J. Plasma Fusion Res. SERIES, Vol(2001) 5t4-518

Field and Plasma Density on Nonlinear
Waves in a Relativistic Plasma

Effect of MagSol

MALIK Hitendra K.* and NISHIDA Yasushi
EnergyEnvironmental Science, Graduate School of Engineering'
U tsunomiya lJniversity, Tochigi 32 I -85 8 5' Japan(5 December 2000 / Accepted: 27 August 2001)

Abstract

Vries (mK-dV) equation is derived for perturbed potential and solved in orderre for the propagation of nonlinear solitary waves (solitons) in a magnetizedto study the potential
plasmahaving relativistic ions and electrons. It is found that fast and slow
spatially inhomoge
possible for amodes occur in the plas . and only the fast modes conespond to the solitons. Solitons areg
the magneticofdirection
andvector(anglewave
betweenparticular range of wave propagation angleof ions andspeeds(TilT")
relativisticandratio (milm"), temperature ratiofield) governed by
Modified Korteweg

electrons. Effect of maof soliton get lower andwaves propagating at
higher amplitude andic field is to decrease the soliton width and energy. The amplitude and energy
width gets wider when it propagates into higher plasma density region. Ther angles with the direction of magnetic field correspond to the solitons ofler width.

Keywords:
relativistic ion, relativi

electron, potential, mK-dV equation, magnetic field' inhomogeneity, soliton

plasma taking the Boltzmann distribution for electrons.
Fully relativistic ion fluid equations were also presented
to study the double layers, spiky solitary waves and1. lntroduction
Solitonic excitation isaspects of the nonlinear
plasmas. Since the solitonof the most importantencountered in theure can trap plasma

explosive modes considering the Boltzmann distribution

for electrons [3]. However, Kuehl andZhang [4] suggested that in place of electron Boltzmann distribution
the finite electron inertia should be taken into account
while studying the propagation of solitary waves in a
relativistic plasma, since the wave velocity can becomparable to the electron thermal velocity. Later'
taking account of finite electron inertia and ion
temperature, a limit on ion drift velocity was obtainedfor existence of the solitons in a relativistic plasma [5]'Recently, the effect of electron inertia has been shown

distances, they canparticles and convect them overport of particles andcontribute to anomalousin laboratorv, astroenergy from one region toefore, the subject ofphysical and space plasmas.
received a great deal
soliton propagation in Plasmas
of interest and the nonlinear litary waves have been
plasmas as well.
extensively studied in relativi
Considering only the relat istic speed of ions, Das

and Paul [1] derived ag-deVries (K-dV)

the possibility ofequation for a cold plasma andmore solitons.
manywaveion
acousticofbreaking
ontemperatureion
initeofeffect
Nejoh [2] studied therelativisticincollisionless
ion acoustic solitarY waves*Corre sponding author's

to decrease the phase velocity, amplitude and energy ofthe solitons in a relativistic plasma [6].

: hkmalik@ plasma. ee s.utsunomiya-u.ac.ip
514@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Malik H'K' et al.' Effect of Magnetic Field and Plasma Density on Nonlinear Solitary

Soliton characteristics are altered when a soliton

Waves in a Relativistic plasma

to magnetic field pressure is taken to be small. Kinetic

propagates in a nonuniform plasma and the K-dV equa-effects such as Landau damping, heat conduction,
viscosity etc. are neglected. The electron Debye lengthis taken to be much smaller than the dimensions of thesystem and the plasma is quasineutral. Under theseconditions, the following basic fluid equations aretion describing the soliton behaviour is modified either
by varying coefficients or by some additional terms.
Assuming that the soliton width is small compared withthe scale length of plasma inhomogeneity, arbitrary
amplitude soliton propagation in nonrelativistic plasma[7] and small amplitude soliton propagation in

obtained in normalized form.
For ion fluid:

relativistic plasma [8] have been studied. It has beenshown that the density inhomogeneity considerably
modifies the soliton behaviour in relativistic plasmas

ry+(nv)x+(nvr)r-O

t8,el.

(Tiv,), + v,(Tiv,), + v r(yp,), + Q,

(1a)

It can be noted for most of the analyses on relativistic plasmas that only the ions are taken to be re-- ({2i/r4)v, + 2on,ln = O(1b)lativistic. However, in space plasmas of solar wind [10],boundary layer of the Earth's magnetosphere [11] etc.,
high-energy ions as well as electrons are found to be
streaming with relativistic speeds. Also, under the(vr), + v,(vr), + vr(vr), + (Qilrtt )v, = o(1c)(vr),+ v,(vr),+ vr(vr)r+ Qr+ 2onrln = 0(1d)influence of high-power laser radiation the plasma
particles may attain relativistic speeds. Therefore,relativistic effects of electrons should be consideredwith the ions to understand truly the behaviour ofFor electon fluid:(n"), + (n"u,), t (n.ur), = Q(1e)(m"/m) f(y"u,), + u*(y"u,), + ur(y"u,)rl

solitary waves in relativistic plasmas. Considering therelativistic effects of both ions and electrons, however,
Nejoh and Sanuki [2] studied the large amplitude
Langmuir and ion acoustic waves and investigated theconditions for their existence in a relativistic plasma.
The analysis carried out in Ref. [12] with relativistic ions and electrons was restricted to homogeneous and unmagnetized plasma. But invariably weencounter inhomogeneous plasmas under actual con-

- Q,+ (Ai/qi)ur+ (n")"|n"=Q (lf)(mJm) f(ur), + u"(ur)" + u,(ur),)- (d\lap)u, = O(1e)(mJm) [(u,), + u,(u,)" + u,(ur),]- Q, + (n")r/n" = Q

ditions, e.g. more so closer to the sheath regions, and the(1h)Poisson's equation:

soliton behaviour is also modified under the effect of

Q*+Qo-ne+n=O

magnetic field [3-15]. Therefore, in the present paper,
we study the effect of magnetic field and plasma density( 1i)

In these equations, ion and electron densities arenormalized by the unperturbed plasma density n6, fluid
velocities by the ion acoustic speed, potentialby KT"/e,on the soliton propagation in a plasma having both
relativistic ions and electrons. Using reductive perturbation technique, we derive a modified K-dV equation

where K is the Boltzmann constant. Time and space arenormalized by the ion plasma period and electron Debye
length, respectively. Subscripts t, x and z represent thefor the perturbed potential and find that it admits thepotential structure of compressive solitons for a particular range of wave-propagation angle.

differentiation. O1 is ion gyrofrequency and ooi is ion
plasma frequency and their ratio Qlatoi= Bo"[eo/no^r.

2. Phase Velocity and Limit on Wave-

Ion to electron temperature ratio (TilT.) is taken as c. f1

Propagation Angle- (l-voz1rz1-rr2 and y. = (l-uo2lc2)-tl2, with y0 and ll0 as

We consider a magnetized collisionless spatiallyweakly inhomogeneous plasma consisting of relativistically streaming warm ions and hot electrons. A

the ion and electron streaming speeds in.r-direction withweak relativistic effects, i.e. v6, us<.c. For the sake ofsimplicity, we have assumed that the ion and electron
speeds are relativistic only in the _r-direction. It can benoted from ion momentum equation (lb) that thespecific heat ratio for adiabatic ions is taken as 2, sincesmall amplitude ion acoustic wave is assumed to propagate in (x,z) plane at an angle 0 with the direction ofmagnetic field B (= Boz). The ratio B of particle pressure515

Malik H.K. et al., E,ffecr

Field and Plasma Density on Nonlinear Solitary Waves in a Relativistic Plasma

the number of degrees offor the present case is

2. Further, the continuitY andused for electron fluid, since telectrons are taken as

tan203mi(l + Zo)lm.(vs- uo)2 .

equatrons are

For positive Ls, the first term of R.H.S. of Eq.(4) should
be greater than the second term. Considering this point'
we find the following condition for positive Le.

relativistically streaming speci

ve perturbation techFor employing thestretched coordinates
nique, we use the following set

tan20> (I + 2o)l(vo2 + us2m.lmi)

t8,14,161.

ion acoustic wave will propagate if the following= ett2(xsin9lLs * Zcos9l

inequality is satisfied in the plasma conditions.
(1 + Zo)l(voz + u62m"/mi)(2)

t = €3t2(k'r) = e3t2(xsin? +<tan20 < mi(l + 2o)lm"(vs - us)z

10n acoustrc wave ln
Here, ia is the phase velocity
ionless expansion(f,r) space and e is a small d
the direction of wave
parameter. ft is unit vector ang an angle 0 with the
propagation in (x,z) Plane
direction (z-axis) of magnetic
and 0 are expanded
The quantitigs n, ne, v,

fore, it is deduced that the fast and slow modes
propagate in the plasma when the angle 0 falls in theexpansion is given by

7=fo@,2) + efi(x,z,D + e2" (x,z,t) + ...,

range assigned by Eq.(6).

g = e g {x,z,t) + e2 g2(x,z,t)

3. Modified Korteweg-de Vries Equation in
Perturbed Potential

h = e3t2hr(x,z,t) + e2hr(x,z...,(3)The reductive perturbation treatment gives differenth = (vy, ur) and s = v or

equations in second-order quantities ft2, tte2, Q2, v,2, vyz,rzz, utz, uy2 and ur2. Elimination of these quantities withthe help of phase velocity relation and equations in first-,z,t) +u.

order quantities yields the following modified
Korteweg-deVries (mK-dV) equation in perturbedtEq.(2)l and expansion
in basic fluid equationslEqs.(l)l and find a set of equ ons in different ordersof e. The equations obtained i first-order quantities n1,Now we use the stretchi

of dependent quantities [Eq.(3na, Qr, vs, ryr,'t4, t't1, Ityl iphase velocity relation for the

potential @1.

Qr,+ aQrQrr+ Fh6aq- arjQtEno,- a2(ng,,= 0 (7)The coefficients appearing in the mK-dV equation (7)2.1 yield the following

acoustlc wave.

are given by)"0= (vo+ uom"l mr)sin0

s1= {CfD2 - (milm")Glcos20}/t / 1t + 2o)cos20- m"( o- uoyzsin'o/m, (4)This equation indicates that the fast mode. corres-2)"02(vo -us)DEsin?

mode, corresponding
ponding to plus sign, and theplasma. Also, the phaseto minus sign, can occur instreaming speeds
velocity depends on ion and e(relativistic effects), theirand temperature ratios
The phase velocity
and angle of wave Propagat
increases with increasing ion

ing speed and wave-

propagation angle, but it
electron streaming sPeed.

for higher values of

For real phase velocity,
the square root terrn of Eq.(4)(6)

Now we use some typical values of v0, ilg and o tofind the limit on wave-propagation angle 0 for existenceof the wave in the plasma. For a relativistic spaceplasma [5,17], we select ve = 70, uo = 35, o = 0.04 formylm" = 1836 and find that 0.850 < 0 < 5l'830. There-around the equilibrium state n terms of e and their

where,/: (n, n., Q), I = (vz, uz(5b)

On the basis of Eqs.(5a) and (5b), we can say that the€=ett'(k'rlLo-t)s" = ss * t3t2ss(x,21) + ez

(5a)

B= lCDlb2E(milm")+ (milm)nsC(sin2g + Yrcos207- nsG(sinz? + y.cos2e111l2b2(m,lm") )'oo ro(uo - us) sin0cos20
a1 = fFCDz + F(my'm") GC- 4o)'sD2 cos20- 2C DEussin0ll2)"sno(vs - us) DE sinO

quantrty appeanng ln
be positive. which

az= LolF(vo- u)CDsin9

reads516

Malik H.K. et al., Effect of Magnetic Field and Plasma Density on Nonlinear Solitary Waves in a Relativistic plasma- G(Eussin? - FD) - Zo).oD2cos20f/

1.02nsCD (v s - ao) sin0cos20

0.99,n (uo/c=0.0'161)

Here, b = d\lrtei C = ho- v6sin0; D = Lo
C2- u{)sinl; E =e-r

- 2ocos20; F - 2ocos20 + Cvssing; G = (l +2o)cos20 - C.

0.8

0.7

Now we solve the mK-dV equation in order tostudy the structure of perturbed potential fi. The use oftransformation ( = 6 - Vr , where Vis a constant, gives
the following solution of Eq.(7) at constant density

0.6

0.5

0.6

0.7

0.80.9Do

gradient [8,9].
Q, (() = 13 (V + a, ( n o)t u

Fig.1 Effect of plasma density n0 on the soliton
amplitude @, and soliton width L. Here, vorc =0.0161 (for uolc = 0.0322l,, volc = 0.0322 (for uotc =0.0161 ), no = 0.8, no,= 0.002,5 = 0.01, V= 0.0009, b= 0.0015 (4 = 5.8 x 10-iT), o= 0.04 and 0=20o.

lseclf{(,/O;;3;;W}(8)

Inspection ofEq.(S) reveals that it describes a potential
structure which reaches a peak at ( = 0 and vanishes at(-+ t-, with the peak value = 3(V + a1(ns,ya(= S^, sayland half width = ^f4fliOlZAJ (= Z, say). This
structure is known as soliton with amplitude (@- andwidth Z. Soliton energy E, [8] can be calculated using
Eq.(S) and we find that 4 - 40^2L8.

2.5

2.O

1.5 m

4. Results and DiscussionJ

1.0 oN

It is observed for slow ion acoustic mode that thesoliton width is not real, since the dispersion coefficientB attains negative values. Therefore, the soliton propagation is not possible for the slow mode. However, isB0r|456789always positive for the fast ion acoustic mode and||

||

lO.O

thereby gives the real values of soliton width. Further,Bo (10-?T)

the nonlinearity coefficient a is positive for the fastFig. 2 Effect of magnetic field Bo on the soliton width I
and soliton energy E". Here, volc=O.O322, uolc=0.0161 and other parameters are same as in Fig.1.

mode and therefore it is concluded that the compressive
solitons are possible only for the fast mode. For equal
speed of ions and electrons (i.e. vo = ao), it is observedthat a, B and a1 -+ - which make the soliton amplitude
and width indeterminate, indicating that the soliton
cannot occur in the plasma having ions and electrons ofwill also decrease with increasing plasma density. The
effect of relativistic electron speed uslc is found todecrease the soliton amplitude and to increase theequal speeds.
Effect of plasma density n6 or the soliton amplitude
p- and soliton width Z is shown in Fig.l, for different

soliton width (Fig.1). However, we observe on the basisof expressions for soliton characteristics that the soliton
behaviour with relativistic ion speed is opposite to thatwith the electron speed.

values of relativistic electron speed uslc. This figure
shows that the soliton amplitude gets lower and its
width gets wider for increasing values ofze. This can be
explained on the basis of coefficients a, B and a1.
Numerical calculations reveal that B and a1 decrease
with increasing zs, but dremains constant. Further, thedecrease in a1 is faster than the decrease in B, whichcauses the soliton amplitude to decrease and the width
to increase for higher values of plasma density. Since
the soliton energy E" is directly proportional to Q^2, it

Effect of magnetic field ,86 on the soliton
characteristics is shown in Fig.2, for different values ofwave-propagation angle 9. It is clear from the figure thatthe soliton propagates with smaller width (Z) and lower
energy (d) under the effect of stronger magnetic field.
This can be seen from the expressions of a, p and a1
that only the dispersion coefficient B is a function of b.

It means B depends on the magnetic field, since b = d)ilar, = Bo'[eolnor4. It is observed that the dispersion517

Malik H.K. et al., EffectField and Plasma Density on Nonlinear solitary waves in a Relativistic Plasma

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coefficient decreases for hig

soliton energy also decreases wi

is directly proportional to withat the soliton width becomes
gets higher for increasing
the basis of numerical calc

increasing 0 that the soliincreasing -86, since it
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Acknowledgement
Hitendra K. Malik thanks
Promotion of Science (JSPS,support and I. E. T.

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tl3l E. Witt and W. Lotko, Phys. Fluids 26, 2176

- 226 021. India for(1e83).

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