b l a s t wave as the ' f o r w a r d ' r a r e f a c t i o n wave r e m o v e s energy f r o m the Peak of the s i g n a l due to d i s p e r s i o n . T h i s , however, will be a e o m p a r a t i v e l y slow p r o c e s s as the slope of the f o r w a r d p a r t of the wave is much l e s s than that at the back. T h i s s i m p l e q u a l i t a t i v e d i s c u s s i o n has e n a b l e d us to deduce s o m e o r d e r of m a g n i t u d e v a l u e s for the q u n a t i t i e s in this phenomenon. A m o r e s o p h i s t i e a t e d a n a l y s i s , however, p r e s e n t s m a n y p r o b l e m s - in p a r t i c u l a r the effects of
7 September 1970
non-linear refraction are calculated from the pulse itself whereas those of dispersion require a Fourier transform of the signal. That the approach implicitly used here, namely, the representation of the wave-packet by the superposition of many small wave-packets, is valid and is due to the fact that the packet itself is equal to the sum of the transforms of the incremental packets.
References [1] P. L. Kelley, Phys. Rev. Letters 15 (1965) 1005. [2] R. Y. Chiao, E. Garmire and C. H. Townes, Phys. Rev. Letters 13 (1964) 479. [3] M. E. Mack, R. L. Cannon, J. Reintjes and N.Bloembergen, Appl. Phys. Letters 16 (1970) 209. [4] P. D. Maker and R. W. Terhune, Phys. Rev. 137 (1965) A801. [5] L. D. Landau and E. M. Lifshitz, Fluidmechanics (Pergamon Press, Oxford, 1962) p. 372. [6] J. D. Jackson, Classical electrodynamics (John Wiley and Sons, Inc., New York, 1967) p. 215. * * * * *
TRANSVERSE
INSTABILITIES
IN S T R E A M I N G
COLLISIONAL
PLASMAS*
B. BUTI and G. S. LAKHINA
Department of Physics, Indian Institute of Technology, New Delhi, India Received 17 July 1970 The Coulomb collisions are found to stabilize the system comprising of contrastreaming plasmas by increasing the value of the streaming velocity required to excite the transverse instability in these plasmas.
T w o - s t r e a m i n s t a b i l i t y in s t r e a m i n g p l a s m a s with and without c o l l i s i o n s has b e e n studied by s e v e r a l a u t h o r s . Recently B u n e m a n n et al. [1-5] i n v e s t i g a t e d the i n s t a b i l i t y of the t r a n s v e r s e waves in s t r e a m i n g p l a s m a s in the a b s e n c e of any c o l l i s i o n s . Here in this c o m m u n i q u e , we find the effect of the Coulomb c o l l i s i o n s on these u n s t a b l e t r a n s v e r s e waves in c o n t r a s t r e a m i n g p l a s m a s by m e a n s of F o k k e r - P l a n c k equation. Let us c o n s i d e r two homogeneous c o n t r a s t r e a m i n g p l a s m a s with ions i m m o b i l e (providing only a n e u t r a l i z i n g background) and the e l e c t r o n s obeying the F P equation. It was shown by Buti et al. [6, 7] that in p l a s m a s with i n f r e quent c o l l i s i o n s for s m a l l wave n u m b e r s , the
* Work partially supported by ESSA (USA). 518
e l e c t r o n - e l e c t r o n (e-e) c o l l i s i o n c o n t r i b u t i o n is n e g l i g i b l e c o m p a r e d to e - i contribution. Since the t r a n s v e r s e i n s t a b i l i t i e s , in which we a r e int e r e s t e d , a r e i m p o r t a n t ff (ck Iwp) £ 1 we shall n e g l e c t the e - e c o l l i s i o n s . Consequently the l i n e a r i z e d F P equation [8] which g o v e r n s the m o tion of the e l e c t r o n s is given by a/a afa -~F+ V . a X
Ne(E+ m
1 c VxB)"
aYoa
a--V'
("
where F = (47re 4 / m 2) In A, f o a is the e q u i l i b r i u m d i s t r i b u t i o n function n a m e l y , f o a = (2~V:a)'3/2 e x p [ - ( V - V ~ ) 2 / 2 V 2 t a ] '
(2)
Volume 32A, number 7
PHYSICS LETTERS
and a l a b e l s the two s t r e a m s . In w r i t i n g eq. (1), we have taken ions to be cold. On i n t r o d u c i n g the e l e c t r o n c o l l i s i o n f r e q u e n c y Uc = NF/Vt3 and on taking the F o u r i e r - L a p l a c e t r a n s f o r m of eq. (1) and of Maxwell equations, we get the following d i s p e r s i o n r e l a t i o n to the lowest o r d e r in Vc;
7 September 1970
~.
..
~/kVt,o.o
ikv,.oo
///
/jj
1,2
__ (_
+
2k2) ?
2
a= 1
2
pa
fdV
V[Z+i
UcV:a (/(-V--w)
aV i
(3)
with 0.4.
(4) N o w for wave propagation transverse to the streaming e.g. k = k ~x and U= U@z, for identical counterstreaming (U 1 = - U 2 ==/] say) plasm a s the non-diagonal elements of R vanish. Rxx and Ryy are not modified by streaming [5]. Let us consider the m o d e Rzz = 0 which is affected by streaming; for the case of marginal instability (00 ---0) this yields c2k 2
w2 -42
2~ v c y
kVt
exp(-Y) Io(Y) ,
(5)
1.,
where y = Uc2 / 4 V 2 and Io is the modified B e s s e l function with i m a g i n a r y a r g u m e n t . Uc given by eq. (5) defines the b o u n d a r y between the stable ( U < Uc) and the u n s t a b l e ( U > Uc) r e g i o n s . Fig. 1 c l e a r l y shows that Uc i n c r e a s e s with v c i.e., Coulomb c o l l i s i o n s have a s t a b i l i z i n g effect on the t r a n s v e r s e waves. T h i s effect is m o r e p r o nounced for l a r g e r v a l u e s of (c2k2/w2p); however our model b r e a k s down for l a r g e v a l u e s of this p a r a m e t e r s in which case one should include the e - e c o l l i s i o n s . The detailed a n a l y s i s of the gene r a l p r o b l e m taking into account the hot ions and the e - e c o l l i s i o n s will be r e p o r t e d e l s e w h e r e .
0.4
0!8
,.2
1.6
2 2 versus c2k2/~02 for Fig. 1. Variation of Uc/Vt Vc/kVt = 0.0, 0.05, 0.10. We a r e indebted to ESSA (USA) for supporting this work. One of u s (G. S. L.) is thankful to Council of Scientific and I n d u s t r i a l R e s e a r c h of India for the g r a n t of a fellowship.
References [I] D. Bunemann, Ann. Phys. 25 (1963) 340. [2] H. Momota, Progr. Theoret. Phys. 35 (1966) 380. [3] Kai Fong Lee, Phys. Rev. 181 (1969} 447. [4] B.Buti, Phys. Rev. A1 (1970) 1772. [5] B. Buti and G. S. Lakhina, J. Plasma Phys., to be published. [6] B.Buti and R.K.Jain, Fluids 8 (1965) 2080. [7] B.Buti and S.K.Trehan, Ann. Phys. 40 (1966) 296. [8] M. N. Rosenbluth, W . M . McDonald and D. L. Judd, Phys. Rev. 107 (1957) 1.
519
PHYSICS LETTERS
(a'cE/n'o 2) 6t> ,/(",Xo) 2 + (5) 5t/ZXXo) 2 - zXxo
(3)
w h e r e z~xo is the i n i t i a l width of the wave packet. T h i s condition should be c o m p a r e d with (1) and it will be noted that it is s i m p l y that the w a v e packet is l e s s steep than the 'light shock'. The 'light shock' will f o r m in a time.
~" ~ no2 LXXo/COL'E
(4)
The s u b s e q u e n t s t r u c t u r e of the w a v e - p a c k e t will be s i m i l a r to a b l a s t wave as the ' f o r w a r d ' r a r e f a c t i o n wave r e m o v e s energy f r o m the Peak of the s i g n a l due to d i s p e r s i o n . T h i s , however, will be a e o m p a r a t i v e l y slow p r o c e s s as the slope of the f o r w a r d p a r t of the wave is much l e s s than that at the back. T h i s s i m p l e q u a l i t a t i v e d i s c u s s i o n has e n a b l e d us to deduce s o m e o r d e r of m a g n i t u d e v a l u e s for the q u n a t i t i e s in this phenomenon. A m o r e s o p h i s t i e a t e d a n a l y s i s , however, p r e s e n t s m a n y p r o b l e m s - in p a r t i c u l a r the effects of
7 September 1970
non-linear refraction are calculated from the pulse itself whereas those of dispersion require a Fourier transform of the signal. That the approach implicitly used here, namely, the representation of the wave-packet by the superposition of many small wave-packets, is valid and is due to the fact that the packet itself is equal to the sum of the transforms of the incremental packets.
References [1] P. L. Kelley, Phys. Rev. Letters 15 (1965) 1005. [2] R. Y. Chiao, E. Garmire and C. H. Townes, Phys. Rev. Letters 13 (1964) 479. [3] M. E. Mack, R. L. Cannon, J. Reintjes and N.Bloembergen, Appl. Phys. Letters 16 (1970) 209. [4] P. D. Maker and R. W. Terhune, Phys. Rev. 137 (1965) A801. [5] L. D. Landau and E. M. Lifshitz, Fluidmechanics (Pergamon Press, Oxford, 1962) p. 372. [6] J. D. Jackson, Classical electrodynamics (John Wiley and Sons, Inc., New York, 1967) p. 215. * * * * *
TRANSVERSE
INSTABILITIES
IN S T R E A M I N G
COLLISIONAL
PLASMAS*
B. BUTI and G. S. LAKHINA
Department of Physics, Indian Institute of Technology, New Delhi, India Received 17 July 1970 The Coulomb collisions are found to stabilize the system comprising of contrastreaming plasmas by increasing the value of the streaming velocity required to excite the transverse instability in these plasmas.
T w o - s t r e a m i n s t a b i l i t y in s t r e a m i n g p l a s m a s with and without c o l l i s i o n s has b e e n studied by s e v e r a l a u t h o r s . Recently B u n e m a n n et al. [1-5] i n v e s t i g a t e d the i n s t a b i l i t y of the t r a n s v e r s e waves in s t r e a m i n g p l a s m a s in the a b s e n c e of any c o l l i s i o n s . Here in this c o m m u n i q u e , we find the effect of the Coulomb c o l l i s i o n s on these u n s t a b l e t r a n s v e r s e waves in c o n t r a s t r e a m i n g p l a s m a s by m e a n s of F o k k e r - P l a n c k equation. Let us c o n s i d e r two homogeneous c o n t r a s t r e a m i n g p l a s m a s with ions i m m o b i l e (providing only a n e u t r a l i z i n g background) and the e l e c t r o n s obeying the F P equation. It was shown by Buti et al. [6, 7] that in p l a s m a s with i n f r e quent c o l l i s i o n s for s m a l l wave n u m b e r s , the
* Work partially supported by ESSA (USA). 518
e l e c t r o n - e l e c t r o n (e-e) c o l l i s i o n c o n t r i b u t i o n is n e g l i g i b l e c o m p a r e d to e - i contribution. Since the t r a n s v e r s e i n s t a b i l i t i e s , in which we a r e int e r e s t e d , a r e i m p o r t a n t ff (ck Iwp) £ 1 we shall n e g l e c t the e - e c o l l i s i o n s . Consequently the l i n e a r i z e d F P equation [8] which g o v e r n s the m o tion of the e l e c t r o n s is given by a/a afa -~F+ V . a X
Ne(E+ m
1 c VxB)"
aYoa
a--V'
("
where F = (47re 4 / m 2) In A, f o a is the e q u i l i b r i u m d i s t r i b u t i o n function n a m e l y , f o a = (2~V:a)'3/2 e x p [ - ( V - V ~ ) 2 / 2 V 2 t a ] '
(2)
Volume 32A, number 7
PHYSICS LETTERS
and a l a b e l s the two s t r e a m s . In w r i t i n g eq. (1), we have taken ions to be cold. On i n t r o d u c i n g the e l e c t r o n c o l l i s i o n f r e q u e n c y Uc = NF/Vt3 and on taking the F o u r i e r - L a p l a c e t r a n s f o r m of eq. (1) and of Maxwell equations, we get the following d i s p e r s i o n r e l a t i o n to the lowest o r d e r in Vc;
7 September 1970
~.
..
~/kVt,o.o
ikv,.oo
///
/jj
1,2
__ (_
+
2k2) ?
2
a= 1
2
pa
fdV
V[Z+i
UcV:a (/(-V--w)
aV i
(3)
with 0.4.
(4) N o w for wave propagation transverse to the streaming e.g. k = k ~x and U= U@z, for identical counterstreaming (U 1 = - U 2 ==/] say) plasm a s the non-diagonal elements of R vanish. Rxx and Ryy are not modified by streaming [5]. Let us consider the m o d e Rzz = 0 which is affected by streaming; for the case of marginal instability (00 ---0) this yields c2k 2
w2 -42
2~ v c y
kVt
exp(-Y) Io(Y) ,
(5)
1.,
where y = Uc2 / 4 V 2 and Io is the modified B e s s e l function with i m a g i n a r y a r g u m e n t . Uc given by eq. (5) defines the b o u n d a r y between the stable ( U < Uc) and the u n s t a b l e ( U > Uc) r e g i o n s . Fig. 1 c l e a r l y shows that Uc i n c r e a s e s with v c i.e., Coulomb c o l l i s i o n s have a s t a b i l i z i n g effect on the t r a n s v e r s e waves. T h i s effect is m o r e p r o nounced for l a r g e r v a l u e s of (c2k2/w2p); however our model b r e a k s down for l a r g e v a l u e s of this p a r a m e t e r s in which case one should include the e - e c o l l i s i o n s . The detailed a n a l y s i s of the gene r a l p r o b l e m taking into account the hot ions and the e - e c o l l i s i o n s will be r e p o r t e d e l s e w h e r e .
0.4
0!8
,.2
1.6
2 2 versus c2k2/~02 for Fig. 1. Variation of Uc/Vt Vc/kVt = 0.0, 0.05, 0.10. We a r e indebted to ESSA (USA) for supporting this work. One of u s (G. S. L.) is thankful to Council of Scientific and I n d u s t r i a l R e s e a r c h of India for the g r a n t of a fellowship.
References [I] D. Bunemann, Ann. Phys. 25 (1963) 340. [2] H. Momota, Progr. Theoret. Phys. 35 (1966) 380. [3] Kai Fong Lee, Phys. Rev. 181 (1969} 447. [4] B.Buti, Phys. Rev. A1 (1970) 1772. [5] B. Buti and G. S. Lakhina, J. Plasma Phys., to be published. [6] B.Buti and R.K.Jain, Fluids 8 (1965) 2080. [7] B.Buti and S.K.Trehan, Ann. Phys. 40 (1966) 296. [8] M. N. Rosenbluth, W . M . McDonald and D. L. Judd, Phys. Rev. 107 (1957) 1.
519