Anil Kumar, C P

George, B

Renuka, G

Venugopal, C

IACS Institutional Repository

■„dum.l Phy^- 77B (5), 561-566(2003)3:< IJPB  %Linear instabilities in multi-ion plasmasC P Anil Kumar'*, C Venugopal', B George’ and GRenuka^S c h o o l  o f  P u r e  &  A p p l i e d  P h y s i c s .  M a h a t m u  G a n d h i  U n i v e r s i l y ,  P r i y a d a r s h m i  H i l l s ,  K o U a y a m - 6 8 o  5 6 0 .  K e r a l a ,  I n d i aD e p a r t m e n t  o f  P h y s i c s ,  l l i i i v c r s i t y  o f  K e r a l a ,  K a n y a v a t l o m  C a m p u s  T h i r u v a i i a n t h a p u r a m  6 ^ 5  5 8 1 ,  K e r a l a ,  I n d i aE - m a i l  , c p a n i K ^ c i i s a t . a c  in  R tvv ived  5 J i i l \  2002, at cvpied 29 May  200 .1\ h s t r a c l  In t h i s  p a p e r ,  t h e  . s ta b i l i t y  o t  i o n - c y c l o t r o n  w a v e s  in  m u l l i - i o n  p l a s m a  c o n s i s t i n g  o l  e l e c t r o n s ,  p o s i t i v e l y  a n d  n e g a t i v e l y  c h a r g e d  o x y g e n  niMs ,ind h y d i o g c n  i o n s  a i e  e x a m i n e d  b y  s i m u l a t i o n s .  In  o r d e r  t o  i d e n t i f y  th e  i n o t l e s  in  a  m i x e d  p l a s m a  a n d  t o  g a m  p h y s i c a l  i n s i g h t  i n t o  t h e  n a t u r e  o f  itk i i is iab il i iy ,  c o l d  p l a s m a  I h c o i y  is u s e d ,  w h i l e  t h e  w a r m  p l a s m a  t h c o i y  a c c o u n t s  f o r  f i n i t e  t e m p e r a t u r e  e f f e c t s  T h e  e x p r e s s i o n  f o r  t h e  f r e q u e n c i e s  m! h y b i i d  a n d  l o n - i o n  h y b r i d  m o d e s  a r e  d e r i v e d  a n d  s o l v e d  b y  n u m c i i c a l  m e t h o d s  F r o m  th e  . s i m u l a t i o n  s t u d i e s  i t  is  c o n c l u d e d  th a t  l o w e r  h y b r i d  (lukJc IS a w e a k l y  u n s t a b l e  m o d e  w h i l e  t h e  l o n - i o n  h y b r i d  m o d e  is a l m o s t  f i e e l y  p r o p a g a t i n gK i v w o r r i s  , I o n  c y c l o t r o n  w a v e ,  B u c h s b a u m  m o d er\rs  N o . .52 1,5 Qz.1. IntroductionInriLvcloimn waves were first studied by Kindel and Kennel 11 ] whn .sh(^ )wed that for current carried by a streaming electron tlisinhution in the presence of stationary Maxwellian ions, ion- cydoiion waves have the lowest threshold among the current Jnven instabilities in an isothermal plasma (T = 7'^ ). Later, Abdalla (tl \2 \ showed that the addition of heavy ions in the plasma tlispLMsion modified the lower - hybrid mode and also allowed in lon-ion hybrid mode /.c the Buchsbaum mode to exist [3,4J.I he ion-ton mode has a frequency between the proton and heavy "HI gyro-frequency and is driven unstable at oblique wave "nglcs. by an electron beam.fhe plasma wave probe carried  by the spacecraft f^iO nuaional Cometary Explorer (ICE) detected a number of ""Tacoustic and ion-cyclotron waves in the environment ot "^Hncr Giacobini-Zinner. During ICEs nearest approach, the plasrnu wave instrument detected broad-band electrostatic noise a changing pattern of weak electron plasma oscillations forAuthorlo r  C ' o m m i n c a i o n  :P^^niiL-ni of Electronics, Cochin University of Science & Technology.Cochin. Trikkakara-682 022. Kerala, Indiathe outer layers of the cold plasma tail. Near the tail axis, the plasma wave instruments also detected whistler mode (/ < 100 Hz) and lower hybrid mtxle 0  < 10 Hz) [5).The comet literature contains many theoretical suggestions invt)lving the generation of various plasma instabilities [6-15]. These studies either modelled cometary environments or were easily adaptable to them. The analysis presented here, illustrates the impressive degree to which ion-cyclotron instability and newborn ion species interact over a region, that extends beyond the comet boundaries (i.e, tail). The expressions for the frequencies of the 'lower hybrid' and 'ion-ion hybrid' modes are derived using cold plasma theory and considered the stability of these modes. It is observed that the lower hybrid mode is weakly unstable, while the ion-ion hybrid mode is an almost I’reely propagating mode. The model approximates the plasma environment of the tail of comet Giacobini - Zinner.2. Method of analysisIn order to identify the modes in a mixed plasma and to gain physical insight into the nature of the instabilities, cold plasma theory can be used. The result from this analysis was then used to study the stability of these modes, using warm plasma theory.©2003IACS(a ) Cold plasma theory :5 6 2 C P Anil Kumar. C Venugopal, B George and G RenukaWe consider mixed plasma that includes electrons, negatively and positively charged oxygen ions (0 “ and O"^ ) and hydrogen ions (H) which drift with a velocity V with respect to the 0 “*^ ions. For simplicity, in the derivation of the frequencies of the mode, let U = 0. The cold plasma electrostatic dispersion relation for wave propagating perpendicular to the magnetic field is well known and is given by an extension of the result [16] asIn eq (1), the species indices are given by 'e for electrons, H for hydrogen and and O ' respectively for positively and negatively charged oxygen ions. The gyro-trequencies for species a  is denoted by with B! M^ J^C (r/^, isthe charge of the particle; /?, the ambient magnetic field; the mass of the particle and C, the velocity of light). The plasma frequency is defined as with T^ e^frequency is indicated by cj). For low frequencies such that 0)^ « Q I  and assuming co}.  ^ «  , eq. ( 1 ) reduces to1 +- T + - T-f- , - 0 . (2)'O ' .......................0The two positive solutions to (2 ) are given by(3)and(O — 4^3^+ (« ; + n ; ) i 2 ^. (4)] +0) - oo + nQ,_ar« jIn iYafa:-A',1 +(0-K«U  ..KTH^ (O + n Q j f - K M ^THA.O'h )' - 0(5|In (5), the species summation a  is over electrons, 0^ and 0 “ ; the hydrogen contribution has been written out separately K is the wave vector which has parallel components /C| and perpendicular to the magnetic field. Z is the plasma dispersion function, which arise from the dVj| integration and is defined as [ 18)Jrr J x - C (6)while /„(^//) IS the modified Bessel function which arises tiom the dV^ integration with an argumenty,t = a  = H, 0 ^  O ■ or e (7)The temperature of are related to the thermal velociiy V'Ta by the relationV r a = i f -  <x = i i . O \ 0 -  e. (8)The densities of H*. O* and O 'are  given respectively by K  andH, .Eq. (3) IS the expression for the lower hybrid frequency in the presence of the heavy ions O'*' and 0~, while eq. (4) represents the ion-ion hybrid (/.e. Buchsbaum) model. The mode is present only when another ion component is added to the plasma, as is evident from cq. (4).(b) Warm plasma theory :The cold plasma theory is very valuable for the mode identification. However, it cannot account for the finite temperature effect. Therefore, we consider the fully magnetized kinetic dispersion relation for electrostatic waves. The plasma, as already mentioned, is made up of hydrogen (H), oxygen ions (O* and O") and electrons. Thus, generalizing the well-known dispersion relation of [ 17] for electrostatic waves, we haveAll the four constituents of plasma have been modelled by isotropic Maxwellian distributions (the hydrogen ions alone being modelled as an isotropic, drifting component). Assume that 0) is complex and can be written ascu = cu^  + 1/ .Expanding the plasma dispersion function as a Taylor series :Ct) + n i2 „ -  7 (Or + n£2a 0)^+nS2ayl2K^Vra J= Z0), +n£2a 2iy ( n  V(Or TnWer_V2 J^ „VVa J >/2 ^yVV„ V 2^ iV7« [pK ^V ra  JJ(9)Linear instabilities in multi-ion plasmas Substituting (9) in (5), we can write down the dispersionrelation asa:Z (D  + - c o , ( ( 0 ,  + n £ 2 „ )/ 2 ^ )■ z ( n/ 2 ^ Tf/^// =I 0)V2A' ~  ^ 1 1 ^  If .fCV;77/Z ( D - 2 -_ 1 (o^  + nQ^1^1j, '" ” . r „  . f tW,-v'^.. y563( ( 0 , - K . U )  ( u ) ,  -  K M  +  i i Q  „ )\  -  -  i''  ^ — ------5---------- ( 10) ‘3 ft1V2 A'li V, (13)In eq. (13), 0 -• is the ratio of the wave vector paralleland perpendicular to the magnetic field, is the species density while the remaining parameters are defined earlier.oj^+n£2^ {oj, -  Kfj^ U + n£2 „)where / -  -  , ■ =  and I = ------- , " ------.Incq (10), again the summation a  includes e, and o“. , . . .In the denominator oi ( 1 2 ), since -  L „ (x) , theOne simplifying assumption has been made with regard to summation over n form ~ oo to + «> can be reduced to athe electron contribution, that the species arc over magnetized; summation from 0 to oo. Thus, carrying out the summation andihat ls^ 2, »  K^Vr,. Thus, « i  and among the infinite writing down the contribution, we getnumber o( terms, only one survives, namely = 1 ; theother terms are not contribuiive as, = 0 for/i ^ 0. Intheasymptotic expansion for electrons and hydrogen ions, consider only the case where both and o~ ions are cold, thus necessitating the asymptotic expansion for Z for these ions.I he asymptotic expansions for the plasma dispersion function is|18|1 n i,. 7;, 1-  -'«0-T,:2n, T,i w „ . T„3+ —r( 11) . ' ' ' y . i f / i )nA)Z* f01^„Y’ z :+ 3 Z .n  Z* - n  V 2 /  -  n jEq. ( 1 1 ) is now substituted in ( 10 ) and the real and imaginary ^^ o^ntributions of each species to the dispersion relation are ‘•eparated. Writing each contribution in full and carrying out the involved algebra, the expression for the growth/damping rate ^hich IS normalized against the ion gyro-frequency, can be ’^blamed as, / . *  + 10Z .'m - + 52 ,w ^  „ „  Z . ( Z ? + w ^ )UYij'( z ? - / . ^ rfhe contributions are;1  , Z (Z * + 6Z,-ir m iM. I t o y A A ------------- J— La  .0^ withZ. =co^CO, K^ i^l Q q £2^7 2  K^Vr, '5 6 4 C  P Anil Kumar, C Venugopal, B George and G Renuka, Tr „ ,  = 2 - ^  '/7=.0oV yZ % 3 Z  / r3(^1, )5 Z ^ + 1 0 Z ^ r+ ,‘)Z ;/ z ( Z -4 « - )" ( z ^ - r rand(z ^ - ,r - )3 /,^ , Z(Z'‘ 4 fiZ-zr+zj-*)( /  )r  / fvM ^0 n 0"  Z - - n -  2  J  [ z - - , r f3 -  O Z -U l0 Z V + 5 ZV Z (z -H ;r )( z - - « - f  ' ( z ' - '•-)■4 K  )3 z(z-* 4 6Z - « - +//■*)(14)found to be good. The programme was then completed lo calculate the other terms in (13) and (14), and computation made. Though eqs, (13) and (14) contain an infinite summation over the modified Bessel functions, the summation was restricted to only twelve terms, from zero to eleven, since the value* /„ (a) of beyond this was found to be extremely smallWe first consider the stability of the lower hybrid wave corresponding to (3). Figure 1 is a plot of the growth rate vm/o frequency as a function of 0 ( -  K^  ^ / K^  ^ = 0 .6, 0.7 and 0.8) foi ;i,, = 3.0, -  I.OkcV, 17= 10.0, Vt , - . = V  = T . .  /-  0.03eV and ^  1.0 KeV. We find that the instability u t  the wave shifts towards higher frequencies with increasing for increasingly parallel propagation. Also the growth rate, m general, increases with d.W 2 ^ 1  '^ (twith Z -  — ^  and .v« , where a  -  H, O^and O '3. ResultsThe computation of the expression for the growth / damping rate with the density of hydrogen ions /i^^= 3.0 at a temperature equivalent of = 3.0 keV and a drift velocity of U -  10 has been considered. The oxygen ions and wereassigned a value of 1,0 each and a temperature equivalent of0.03 eV. In addition, the electrons were assigned a temperature equivalent of T = 1.0 keV; the number densities being calculated from the charge neutrality condition, + r?/y. Themagnetic field used was the observed value of 60 x 10 '^  Gauss[13].The routines for the calculation of the modified Bessel functions that occur in the expressions (13) and (14) were obtained from Ref. |191. The values of e~’U„{x) were then checked against their values in Ref. [20], the agreement wasF i g u r e  1.  P l o t  o f  g r o w t h  r u t e  verMts f r e q u e n c y  f o r  =  3 0  (T jf  ^  1 0 Kt\  U  =  1 0  ) , n , /  =  =  I 0  -  7 ^ -  0 . 0 3  e V  ) a n d  T.  =  I 0  KcV.a f u n c t i o n ' o f  0  ( =  K^^/K^) = 0 . 6 ,  0  7  a n t H )  8The stability of the wave was studied as a function of ihc hydrogen density Figure 2 depicts the variation of the growth rate versus frequency as a function of/i^ (= 2.5,3.0 and 3 ..5) wiih 0 = 0 .6 ; the other parameters being the same as in above case We find that the lower frequency has a marked effect on tht stability of the wave, the growth rate increasing with hydrogen densities. As the frequencies increase, the lighter ion doc.snot have any effect on the growth rate of the wave.Figure 3 is a plot of the growth rate versus frequency as a function of the ratio of the temperatures T + to (= 2 O'  ^  ^and 0.5), for =1 and s  = 0.6; the parametershydrogen and electrons are the same as in Figure ILinear instabilities in multi-ion plasmas 5 6 5insitability of the wave starts at lower frequencies when the positively ionized oxygen is hotter = 2.0). Instabilitye^is in at the same frequency when = 1.0 and 0 .5 .However, the inspection of three curves shows that the growth laic IS. in general, larger when 7^, * T^..FrequencyI if jurc 2.  r i n l  o f  g r o w U i  r a l e  i r n / M  f i e q u e n u y  au a  f u n c t i o n  o f  ( -  2  5.  ' (I aiKl 3 5 )  w i l h  'I fi = 1 0  K e V ,  =  1 0  a n d  =  0  6 ,  i h c  o t h e r  p.ii.iim iL-is b e i n g  t h e  s a m e  a s  in  F i g u r e  1occurs at lower frequencies, with a correspondingly low growth rate.I'iRUrt* 3. Plot of growth rale versus frequency as a function of the i^ 'mix’iaturc ratio T , IT . ( = 2 0, 1.0, 0.5) with n , =n = 10 . n„ =0 I) "i'll, 1.0 KeV, f/ = 10 ) Tc = 1.0 KeV and 0 “ 0 .6 .Figure 4 depicts the variation in the growth rate versus l^equency as a function of [defined in (7)1 for 0 = 0.6, = ^0 (7„ = 1 .Kev. f/ = lO.OV 7^,). = «„- = *0 03eV)and7^sl.OKeV,Yo*isequalto/f^y^, where7 ,^ . is •be larmour radius of the O* ions. For low , the instabilityF r e q u e n c yF i g u r e  4 .  P ln l  o f  g i o w i h  r a l e  f r e q u e n c y  a s  u f u n c t i o n  o (  ( =.0 5 ,  0 7 5 ,  0  1) fo i  0 =  0  0 .  / i „  =  3 . 0 ,  =  I 0  K e v ,  (J =  1 0 . 0„ == n =  I 0  (7; , .  yjj. =: 0 3 c V )  a n d  1 0  K e VFinally, in Figure 5, the growth rate versus frequency as a function of , the negatively ionized oxygen density is shown. The other parameters for the figures a r e =1.0(7^,  “ 7"^-“ 0.03eV),A/,/ = 3.0 ,(r^= l OKcV andf;= lO.OV 7J^.)and7 -  l.O KeV. It IS noted that when = 0, the growth rate is low with instability setting in at the higher end of the frequency spectrum. However, when ^0{n^  = (175 and J .5), the instability sets in at very low frequencies. For the aforesaid parameters, the Buchsbaum mode given by (4) is almost a freely propagating mexie.Figure 5, Plot of growth rate versus frequency as a function ot (= 0 .0, 0 75 and 1 5); the other parameters being * l.O (T  ^= T',. = 0.03eV), Ai„ = 3 0 and T„ = IKcV and V » 10.0 = 1.0 KeV and 0 = 0.64, Conclusions566 C  P Anil Kumar. C Venugopal. B George and G RenukaW c  s t u d i e d  t h e  s t a b i l i t y  o f  e l e c t r o s t a t i c  w a v e s  i n  a  m u l t i - i o n  p l a s m a  m a d e  u p  o f  e l e c t r o n s ,  p o s i t i v e l y  a n d  n e g a t i v e l y  c h a r g e d  o x y g e n  i o n s  a n d  h y d r o g e n  i o n s  w h i c h  d r i f t  w i t h  a  v e l o c i t y  U  w i t h  r e s p e c t  t o  t h e  0 ^  i o n s .  T h e  t w o  w a v e  m o d e s  w e r e  c o n s i d e r e d :  t h e  l o w e r  h y b r i d  m o d e  a n d  t h e  l o n - i o n  h y b r i d  ( i.e .  B u c h s b a u m )  m o d e  w h o s e  f r e q u e n c i e s  w e r e  d e r i v e d  u s i n g  t h e  c o l d  p l a s m a  t h e o r y .  T h e  s t a b i l i t y  o f  t h e  l o w e r  h y b r i d  w a v e  i n c r e a s e s  w i t h  p a r a l l e l  p r o p a g a t i o n  a n d  h y d r o g e n  d e n s i t i e s .  I t  IS l a r g e r  w h e n  ^  t h a n  f o r  t h e  c a s e  w i t h  a n d  i ti n c r e a . s e s  w i t h  i n c r e a s i n g  ( w h e n  y ,  i s  t h e  p e r p e n d i c u l a r  w a v e  v e c t o r  n o r m a l i s e d  w i t h  0 +  i o n s  L a r m o u r  r a d i u s ) .  T h e  m o d e  IS d r i v e n  u n s t a b l e  a t  l o w e r  f r e q u e n c i e s  i n  t h e  p r e s e n c e  o f  n  a n d  n o  v a r i a t i o n  i s  l o u n d  w i t h  r e s p e c t  t o  t h e  d r i f t  v e l o c i t y  U  o f  t h e  h y d r o g e n  i o n s .  I t  m a y  b e  d u e  t o  t h e  f a c t  t h a t  n o  r e s o n a n c e  c o n d i t i o n s  w e r e  s a t i s f i e d  f o r  t h e  s e t  o f  p a r a m e t e r s  c o n s i d e r e d .  T h e  o t h e i  m o d e ,  t h e  B u c h s b a u m  m o d e  i s  a  f r e e l y  p r o p a g a t i n g  m i x i e .Rcrcrciu'c.s|l) J M Kindfl and C V  Kennel ./ Geophys Hes 76 3055 (1971)|2] M Ashoui Ahdalta, It Okuda and ,S Y Kim d c o n lm  U-U 14 .175 (l<>87)|.1] S J Duchsbaum Phy\. Fhnds .X 418 (!%())141 T  l \  T lw  Theory  o f  P laM iu i Waves (New York Mct'naw- Hitts)I5j(61(7|l«l[9|| 10 |III]|I 2 |[13]114][15]|I 6]|17|[18][I'l][20]F L Scarf, F V Coroniti, C F Kennel, P A Gumeti W H in J Smith N a tu re  321 377 (1986)  ^ ^S P Gary. C W Smith, M A Lee, M L Goldstein and D W l-v,Phys F lu id.'! 27 1852 (1984) ^ “"'10 P Sharma and V L Patel J  G e o p im . Re.'!. 91 1529 (loof,)S P Gary and D Winskc J  GVop/iy.v. Res. 91 13 699 A L Brinca and B T Tsurutani J. C eophys. Res. 93 243B T Tsurutani, A L Brinca, E J Smith, R M Thorne. F L ,S r 1 1 Gosling and F M Jpavich A.stroii. A s lrop lty.\. 187 92A L Brinca and B T Tsurutani G e o p im .  Res l e u  14(1987) ■V W C’how and M Ro,senberg P lm  Plasmas 3 1202 (I99(,)P Chaizy, H Reme, J S Saiivaud. Cd'u,ston, R P Lin, D E Larson d L Mitchell, R A Anderson, C W Carlson, A Korit and D A Mm 1 Mfl/iw .349 393 (1991)R Z Sagdeev, J Blamnul, A A Galcev V 1 Mcroz, V n Shapiro, V | Sherchenko and K Szego N a tu re  321 259 (1986)R Z Sagdeev, V 1) Shapiro and V J Szego Shovchenko Gea,./,u Re.s L e tt 13 85 (1986)F F Cheu I n t r o i l i i i t io i i  to P lasm a Phy.sic.s and  C o n iro lle k  Fiok/h (New York * Plenum) (1974) \P G Harries P hy\. Rev Le tt 2 34 (1959) iB 1) Fried and S D C’ontc The P lasm a D ispers ion  FunvUot^ (,San Diego, California Academic) (1961)w H Press, B P Flanneiy and W T Vatterling The A n  of Saeniili. C om p u tin s  (Camhiidgc Cambridge University Pre.sM f|9fit,lM Abramowiiz H and book o f  M a ih e m u tu a l F un riio ns  (New York Dover Publications) (1970)

Linear Instabilities in Multi-ion Plasmas

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Linear Instabilities in Multi-ion Plasmas

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