J. Plasma Fusion Res. SERIES, Vol.5 (2002) 454458

Studies on Modes in Weakly Relativistic Magnetized Plasmas
MALIK Hitendra K. and NISHIDA Yasushi
Energy and Environmental Science, Graduate School of Engineering
Utsunomiya University, Tochigi 321-8585, Japan(Received: ll December 2001 / Accepted: 18 June 2002)

Abstract
Considering magnetized homogeneous and inhomogeneous plasmas with weakly relativistic ionsand electrons we derive the dispersion relations by carrying out the usual mode analysis of the ion andelectron fluid equations. In addition, the reductive perturbation technique is employed to derive theexpressions for the phase velocities of the modes by using a small dimensionless expansion parameter t
for the dependent quantities. Three types of modes in inhomogeneous plasmas and two types of modes inhomogeneous plasmas are found to occur and usual mode analysis confirms their acoustic nature. In
homogeneous plasmas, the possibility of instability is predicted on the basis of mode analysis. The
conditions on modes propagation in the plasmas are achieved and the effects of various parameters on
their phase velocities are analyzed. It is suggested that all these modes should be taken into account forthe studies on solitons.

Keywords:
magnetized plasma, relativistic ions, relativistic electrons, modes, dispersion relation, reductive perturba-tion technique, phase velocity

solitons in different plasma systems including therelativistic plasmas [5-10] which can be realized in theplasma sheet boundary layer of the Earth's magnetosphere, solar wind, solar flare and under the influenceof high-power laser radiation. However, most of thestudies on solitons are limited to some particular mode
(mainly fast mode), whereas other modes may also
occur in the plasmas under the effect of magnetic field

1. Introduction
There are a variety of equations that govern theevolution of nonlinear phenomena in plasmas. The
nonlinear phenomena that may be observed in higherregions of the near-Earth plasma can be of different
kinds e.g. heating type, parametric type etc. and anotherimportant class of such phenomena is solitons [1].
Production and propagation of ion acoustic solitons
were studied by Ikezi [2], and Dahiya et al. l3l observed
their partial reflection from a negatively biased gridand can correspond to the solitons of different
behaviour. Therefore, our main concern in the present

immersed in the plasma. Later, Nishida [4] madepaper is to study the different types of modes inextensive studies on the soliton reflection from a planar
metallic plate, glass plate and metallic mesh.

relativistic magnetized plasmas in view of soliton
propagation.

For a weakly nonlinear and dispersive plasma, a2. Basic Formulation: Fluid Equations

time-dependent perturbation leads to the well-known
Korteweg-deVries (K-dV) equation which describes
one-dimensional solitary waves (solitons). Using K-dV
equation, many researchers have studied the ion acoustic

We consider a magnetized and spatially inhomogeneous plasma having weakly relativistic ions and@20O2 by The Japan Society of Plasma

Corresponding author's e-mail: hkmalik@ plasma.ees.utsunomiya-u.acjp
454

Science and Nuclear Fusion Research

Malik H.K. et al., Studies on Modes in Weakly Relativistic Masnetized Plasmas

and ft for the present case where ions as well electrons
are taken to be weakly relativistic in an inhomogeneous
magnetized plasma. If we consider a simplified casewhere ions and electrons are nonrelativistic (v; = 0 andTi = l) and the plasma is homogeneous (ze. - 0 and @s.
= 0) and unmagnetized (Q = 0), the above equationelectrons. The ratio ofparticle pressure to magnetic fieldpressure (F) is taken to be small so that the diamagnetic

effect is small and a uniform extemal magnetic field B(= Boz) along z-axis is assumed. The wave propagationis considered to be in the (.x, z) plane. Under these
conditions, the following basic equations are obtained
for the ion and electron fluids.

0nt/0t+\'.(n1v)=O,n1 m1d(y1assumes the following form

(1)

I-@pj2lf@2-PcrTrlmrl- awzlla2 - k2crT"lm"] = o .

- C1T1Yn1,- eoY2Q = e (ni - n") .

If the electron mass ,??e is neglected in eq. (5a), the following relation between a and k is obtainedQ)(3)

61ft=fC{ilmi+ C"T"lm1(l + l8L*2)lrt2. (5b)Here, eq. (1) is the continuity equation and eq.
(2) is the momentum equation for ion ("t = i and s = +)and electron (,1 = e and s - -) fluids, with n, as thedensity, mi as the Dass, v; as the fluid velocity, Ti = (l v2lc2)-u2 as the relativistic factor, Q as the specific heatratio and { as the kinetic temperature. Equation (3) isthe Poisson's equation with @ as the electrostatic
potential. It can be noted that the fluid equations forelectrons with rclativistic effect have been introduced inplace of their Boltzmann distribution, since the wave
velocity in the relativistic plasma can be comparable tothe electron therrmal velocity [7].
Now we carry out the usual mode analysis of eqs.
(1)-(3) to find the dispersion relation for linear waves.
The wave propagation is taken to be almost perpendicular to the direction of magnetic field (z-axis) sothat the ions and electrons can preserve the charge
neutrality. We assume all the perturbations of the form

This relation is same as obtained by Chen [eq. (4-48) of ref. [1]l for ion waves in a plasma when thePoisson's equation is taken into account and theneutrality condition flt = fte is not used, except theappearance of specific heat ratio C" which he took unity

for isothermal electrons. This relation is a dispersion
relation for ion waves showing their acoustic nature
where the inertia is provided by the ion mass and thepressure by the electron temperature. However, thecontribution of finite electron mass to these ion acoustic
waves in the plasma is reflected through eq. (5a).

Moreover, since eq. (4) is a fourth-order equation in al
and ft, it will give four dispersion relations with due

corrections of the effects of finite electron mass,relativistic speeds, inhomogeneity and magnetic field.
Subsequently, it is expected that four types of modes

will appear in the present plasma model.
Now to find the phase velocity relations of theof expft(kx - rot)l in (x, z) plane. Under theseconditions, we find the following equation when thewave width is taken to be smaller in comparison to thescale length of density inhomogeneity (dnslnyax, z <<expected modes, we employ the reductive perturbation
technique ofnormalized basic fluid equations separately

for the homogeneous and inhomogeneous plasmas alongwith the same normalization as used in ref. [10].

ft)

1(5a)t')ldt = - s ryeYQ + s n1e(vi x B)

- I;=r" rooft{y,(a- kv)2
- aq2t@- kv) - lecjTjtmj- lse {lf /n()m,1a - kv)2)no,Qo. } = 0 .

3(a). Homogeneous Plasma
For the homogeneous medium, the following(4)

stretched co-ordinate system is introduced

In eq. (4), ne is the unperturbed plasma density and ari
and Q are tho plasma frequency and cyclotron frequency of the jth species, respectively. vr = u (v) is theinitial ion (electron) velocity in the,r-direction and ns.
and Qs. are the density and potential gradients in the z-€=ett2(k'r-A.t)= eu2(rsinO + zcos? - Lt) ,Tl =t3t2t.

(6)

Here, ft is the unit vector along the direction of the wave

direction, respectively.
Solution of eq. (4) will give the relation between a)

propagation that makes an angle 0,with the direction of
the magnetic field. /, is the phase velocity of ion acous455

Malik H.K. et 41., Studies on Modes in Weakly Relativistic Magnetized Plasmas

perturbation of dependent quantities as taken in ref.
[10], we can obtain the following relations at different
orders of e.

tic wave in (6, n) space and g is a small dimensionless
expansion parameter.

The quantities n, ne, Q, u,, ur, trz, vx, vy and v. canbe expanded around the equilibrium state in terms of t

At order et/2 : noE= Loe = QoE= 0 .

as below

f =l+efr+e2fr+...,
8 = €8t + t2g2+ ... ,h = E3t2hr + eh2 + ... ,p1 =p + E'''pjr+ €2pjz+...,(10a)This equation indicates that the zeroth-order (unperturbed) quantities vary only with space not with time,
implying the inhomogeneity present in the plasma.

(7)
At order E:

flt = fiet ,(10b)where/: (n, n"), I = (Q, u,, v,), h: (u, vy) and pi = (u",v,) along with p - u or v.
Now the normalized form of the basic fluid eqs.
(1-3) gives the following relations in first-order
quantities at different orders of e along with the use of

At order t3t2 :- (Lo - bsin9)ntry lLs

eqs. (6) and (7).

Atordere: fl1-n.1=Q.

+ nscosO b,1g/Ls+ Dsin0 nor=0,(10c)

brr = sinO Qg lALo + ssrsin0 n11g lAnshs+ sin9 QsrlA + ssrsin? nsrlAns ,(10d)(8a)- no(Lo - bsin0)b41 l),o + srmcos9 n1c l),sI smnscos9 Q14 /Lo + srmcos9 ns,

At order t3t2 :- Q, - bsin0)n1r6 * cos0 baE= 0,(8b)szrsin0 Q4- smAby + srm sin9 nig= 0 ,(8c)

At order E2 : ns sin9 bac l),o = 0 .

(8d)n1 , we obtain- (L - bsin0)b41t srm cos9 niry+ sncos0 Qg,=0 .

I smnocos9 Qort = 0 ,

(10e)

(100

When we solve eqs. (10) for the first-order quantity

nr= {(fmcos? + (Lo- vsinO)vsinO/cos0

In the above equations, b = u (v) is ion (electron) fluid
velocity, s = + (-) for ion (electron) fluid, m = | (m,) forion (electron) fluid and sc = 2o (1) for ion (electron)
fluid, where rrt, = rrtilrrT", o = TtlT" and A = C)1/a;01. The
integration of eqs. (8) under the boundary conditions+ 2omcos0 + (ho - usin9)musin9lcos9f

x nsnlllmcos1lLo - (Lo - vsin0)21),scos9+ 2omcos1l),o - m(Lo - usin0)21)'scos0l .

that n, n. -+ I and Q, u* u", vy, vz -) 0 as { -+ *- yields

Here, it can be noted that the left hand side is a first-order term and the right hand side contains only the zeroth-order terms. Since a first-order term cannot be ex-

the following phase velocity relation for the ion acoustic
wave-

plicitly expressed in terms of only the zeroth-order

L= l(u + vm"lm)sin9+ [(l + m,lm)(l + 2o\cos20- (m"lm) (u - v)2sin20ltt21/71 + m"/mt1 (9)terms, the right hand side is made indeterminate by putting separately the numerator and denominator equal tozero. This process yields three relations for the phasevelocity /,s, out of which the two relations: corresponding to the denominator, are the same as obtained in caseof homogeneous plasma [eq. (9)]. The third mode: corresponding to the numerator, has the following phase

It is evident from eq. (9) that two types of modes,
namely fast mode (Lp): corresponding to plus sign andslow mode (.1,s): corresponding to minus sign are
possible in a magnetized homogeneous plasma having
weakly relativistic ions and electrons and their phasevelocity relation
L1 =f(u2 +v2m"lm)sin29- (1 + 2o)cos201l1u + vm"lm)sin9. (11)velocities depend on u, v, o and 0.

3(b). Inhomogeneous Plasma
Usine the same stretched co-ordinates and the456

Malik H.K. et dl., Studies on Modes in Weakly Relativistic Magnetized Plasmas

4. Results and Discussion: Propagation ofModes
To examine the propagation of modes, we analyze
phase
thevelocity relations (9) and (11). Relation (9)shows that the phase velocity of the fast mode (2p) and28.425 lr

b 26.62
'6

28.42

'6

o
28.415 ooo
28.41
(!

c,Eo
26.61
oG.Co' 26.6

slow mode (/.r;) will be real if the quantity appearing inthe square roo1. term be positive, which reads
tan20 < (I + milm")(l + 2o)/(u - v)2 .

U' 26.63.C

28.40so.

45 50 55 60 65 70(r2)

Electron Streaming Speed v
For the slow mode, the phase velocity 15 will be positive if the first term of R.H.S. of eq. (9) be greater thanFig. 1 Weak dependence of the phase velocities i, and,1. of fast mode and slow mode, respectively, on
the electron streaming speed v. Here, o= 0.05, 0 =30'and u = 55.

the second one. This requirement leads to the followingconditiontan20) (1 + 2o)l(u2 + v2m"lm1) .

(13)
dependence on the electron speed, which is caused by

The third type of mode shall propagate in the plasma if
the first term of eq. (l l) be greater than the second one.
This yields the same limiting condition on the wave
propagation angle 0, as given by eq. (13). Therefore, we

the electron mass r??e appearing together with v in thephase velocity relations. It is evident from the figure thatin (1s) attains maximum (minimum) value when the1. For the propagation of the fast mode, the wave
propagation angle 0 has an upper limit, given by eq.
(12). For example, when o = O.O5, u = 45 (7O) and v =55 the upper limit of the angle is 77.458o (71.546").

electron speed becomes almost equal to that of the ions.
This may be attributed to the energy exchange mechanism with the modes; however, one would have to takeup kinetic approach for the present plasma model toexamine such mechanism. For the third type of modealso, it is seen that ).7 increases slowly from 27.4681 to27.4752 when v is raised from 45 to 70. If we analyze
the effect of ion speed, the numerical calculations show

This limiting value is decreased for the higher ion

that,?,p increases rapidly from23.414 to 35.906 and.1.5

speeds and lower ion temperatures.

from 21.614 to 34.123 and L-1 from 22.466 to 34.973
when a is raised from 45 to 70. It can be noted here thatthe magnitudes of ,1,p, 25 and /,1 follow lF > Lr > Ls.
Strong dependence of the phase velocities of all theconclude the following on the basis of above inequalities2. The slow mode can propagate only for the certain
range of angle 0 prescribed by eqs. (12) and (13). For
example, when o = 0.05, u = 45 (70) and v = 55, therange is 1.330'< e<77.458" (0.858'< 0<71.546").
This range of t? is decreased for the higher ion speeds.

modes on wave-propagation angle 0 (in degrees) is
portrayed in Fig. 2. This figure shows that the phasevelocities of the modes get higher for the larger wavepropagation angles and the magnitudes of the velocities
follow the same trend i.e. hp > h, > hr.Further, it iseasy to examine the effect of ion temperature (o) on ,1,p,
1.5 and ,l"t through eqs. (9) and (11). For higher ion
temperatures, the phase velocity of the fast modeincreases and those of the slow and third type of modes

3. The third type of mode propagates only when thewave propagation angle 0 exceeds the lower limit given
by eq. (13). For above set of values, this lower limit is
1.33o for ion streaming speed a = 45 and it is decreased
to 0.858' when u is increased to 70.

get lower.

Now we study the effect of ion and electron
streaming speeds, ion to electron temperature ratio (o)and the wave propagation angle 0 on the phase

Now we analyze the results of reductive
perturbation technique and those of normal mode
analysis. Normal mode analysis shows the presence offour types of modes in the present plasma model. Since

velocities of the above three types of the modes. Figure
I shows the opposite behaviour of the phase velocities
of fast and slorv modes with electron streaming speed v.
Here, it is clear that the phase velocities have weaktwo real dispersion relations are inferred by thereductive perturbation technique for the homogeneous

plasma, the other two roots may be complex and457

Malik H.K. et al.. Studies on Modes in Weakly Relativistic Masnetized Plasmas

45

Acknowledgement

40

Hitendra K. Malik thanks the Japan Society for thePromotion of Science (JSPS, No. P- 00100) for financial'E rs

support and I. E. T. Lucknow - 226 O2l, India forproviding the leave.

>30o9.oe:3reReferences[U Ya. L. Al'pert, Space Plasma (Cambridge Univer-20sity Press, Cambridge, 1990) vol. I, 135.

1520304012) H.Ikezi, Phys. Fluids 16, 1668 (1973).

50[3] R.P. Dahiya, P.I. John and Y.C. Saxena, Phys. Lett.

Wave Proapagation Angle 0 (Degree)

A 65,323 (1978).

[4] Y. Nishida, Phys. Fluids 27,2176 (1984).
[5] G.C. Das and S.N. Paul, Phys. Fluids 28, 823

Fig. 2 Variation of the phase velocities LE, Ls and i1 offast mode, slow mode and third type of mode, re-spectively, with the wave propagation angle 0.

(198s).

Here, o= 0.05 and u = v= 55.

[6] S. Singh and R.P. Dahiya, Phys. Fluids B 2, 901(1990).

therefore instability may be expected in such a plasma
model. However, in inhomogeneous plasma, we cannot

t7l H.H. Kuehl and C.Y. Zhang, Phys. Fluids B 3,26

expect the instability because three real dispersion
relations are obtained. Further, it is suggested that all themodes should be taken into account while studying thesolitons in magnetized plasmas having weakly
relativistic ions and electrons, because they may[8] Y. Nejoh, Phys. Fluids B 4,2830 (1992).
t9l H.K. Malik, S. Singh and R.P. Dahiya, Phys. Plas-correspond to the solitons of different behaviour.

[11] F.F. Chen, Intoduction to Plasma Physics andContolled Fusion (Plenum Press, New York andLondon, Second Edition: 1990) vol. l, 98.

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[10] H.K. Malik and Y. Nishida, J. Plasma Fusion Res.
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