Dust-acoustic waves and stability in the permeating dusty. plasma: II. Power-law distributions

Transcription

1 arxv:.645 Dust-acoustc waves an stablty n the permeatng usty plasma: II. Power-law strbutons Jngyu Gong, Zhpeng Lu,, an Juln Du a) Department of Physcs, School of Scence, Tanjn Unversty, Tanjn 37, Chna Department of Funamental Subject, Tanjn Insttute of Urban Constructon, Tanjn 3384, Chna Abstract The ust-acoustc waves an ther stablty rven by a flowng usty plasma when t cross through a statc (target) usty plasma (the so-calle permeatng usty plasma) are nvestgate when the components of the usty plasma obey the power-law -strbutons n nonextensve statstcs. The freuency, the growth rate an the stablty conton of the ust-acoustc waves are erve uner ths physcal stuaton, whch express the effects of the nonextensvty as well as the flowng usty plasma velocty on the ust-acoustc waves n ths usty plasma. The numercal results llustrate some new characterstcs of the ust-acoustc waves, whch are fferent from those n the permeatng usty plasma when the plasma components are the Maxwellan strbuton. In aton, we show that the flowng usty plasma velocty has a sgnfcant effect on the ust-acoustc waves n the permeatng usty plasma wth the power-law -strbuton. a) Electronc mal: julnu@yahoo.com.cn

2 I. INTRODUCTION As revewe n paper I, the permeatng (nterpenetratng) usty plasma system conssts of two parts; one part s the flowng usty plasma, the other s the target usty plasma beng n a relatve statc state. The flowng usty plasma moves through the target usty plasma when they are encounter n space. The ust-acoustc waves an ther stablty n the usty plasma can be analyze on the bass of the relate statstcal theory. It has been observe that the electrons, ons, an ust grans ether may be the Maxwellan strbuton n some physcal stuatons near eulbrum, an they may also be the non-maxwellan strbutons n many physcal stuatons far away from eulbrum -8. If the components of usty plasma are the non-maxwellan strbutons or the power-law -strbutons that can be escrbe by nonextensve statstcs, the characterstcs wll be fferent from those escrbe n the realm governe by the tratonal statstcal mechancs 9-3. Recently, the plasma wth the power-law -strbutons have been nvestgate n the framework of nonextensve statstcal mechancs, such as the basc characterstcs of the plasma 4-3, the solar wn an space plasma 4-3, an the astrophyscal plasma 3-36, etc. It has been known that the power-law -strbutons have represente some basc characterstcs of the plasma n complex systems away from eulbrum. In paper I, we have nvestgate the ust-acoustc waves n the permeatng usty plasma an ther stablty f the components of the usty plasma obey the Maxwellan strbutons n the tratonal statstcs. In ths paper, we nvestgate the above characterstcs of the permeatng usty plasma f the components obey the power-law -strbutons n nonextensve statstcs. The basc theory for the ust-acoustc waves of the plasma s escrbe n Sec.II. The characterstcs of the permeatng usty plasma wth the power-law -strbuton are nvestgate n Sec.III. In Sec.IV, we present the numercal nvestgaton on these characterstcs, an fnally n Sec. V, we summarze the conclusons.

3 II. THE BASIC THEORY FOR THE DUST-ACOUSTIC WAVES IN THE DUSTY PLASMA Frstly, let us present a bref revew on the basc euatons an the sperson functon for the ust-acoustc waves n a collsonless, unboune, unmagnetze usty plasma, compose of electrons, ons an usty grans. The knetc euaton for the partcles can be wrtten by the lnearze Vlasov euaton 36, where f (,, t) f t Q + v f + E f =, () v m rv enotes a perturbaton about the eulbrum strbuton f ( v) of the partcles, wth = e,, stans, respectvely, for the usty gran, the on an the electron. E s the electrc fel prouce by the perturbaton, an Q s the charge of the component. Corresponngly, the lnearze Posson euaton s E Q f v. () = ε Because f ( rv t) ( k r t an ( t ),, exp ) E exp k r, by makng Fourer transformaton for r an Laplace transformaton for t n the Es.() an (), one obtans f + k vf + E f =, (3) v m k E f v. (4) = ε If the wave vector k s along x-axs, vx = u, the oscllatng freuency s = an the normalze eulbrum strbuton s fˆ = f n, p nq εm accorng to Lanau path ntegral 38, t s erve that the sperson relaton s + Or, euvalently, t can be wrtten as ˆ p f u u k k u =. (5) 3

4 where the physcal uantty ε (, k ) = + χ =, (6) ˆ p f u χ = u k (7) k u s calle polarzablty. Base on the above theory, we can stuy the ust-acoustc waves an ther stablty n permeatng usty plasma. III. THE DUST-ACOUSTIC WAVES AND STABILITY IN THE PERMEATING DUSTY PLASMA WITH POWER-LAW DISTRIBUTION If the permeatng usty plasma s not near thermal eulbrum, but s far from the eulbrum, the partcles wll evate from the Maxwellan strbuton, an n many physcal stuatons they may be the kappa strbuton, whch can be escrbe euvalently by the power-law -strbuton n nonextensve statstcs 9,4-6. The power-law velocty -strbuton 9,4-6 for one-mensonal case can be wrtten by fˆ = fˆ = A v v ( ) ( ) π v v T T (8) wth an A A = Γ Γ Γ + + = Γ for <, for >. Ths power-law velocty -strbuton recovers the Maxwellan velocty strbuton E.(8) gven n paper I when t s n the lmt. The -parameter n ths strbuton may be relate to the temperature graent T an the Coulomban potental ϕ by the followng relaton 7,9,4-6, B ( ) k T + Q = (9) ϕ 4

5 where kb s the Boltzmann constant, an Q s the charge of the component. In the unmagnetze ust plasma, the magntue of potental graent ϕ s eual to that of electrc fel E, but the sgn s opposte. The electrc fel s etermne by the lnearze Posson euaton E.(). Thus, the nonextensve parameter as well as the power-law -strbuton s shown to escrbe the characterstc of the plasma system beng a noneulbrum statonary-state when t s away from the thermal eulbrum. Uner ths power-law velocty -strbuton, the sperson functon for the plasma s generalze 9,5,6,3 by ( ) A x Z ( ξ) = x () π x ξ The ntegran n E.() has the same sngular pont as that n E.(9) n paper I,.e. x = ξ on the x-axs. Accorng to Lanau ntegral path, we can get ( ) x A ( ) Pr ( ) ξ = + π ξ π x ξ Z x A. () Then the sperson relaton E.(5) can be generalze as ( A ) x + + ξ Pr x A ( ) ξ π + k λd + π x ξ ξ =, () whch can be wrtten as ε (, k) =, also calle the electrc functon. We now conser the same physcal stuaton as that for Es.(4)-(5) n paper I. An we stll let the target plasma be statc, vs = an every component of the flowng plasma be wth the same velocty,.e. v = f vf, an we make use of the conton: vts vφ vts, vtse an v v φ f vts, vts, v, where v Tse φ = k s the phase velocty of the ust-acoustc wave. In ths way, the real part E.(6) an the magnary part E.(7) of the electrc functon n paper I are now generalze to the permeatng usty plasma wth the power-law -strbuton. The real part becomes 5

6 ε r, + k ( k ) an the magnary part becomes ε π 3kv ps Ts ps 4 λd ( 3s ), (3) s s ps ps pse (, k) As 3 3 ( s ) + A s + A 3 3 se 3 3 kv Ts kv Ts kvts kvtse pf pf pfe + ( kv f ) Af + A 3 3 f + A 3 3 fe 3 3, (4) kvtf kvtf kv Tfe = Let E.(3) be ε, =, s se f f fe where λ D λds λdse λdf λdf λdfe we can get the sperson functon of the ust-acoustc waves, ( 3 ) ( + )( 3 ) + ( 3 ) r s psλd = k s s s k λd + + 3v Ts ( + )( 3 ) + ( 3 ) ( 3 ) λ k λ s s s D s ps D ( ) r r k. (5) For the case of vts vφ vts, vtse, an for s /3, we obtan the freuency, r = ps kλ D + k λ D, (6) where we have scare the small fourth term on the rght se of E.(3). Thus we can wrte the normalze freuency as r nseqsets f f s = + ( + se ) + ( + f ) ps k λds ns QsTse ns QsTf n Q T nfqfts nfeqfet s ( f ) ( fe ) ( s ) , (7) nsqstf nsqstfe where the parameter s sappear, whch mples that the -strbuton of the usty gans n the target plasma oesn t nfluence the normalze freuency. The growth rate of the ust-acoustc waves can be obtane from E.(3) an E.(4). It s erve that 6

7 γ ε = ε r (, k ) r = r π = s s 3 s r A 3 s r r As rps 3 3 ( s ) 3 3 kvts kvts kvts kvts ps r Usng v Ts = normalze growth rate as 3 3 Aser pse r r Afpf Afpf Afe pfe v 3 f (8) kvtse ps k k vtfps vtfps v Tfeps ps λ Ds an substtutng E.(7) nto E.(8), we can wrte the γ wth the -relate parameters: ps v f = B C + D, (9) v φ 3 π nsqst s + s + se nseqsets = λds Ds nsqst λ + + s nsqstse B k k + f nf QfTs + f nfqfts + fe nfeqfet s nsqstf nsqstf nsqstfe n Q T n Q T s n Q T n Q T s s s s se se se s k λds + + s s s s s se + f nf QfTs + f nfqfts + fe nfeqfet s nsqstf nsqstf nsqst fe, () C f f f f s f f f f s fe fe fe fe s ns Qs mstf ns Qs mstf ns Qs mstfe A n Q m T A n Q m T A n Q m T = + +, () an ( ) D = A n Q T k λ n Q T s s s s s se se se s s Ds nsqst s nsqstse s s + f nf QfTs + f nfqfts + fe nfeqfet s nsqstf nsqstf nsqstfe 7

8 3 3 s s s s s se se se se s 3 3 s s s s s s s se A n Q m T A n Q m T + +. () n Q m T n Q m T From E.(9) we fn that the nstablty conton of the ust-acoustc waves n the permeatng plasma wth the power-law -strbutons s gven by v D > v + C f φ. (3) As compare wth E.(7) n paper I, the phase velocty v φ, the parameters C an D now epen on three -parameters for the three components wth the power-law -strbutons. E.(7) n paper I can be recovere by E.(3) only when the three -parameters are set to unty. If the ust plasma system s wthout the permeatng phenomenon, E.(8) becomes γ π = + +,(4) r A r r A r p Ae r pe 3 3 ( ) kvt kvt kvt kvtsp kvtsep r where we have use the conton, v v v, v. T φ T Te Agan we conser the physcal stuaton,.e. f the flowng veloctes v f are fferent from each other for each component of the flowng usty plasma, we have vφ v + v, v + v, v + v Ts f Ts f Tse fe, an the growth rate E. (8) s revse as γ π = s s 3 s r A 3 s r r As rps 3 3 ( s ) 3 3 kvts kvts kvts kvts ps r 3 3 Aser pse r r Af pf + + v f 3 kvtse ps = e,, k k vtf. (5) ps Thus E.(9) becomes 8

9 where γ v f = B D + C, (6) ps = e,, v φ Agan usng m C 3 f f f s 3 s s s f n Q m T = Af. (7) n Q m T m, me, then C an C e are small as compare wth C so that they can be scare n E.(6). The nstablty conton of the ust-acoustc waves now becomes v f D > v φ + C. (8) It s clear that the parameters s, s, se, f may play mportant roles n the nstablty conton of the waves. IV. NUMERICAL CALCULATIONS In orer to llustrate our theoretcal results more clearly, we now perform numercal calculatons for the freuency an the growth rate of the ust-acoustc wave, an the crtcal flowng velocty for the nstablty conton n the permeatng usty plasma. For the numercal calculatons on the permeatng usty plasma wth the power law -strbuton, we conser the same physcal case as those n paper I, n whch the usty plasma matches the contons for the partcle s number enstes, masses, an charges,.e. the partcle s number ensty of each component at the eulbrum satsfes ns = nf = n, ns = nf = n, an nse = nfe = ne; the partcle s mass of each component satsfes ms = mf = m, ms = mf = m an m se = m fe = m e, an the number ensty rate between the on an electron s n n = δ at the e eulbrum. Typcal ensty parameters of ust-laen plasma n nterstellar clous 39-4 are n e an cm n cm. The ust grans n nterstellar clous are electrc (ces, slcates, etc) an metallc (graphte, magnette, amorphous carbons, 9

10 etc), an the ra of the ust grans are approxmately r.μm for the ces. The mass rate between the electron, on an ust gran stll may be m m m 4 9 e : : : : eulbrum are = e, an Qe. The charge of the electron, on an ust gran at the Q = ( ) 4 Q = δ e base on the uasneutral conton. In aton, we assume the -parameters for the ust grans, the ons, an the electrons satsfy s = f =, s = f =, an se = fe = e. Uner the contons of Ts Ts T se an Tf Tf T fe, gnorng the nfntesmal uanttes n the relate euatons n Es.(7), (9) an (3), we can wrte the normalze freuency as ps = k λ the growth rate of the waves as kλ Ds ( δ ) + T T s s 4 Ds T f δtf, (9) γ ps = k λ A πkλ Ds ( δ ) δt 3 ( δ ) s s 4 Ds s 4 ( )( δ ) Ts δts k T s 4 + T + T T δt f f ( δ ) + Ts + T s 4 λds T f δtf 3 v f T s A vϕ Tf +, (3) an the nstablty conton for the ust-acoustc waves as v / / f v > + ϕ D C, (3)

11 where the crtcal value can be expresse by D T T C 3 4 ( )( δ ) s s = ( ) T T s 4 f Ts δ + + δts k λds T f δt f. (3) Base on these euatons, the numercal results are shown n Fg.-. In these fgures, we llustrate the nonextensve parameters of the usty plasma an ther nfluences on the normalze freuency, the growth rate of the ust-acoustc wave an the nstablty conton for the waves by takng δ =., T = K, T = K, s f 3 T s = K an 4 T f = K. Fg. an Fg. llustrate the numercal results of the generalze normalze freuency, E.(9), for the ust-acoustc waves as a functon of the normalze waves number kλ Ds, whch s plotte wth fferent values of the nonextensve parameters for the ust grans an for the ons. They show that the generalze normalze freuency s not very sgnfcantly epenent on the nonextensve parameters an. The curves for = = are the normalze freuency for the Maxwellan strbuton (the sol lnes). Fg.3 an Fg.4 llustrate the numercal results of the nstablty crtcal value of the normalze flowng plasma velocty base on E.(3) for the ust-acoustc waves. The nstablty crtcal value s plotte as a functon of the normalze wave number kλ wth fferent values of the nonextensve parameters for the ust grans an Ds for the ons. The cures for = = are the cases for the Maxwellan strbuton (the sol lnes). We show that the crtcal value of the flowng plasma velocty s sgnfcantly epenent on the nonextensve parameters an only when the normalze wave number kλ Ds s small.

12 Fg.. The normalze freuency ps s plotte as a functon of the normalze wave number kλ for fferent values of at =. The curves are rawn for Ds =.8(long-ashe lne), =.9(ashe lne), = (sol lne), =. (ot- ashe lne) an =.(otte lne). Fg.. The normalze freuency ps s plotte as a functon of the normalze wave number kλ for fferent values of at =. The curves are rawn Ds for =.8 (long-ashe lne), =.9 (ashe lne), = (sol lne), =. (ot-ashe lne) an =. (otte lne).

13 Fg.3. The nstablty crtcal values of the normalze flowng plasma velocty vf v φ for the ust-acoustc waves s plotte as a functon of the normalze wave number kλ for fferent values of at =. The curves are rawn for Ds =.8 (long-ashe lne), =.9 (ashe lne), = (sol lne), =.(ot-ashe lne) an =.(otte lne). Fg.4. The nstablty crtcal values of the normalze flowng plasma velocty vf v φ for the ust-acoustc waves s plotte as a functon of the normalze wave number kλ Ds for fferent values of at =. The curves are rawn for =.8(long-ashe lne), =.9(ashe lne), = (sol lne), =.(ot-ashe lne) an =.(otte lne). 3

14 Fg.5-8 are the numercal results of the growth rate γ ps as a functon of kλ Ds of the ust-acoustc waves base on E.(3) for the fferent nonextensve parmeters when we let the rate between the velocty of the flowng usty plasma an the phase velocty be vf v φ = 4, where = s the case for the Maxwellan strbuton. Uner ths conton, the growth rate γ ps s always postve an thus the ust-acoustc waves are unstable. In Fg.5, the growth rate γ ps s plotte for the fferent values of when we take =, whch shows that the nonextensve effects generally lea to the growth rate to become ecrease. In Fg.6, the growth rate γ s for fferent values of when we take =, whch shows that the ps nonextensve effects wll lea the growth rate to become ecrease f > an to become ncrease f <. In Fg.7, the growth rate γ ps s for the fferent values of when we take =, =, whch shows that the nonextensve effects wll s f lea the growth rate to become ecrease f s > an to become ncrease f s <. In Fg.8, the growth rate γ ps s for the fferent values of f when we take =, =, whch shows that the nonextensve effects wll lea the growth rate to s become ecrease f f < an to become ncrease f f >. Fg.9 an Fg. are also the numercal results of the growth rate γ ps as a functon of kλ base on E.(3) for the fferent values of an when we Ds let the rate between the velocty of the flowng ust plasma an the phase velocty be vf v φ = 5. Uner ths conton, the growth rate γ ps s postve an thus the ust-acoustc waves are unstable f the wave number s small; but the growth rate γ ps s negatve an thus the ust-acoustc waves are stable f the wave number s large. In Fg.9, the growth rate γ ps s for the fferent values of when we take 4

15 =, whch shows that the nonextensve effects wll lea the growth rate to become ecrease f > an to become ncrease f <. In Fg., the growth rate ps γ s for the fferent values of when we take =, whch shows that the nonextensve effects have the nfluence on the growth rate only when the wave number kλ s small an lea the growth rate to become ecrease f > an to Ds become ncrease f <. Fg.5. The growth rate of the ust-acoustc waves γ ps s plotte as a functon of the normalze wave number kλ for fferent values of at = an v v φ = 4. Ds f The curves are rawn for =.8 (long-ashe lne), =.9 (ashe lne), = (sol lne), =. (ot-ashe lne) an =. (otte lne). 5

16 Fg.6. The growth rate of the ust-acoustc waves γ ps s plotte as a functon of the normalze wave number kλ for fferent values of at =, Ds v v φ =. The curves are rawn for =.8 (long-ashe lne), f 4 =.9 (ashe lne), = (sol lne), =. (ot-ashe lne) an =. (otte lne). Fg.7. The growth rate of the ust-acoustc waves γ ps s plotte as a functon of the normalze wave number kλ Ds for fferent values of s at =, f =, vf v φ = 4. The curves are rawn for s =.(long-ashe lne), s =.6(ashe lne), s = (sol lne), s =.4(ot-ashe lne) an s =.8(otte lne). 6

17 Fg.8. The growth rate of the ust-acoustc waves the normalze wave number γ ps s plotte as a functon of kλ Ds for fferent values of f at =, s =, vf v φ = 4. The curves are rawn for f =.8(long-ashe lne), f =.9 (ashe lne), f = (sol lne), f =.(ot-ashe lne) an f =.(otte lne). Fg.9. The growth rate of the ust-acoustc waves γ ps s plotte as a functon of the normalze wave number kλ for fferent values of at = Ds an vf v φ = 5. The curves are rawn for =.8(long-ashe lne), =.9 (ashe lne), = (sol lne), =.(ot-ashe lne) an =.(otte lne). 7

18 Fg.. The growth rate of the ust-acoustc waves γ ps s plotte as a functon of the normalze waves number kλ for fferent values of at = Ds an vf v φ = 5. The curves are rawn for =.8(long-ashe lne), =.9 (ashe lne), = (sol lne), =.(ot-ashe lne) an =.(otte lne). We state that Fgs.5-8 an Fgs.9- are nvestgate respectvely for the two rate between the velocty of the flowng usty plasma an the phase velocty,.e. v v φ = an v v φ = 5. By comparng the results n the two cases, we fn that f 4 f the fference n the flowng usty plasma velocty has a ute sgnfcant effect on the growth rate an the stablty of the ust-acoustc waves n the permeatng usty plasma wth the power-law -strbutons. V. CONCLUSIONS In concluson, we have nvestgate the freuency, the growth rate an the stablty of the ust-acoustc waves n the permeatng usty plasma wth the power-law -strbutons. The freuency of the ust-acoustc waves s erve by E.(7), the growth rate s expresse by E.(9), an the stablty conton s obtane as E.(3) when the flowng veloctes v f are consere the same for each component of the flowng usty plasma. When the flowng veloctes v f are fferent from each other for each component, the growth rate an the nstablty 8

19 conton are revse as E.(6) an E.(8). These euatons have expresse the effects of nonextensvty n the permeatng usty plasma as well as the flowng velocty of the flowng usty plasma on the ust-acoustc waves f the usty plasma obeys the power-law -strbuton n nonextensve statstcs, whch s shown to have fferent characterstcs of the ust-acoustc waves from those when the plasma s assume to obey the tratonal statstcs wth the Maxwellan strbuton. We further perform the numercal nvestgatons on the above characterstcs of the usty plasma. The results llustrate fully the nonextensve effects ( fferent from unty) of each component of the permeatng usty plasma on the freuency an the growth rate of the ust-acoustc waves an ther stablty conton. An we o those for two fferent veloctes of the flowng usty plasma, an we fn that the fference n the flowng usty plasma velocty has a ute sgnfcant effect on the characterstcs of the ust-acoustc waves n the permeatng usty plasma wth the power-law -strbutons. ACKNOWLEDGEMENTS Ths work s supporte by the Natonal Natural Scence Founaton of Chna uner Grant No an No.758. References J.Y.Gong, Z. P. Lu, an J.L.Du, paper I: arxv:.64. J. D. Scuer, Astrophys. J. 47, 446 (994). 3 A. Hasegawa, K. Mma, an M. Duong-van, Phys. Rev.Lett. 54, 68 (985). 4 S. R.cranmer, Astrophys. J., 58, 95 (998). 5 M. N. S. Quresh, G. Palloccha, R. Bruno, et. al, n Solar Wn Ten: Proceengs of the Tenth Internatonal Solar Wn Conference, ete by M. Vell, R. Bruno, an F. Malara (Amercan Insttute of Physcs, 3), p R. A. Treumann, C. H. Jaroschek, an M. Scholer, Phys. Plasmas,, 37 (4). 7 J. L. Du, Europhys. Lett. 75, 86 (6). 8 M. R. Coller, A. Roberts, an A. Vnas, Av. Space Res. 4, 74 (8). 9 Z. P. Lu an J. L. Du, Phys. Plasmas 6, 377 (9). T. K. Baluku, M. A. Hellberg, I. Kouraks, an N. S. San, Phys. Plasmas 7, 537 (). M. Trbeche, M. Bacha, Phys. Plasmas 7, 737 (). M. Trbeche, A. Merrche, Phys. Plasmas 8, 345 (). 3 R. Amour, M. Trbeche, Phys. Plasmas 8, 3376 (). 4 J. L. Du, Phys. Lett. A 39, 6 (4). 9

20 5 L. Y. Lu an J. L. Du, Physca A 387, 48 (8). 6 Z. P. Lu, L. Y. Lu, J. L. Du, Phys. Plasmas 6, 7 (9). 7 A. S. Bans, M. Trbeche, an T. S. Gll, Phys. Plasmas 8, 8 (). 8 A. S. Bans, M. Trbeche, T. S. Gll, Phys. Lett. A 375, 59 (). 9 L. A. Gougam, M. Trbeche, Astrophys. Space Sc. 33, 8 (). S. Shakh, A. Khan, P.K. Bhata, Astrophys. Space Sc. 3, 35 (7). M. Trbeche, L. Djebarn, R. Amour, Phys. Plasmas 7, 44 (). H. R. Pakza, M. Trbeche, Astrophys. Space Sc. 334, 45 (). 3 M.-J. Lee, Phys. Plasmas 4, 3 (7) 4 M. P. Leubner, Astrophys. Space Sc. 8, 573 (). 5 M. P. Leubner, Phys. Plasmas, 38 (4). 6 M. P. Leubner an Z. Vörös, Non. Proces. Geophys., 7 (5). 7 G. Lvaots an D. J. McComas, J.Geophys Res. 4, A5 (9). 8 M. Letner, Z. Vörös, an M. P. Leubner, J.Geophys Res. 4, A4 (9). 9 J. L. Du an Y. L. Song, n Proceengs of the Thr UN/ESA/NASA Workshop on the Internatonal Helophyscal Year 7 an Basc Space Scence, ete by H. J. Haubol an A. M. Matha (Astrophyscs an Space Scence Proceengs, Sprnger-Verlag, Berln Heelberg, ). 3 M. Letner, C. J. Farruga, Z.Vor os, J. Atmosph. Solar-Terres. Phys. 73, 9 (). 3 A. Lavagno, P. Quarat, Astrophys. Space Sc. 35, 53 (6). 3 S. Shakh, A. Khan, P. K. Bhata, Z. Naturforsch. A 6, 75 (6). 33 S. Shakh, A. Khan, P.K. Bhata, Phys. Lett. A 37, 45 (8). 34 M. P. Leubner, Nonln. Proces. Geophys. 5, 53 (8). 35 A. M. Scarfone, P. Quarat, G. Mezzoran, M. Lssa, Astrophys. Space Sc. 35, 353 (8). 36 M. Lazar, S. Poets, R. Schlckeser, Mon. Not. R. Astron. Soc. 4, 663 (). 37 D. G. Swanson, Plasma Waves, n Eton. (IOP, Brstol, 3). 38 L. Lanau, J. Phys. USSR, 5 (946). 39 T. W. Hartust, W. Plpp an O. Havnes, Astrophys. Space Sc. 46, 43 (996). 4 D. A. Mens an M. Rosenberg, Ann. Rev.Astron.Astrophys. 3, 49 (994). 4 P. K. Shukla an A. A. Mamun, Introucton to Dusty Plasma Physcs. (IOP, Brstol, ).