Baronia, A

Tiwari, M S

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Indian J. Phys. 73B (3), 499-507 (1999)I J P  Ban international journalP a r t ic le  a s p e c t  a n a ly s is  o f  A lfv e n  w a v eA Baronia and M S TiwariDepartment of Physics, Dr. H. S. Gour University, Sagar-470 003, Madhya Pradesh, India.Received 13 January 1998, accepted 28 December 1998Abstract The particle aspect analysis has been applied to evaluate the dispersion relation and field aligned current in the presence of Alfven wave. The applicability of theory for space plasmas has been indicatedKeywords Alfven wave, magnetosphere, plasma sheet PACS Nos. . 52 35.Hr, 94 lO.Rk1. IntroductionThe Alfven wave is the dominant low frequency transverse mode of a magnetized plasma. The Alfven wave propagates along the magnetic field and displays a continuous spectrum even in a bounded plasma. This is essentially due to the degeneracy of the wave characteristics, i.e., the frequency ((0) is primarily determined by the wave number in the direction parallel to the ambient magnetic field (£,,) and is independent of the perpendicular wave numbers. The direction along which the wave energy propagates, is identical to the ambient magnetic field lines. Therefore, the spectral structure of the Alfvdn wave has a close relationship with the geometric structure of the magnetic field lines. In an inhomogeneous plasma, the Alfven resonance (co~ CA /fj, = 0, CA is the phase velocity of the Alfvj6n wave) constitutes a singularly for the defining wave equation, this results in a singular eigenfunction corresponding to the continuous spectrum.It is of utmost interest to note that, in its simplest magnetohydrodynamic (MHD) description, the Alfv6n wave has a continuous spectrum even in a spatially bounded plasma [1-3]. The interaction of ions with Alfven waves in the solar wind has been extensively studied in space physics literature [4]. For wave observed on auroral field lines, ion trapping may be important and affects the existing models of transverse ion acceleration. The anomalous transport effects associated with the ion cyclotron turbulence can cause dissipation of the Alfv6n wave, leading to an energization of the plasma and the establishment of a double layer on auroral field lines.© 1999 LACS50() A Baronia and M S TiwariThe present analysis of Alfven wave is based on a physical model similar to that used by Dawson [5] in his theory of Landau damping which was further extended by Tcrashima [6], Mishra and Tiwari [7-9], Tiwari et al (10], and Varma and Tiwari (11-13] to the analysis of electrostatic and electromagnetic instabilities. In the present investigation, this well known approach has been applied for the evaluation of charged particle trajectory in the electromagnetic field of the Alfven wave and used to evaluate the dispersion relation to explain the current drive tor auroral acceleration region. The whole plasma is considered to consist of resonant and non-rcsonant particles. Non-rcsonant particles support the oscillatory motion of the waves while the resonant particles participate in the energy exchange with the wave. The Alfvdn wave is assumed to start at t = 0 when the resonant particles arc undisturbed. The trajectories of the particles are then evaluated within the framework of the linear theory. Using these particle trajectories in the presence of a Alfven wave, the field aligned current and dispersion relation of Alfven wave are evaluated with the help of Maxwell’s equations.2. Basic TrajectoriesWe consider a plasma under static magnetic field B % in which collisions between particles arc neglected. We shall discuss the behavior of an Alfven wave of plane polarization in the formKl \Bt). K B  = Q (1)J? = (0,().£| )a n d £  = ( £ k,0,0)with £ ( -  cos(/k|( z -co t) (2)E.K. cand = ---------cos (Kt z -(Ot ) , (3)where Ex and Bx are the electric and magnetic fields of the wave. Here, the frequency (0 is assumed to be real and the amplitude £ , is treated as the slowly varying function oft. c is the velocity of light.1 dE]--------L «  (0Ex dtB0 is directed along the 2-axis and wave propagates in the direction of ambient to the magnetic field. The wave is assumed to start at / = 0 when resonant particles are not yet disturbed. We begin with the equation of motion for the particles asdvm T r q£ ,+ -U x (f i0 + B v ) (4)where q is the charge and m e. the mass of the particle, y is the unit vector along ihc y direction.The Gaussian system of units is adopted in this paper and interactions between particles are neglected. The electric field E{ on the right hand side is considered to be a small perturbation and v can be expressed as a sum of the unperturbed velocity V and the perturbation velocity u, i.e., v = V + u, u is determined by the following set of equation for w.du _ 9- j — i O u + . dt + mV. K„coE, cos (Kt z-cat),Particle aspect analysis o f Alfven wave 501clu.. q E. K,—  = — V . ------ -  cos ( 9 -  Q t) cos (K., z -co t) ,at m co 1 (5)where u  ^ = it +IM. and Q  =_ ^ o/7/CWe solve cq. (5) in the approximation by replacing the coordinates of the particles on the right hand side by those of a free gyration, ux and w> are the perturbed velocities in thexand y directions respectively, E } and B^  are slowly varying quantities and treated as constants. Our method follows that ol Terashima f6J. Wc start taking the trajectories office gyration asVx(t)= VL cos (0 -jR f),Vx(t) = V± sin (0 -1 2 /), (6)Vm (/) = V|( = Constant,where z0 is the initial position of the particles in the z direction, 6 is the initial phase of gyration.To find an oscillatory solution of u (t) for the resonant particles, it is necessary to take into account the initial condition of u (/ -  0) inferred from the assumption slated above.u x(t) =m1-co- f  sin ( /l0/ +-------------si n (K..zn + f i t ) -2 (A0 - a )  110---------------sin(/i'.r’(. - £ 21)2iAn +Q)  110m,(0  = -inMl A', 1j 1! I E.COl - J -  COS (A0t + A '|:0 )s &cos (Knzn + £2l) -  — -----— cos (A",, - , -  £2/)2{A0 -12) 2 (A0 +£2)u , ( t ) =  C1 ’* m o) 2 (A0 -£2)sin (Aaf + KqZq + 0 -  £2t)1(A 0 + O)S\ a 0 - Q )sin (A0l + /TBc() -  0 + Q t )sin ( +  0 ) -(A„ + JQ)sin (K,,7.0 - 0 ) (7)502 A Baronia and M S Tiwariwhere 5 = 0 for the non-resonant particle and 5= 1 for resonant one andA 0 = V ,* ,,-© , al = A20 - Q 2 . (8)It is easy to calculate the true trajectory of the particle to first order by first integrating «(/) and then adding (6) to it. It follows that«(/) with S = 1 increases linearly if VH tf(| -  (o = 0. Therefore, the resonant particle condition is given as V, Kt -  a> = 0. If we again substitute (6) in (7), then u(t) can be written in the forma (r, t) asu (r,t) = — 11- " " -^-sin (K^z-Oit) m 1 (O-------sin(tfllz - t » / -  A nt+ Q t) ----------------- sin(K,,z- cot -  A nt - Qt)2 (A0 - Q )  " 0 2 (A0 + £2) " 0u ( r , t ) -  —  m1 —(0Q— cos (K^z-cot)Ans scosiK^z-cot-A0t+i2t)+-^~i— cosiK^z-wt-  A 0t-£2t)2(A0-£2) 2 (A0+£2)uz(r ,t) = i . S S S lm O) 2------------sin (K»z- (Ot + 9-£2t)(A 0 -£2 ) "sin (Knz - ( O t - 6  + £2t) -  (An + £2) 1sin (K. .z - (Ot-  A nt + 6) (A 0 -C2) 1 0(A0 + O ) Sin(K'Z- “ ' - /' » ' - e) <*>These equations represent the trajectories of charged particle in the presence of Alfven waves. The trajectories has vast application in plasma heating processes, confinement devices and the space plasmas.3. Density perturbationTo find the density perturbation associated with the velocity perturbation, we consider a group of the particles with the same initial condition and let its number density ben ( r j > V ) = N ( V )  + nx(r ,uV) .  (10)Where N  is the zeroth order distribution, n{ is perturbed density which can be derived from the equation of continuity.Particle aspect analysis of Alfvin wave 503The equation of continuity in the presence of Alfven wave can be written asSn.T 7 = - W '> y , “, (11)The expression on the right hand side can be expressed as function of variable f, for instance, V. uy can be written down in terms of “f” and the initial parameters with the.help of equation (6). After differentiating (9) with respect to “r ” and substituting in eq. (11), followed by integration we obtain the following solution for the non-resonant particle density.* j(r ,0  =q V±E, *n m toN(V)2------------=- sin (K„z-(ot + 6 - f 2 t )(A 0 -£2 )2 1sin { K . z - a J t - 6  + Qt )u 0 + a ySimilarly, for the resonant particles( 12)n.(r,r) = - q V±E\ Kl  N(V)m (01(An - £ 2 fsin ( K . z - e o t + 6 - Q t )+ -------— irsin ( K „ z - ( o t - 6 + Q t ) --------1— r-sin(K„z- (0 t -  Anf + 0)(A0 + f})2 " (A0 ~£2)2 "(A0 +f2)* ysin {K^z-cot -  A y t - 0 ) ------- ------ cos ( K ^ z - t o t - A 0t + 0)(A0 -£2)------- ------ cos (K„z- (o t -  A nt - 0 )(A0 + fl)  " 0 (13)To evaluate the dispersion relation and current we can hereafter take the zeroth order distribution N (VO of the formW(V)=W0/ X (VL) / I (VI) >mI n  T,exp j-m  V^2 / 2 T± | ,m2w 7jj jexp j-m  Vy2 /2 7 jJ ,(14)where 7j, and T± are the parallel and perpendicular temperatures with respect to ambient magnetic field and N0 is the plasma density.504 A Baronia and M S Tiwari4. C urrent densityTo evaluate the perturbed current density we use the following set of equations J  = 1^*11* d d j~  Vx e f ttf+ ^ K V + ^ -A T V ]a n d /= / , - / ,  .With the help of equations (9), (12) and (15), we obtain the ionic current asj  _ 3 e Mp, E\ n K\?VTh4 mJ(oQAlSimilarly, the electron current can be written asj  k i vL:e 4 mf(aa 4eSubstituting eq. (16) and (17) into eq. (15) we getj = _ 3^ |  _ ^*» 2 0)1 A  + . “A4 l me ; a) Q * Q 4(m / m j05)(16)(17)(18)and the average value of perpendicular current (l e ., Jx and Jx ) becomes zero. Here q = +e for ions and -  e for electrons (Offi ^ ( 4 n N 0e2 / m tc)U2 is the plasma frequency.VTUe ==(2T1 i e l mi e ) 12 is the perpendicular component of thermal velocity, 12 r is the cyclotron frequency and Ks = K.r5. Dispersion relationTo evaluate the dispersion relation, we use the Maxwell’s equations asc a t  c dtwhere / ,  is the first order current density.In the case of plane polarized Alfvdn wave, the Maxwell’s equation becomes(19)d*Ex _  4n dJx 1 <92£xd z l 2 d t  + c 2 ' d t 2 (20)Substituting the value of Jx in terms of perturbed velocity u{, the dispersion relation for the Alfven wave is evaluated asParticle aspect analysis o f Alfven wave 505(21)which reduces to well known expression of dispersion relation o> = K„ VA for Alfven waves when VTJ| = 0 and VA «  C. The thermal correction due to term VT]] is the finding of our particle aspect analysis theory. Here VA = # 0 / (4n  N0 m )m  is the Alfven velocity and Vrn = (2 T{UJ  me)m is the thennal velocity of electrons parallel to ambient magnetic field.Following the particle aspect approach and considering an anisotropic plasma, we have obtained the total current density carried by both electrons and ions, and evaluated the dispersion relation. In this model the wave frequency cuhas been considered real and the principal part of plasma dispersion function is used. However, considering imaginary parts of either asymptotic series or power series of plasma dispersion function, the dispersion relation would have been complex resulting the growth/damping of the wave. The present model considers the real part only to evaluate the charged particle trajectories and other parameters and the growth/damping rates are estimated by energy exchange method [ 10, 13] which is not the aim of the present paper.In this paper, we have not considered the coupling of compressional mode to the Alfven wave. The pressure variation along the direction of wave propagation has not been considered. The Alfven waves propagate along the direction of magnetic field by tilting the magnetic lines of force similar to the sound waves in a stretched string. However, m the case of perpendicular propagation, the magnetic field is compressed and rarefied resulting the pressure variation and hence coupling with the ion acoustic waves. Thus, in case of B = 0, the dispersion relation reduces to imaginary a), a non-propagating mode.6. Results and discussionWe have evaluated the dispersion relation and current density using the following parameters for auroral acceleration region [14]Expressions (18) and (21) have been evaluated numerically and results are presented in Figures I and 2. Although many theoretical attempts have been made to apply the Alfv<5n wave propagation to magnetospheric plasmas, most of them lack of a careful check as to the applicability of the theory. The Alfven waves have vast application to heat a Tokamak-type plasma to the thermonuclear temperatures.Figure 1 shows the variation of real frequency ruwith K(j for different values oi VT Here we notice that wave frequency increases linearly with but decreases with the increase of Vr[\ . It is also clear that the slope of curve will give the phase velocity of wave and phase velocity decreases with VTl]. These curves show that the phase velocity is less in hot plasmas.Figure 2 shows the variation of current with/f„ for different values of Vr r It is seen that current increases with as well as It is also clear that the current decreases with phase velocity of Alfven wave.= 4300//T, — 412.v . « £ / « ? =  100,E± =  5 0  mV /  m, Vn  f =  3.5  x  10 4 in  /  se c .506 A Baronia and M S TiwariIn the present paper, we have investigated the effect of thermal velocity on the dispersion relation and current driven by the Alfv6n wave. In the past, the Alfv6n waves are mostly analyzed for the cold plasmas ignoring the thermal corrections. The frequency of micropulsations noted at the sub-storm times in the magnetospheric-ionospheric coupling system and field- aligned current may depend upon the thermal particles streaming along the auroral field lines. Further the current driven by Alfvdn waves in Tokamak plasmas may be also affected by the thermal velocity. The trajectory of charged particle so derived may be useful for ion-pickup processes in interplanetary plasmas [4].Fundamentally, great interest of study is in the magnetospheric plasmas and in the auroral acceleration region where field aligned current can be estimated through the use of ourP artic le  a sp ec t analysis o f  A lfven  w ave 507study. In the present analysis, we have investigated the variation of current density in the presence of Alfven wave in hot plasmas.The theory may be useful for the plasma heating processes, confinement devices and space plasmas. The Alfv6n wave is fundamentally important for supplementary heating of Tokamak-type plasmas. With the help of these study, one can explain the formation of the stable auroral arcs as well as viscous interaction between the solar wind and the magnetosphere.AcknowledgmentWe are thankful to MAPCOST, Bhopal for financial assistance.R eferences[1] H Grad Phys Today 22 34 (1969)[2] J P Goedblod Phys. Fluids 18 1258 (1975)[3] T Kako Math Mech. Appl Sci 7 432 (1985)[4] C S Wu, P H Yoon and J K Chao Phys. Plasma 4 856 (1997)[5] J M Dawson Phys. Fluids 4 869 (1961)[6] Y Terashima Progr. Theor. Phys 37 775 (1967)[7] K D Misra and M S Tiwan Phys Scnpta 16 142 (1977)[8] K D Misra and M S Tiwan Can J. Phys 57 1124 (1979)[9] K D Misra and M S Tiwari Physica 98C 325 (1980)[10] M S  Tiwan, R P Pandey and K D Misra J Plasma Phys 34 163 (1985)[11] P Vanna and M S Tiwari Phys Scnpta 44 296 (1991)[12] P Varma and M S Tiwan Phys. Scnpta 45 275 (1992)[13] M S Tiwan and P Varma Planet Space Sci 41 199 (1993)[14] M S  Tiwari and G Restoker Planet Space Sci. 32 1497 (1984)

Particle Aspect Analysis of Alfven Wave

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Particle Aspect Analysis of Alfven Wave

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