nonplanar electron acoustic shock waves

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Advances in Space Research xxx (2013) xxx–xxx

Nonplanar electron acoustic shock waves

Biswajit Sahu a,⇑, Mouloud Tribeche b

a Department of Mathematics, West Bengal State University, Barasat, Kolkata 700126, Indiab Plasma Physics Group, Theoretical Physics Laboratory, Faculty of Sciences-Physics, University of Bab-Ezzouar, U.S.T.H.B B.P. 32, El Alia,

Algiers 16111, Algeria

Received 7 December 2011; received in revised form 29 January 2013; accepted 31 January 2013

Abstract

The properties of cylindrical and spherical electron acoustic shock waves (EASWs) in an unmagnetized plasma consisting of cold elec-trons, immobile ions and Boltzmann distributed hot electrons are investigated by employing the reductive perturbation method. AKorteweg–de Vries Burgers (KdVB) equation is derived and its numerical solution is obtained. The effects of several parameters andion kinematic viscosity on the basic features of EA shock waves are discussed in nonplanar geometry. It is found that nonplanar EAshock waves behave quite differently from their one-dimensional planar counterpart.� 2013 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Electron acoustic waves; Shock waves

1. Introduction

The electron-acoustic wave (EAW) is a high-frequency(in comparison with the ion plasma frequency) wave thatoccurs in a plasma having, in addition to positively chargedions, two electron components with widely disparate tem-peratures (Tokar and Gary, 1984; Gary and Tokar,1985). The idea of EA mode had been conceived by Friedand Gould (1961) during numerical solutions of the linearVlasov dispersion equation in an unmagnetized, homoge-neous plasma. Besides the well-known Langmuir and ion-acoustic waves, they noticed the existence of a heavilydamped acoustic like solution of the dispersion equation.It was later shown that with two species of electrons withwidely disparate temperatures, referred to as hot and coldelectrons with immobile ions, one can obtain a weaklydamped EA mode, (Watanabe and Taniuti, 1977) whichhas different properties than Langmuir and ion-acousticwaves. The relatively cold inertial electrons oscillate againsta thermalized background of inertialess hot electrons

0273-1177/$36.00 � 2013 COSPAR. Published by Elsevier Ltd. All rights rese

http://dx.doi.org/10.1016/j.asr.2013.01.030

⇑ Corresponding author. Tel.: +91 9434317727.E-mail address: [email protected] (B. Sahu).

Please cite this article in press as: Sahu, B., Tribeche, M. Nonplanar electr10.1016/j.asr.2013.01.030

providing the necessary restoring force. EAWs may alsoexist in an electron-ion plasma with ions hotter than elec-trons (Fried and Gould, 1961). There has been a phenom-enal growth in research activities in EAWs to explain thespace observations of solitary waves with either negative(Tagare et al., 2004) or positive potentials (Berthomieret al., 2000; Mace and Hellberg, 2001). It has been arguedthat the broadband electrostatic noise may be explained interms of solitary electron-acoustic structures with negativepotential in two-temperature electron plasma (Mace et al.,1991). There are several papers dealing with electron-acoustic waves (Singh and Lakhina, 2001; Cattaert et al.,2005; Verheest et al., 2005; Verheest et al., 2007; ?; ?; Pott-elette and Berthomier, 2009; Pakzad and Tribeche, 2010;Tribeche and Djebarni, 2010; Younsi and Tribeche, 2010;Bains et al., 2011). Singh et al. (2001) examined EA solitarywaves in a four-component plasma and applied theirresults to explain the Viking satellite observations in thedayside auroral zone. Verheest et al. Verheest et al. (299)showed that the inclusion of the hot electron inertia canlead to positive potential EA solitons. Lakhina et al.(2008) investigated large amplitude ion and electron-acoustic solitary waves in an unmagnetized multi-fluid

rved.

on acoustic shock waves. J. Adv. Space Res. (2013), http://dx.doi.org/


mailto:[email protected]




2 B. Sahu, M. Tribeche / Advances in Space Research xxx (2013) xxx–xxx

plasmas. Kakad et al. (2007, 2009) examined the coexis-tence of compressive and rarefactive solitary structures inmultispecies, unmagnetized plasmas. Lakhina et al. (2009,2011) proposed a model based on electron-acoustic solitonsand double layers to explain the solitary waves observed byCLUSTER satellite in the magnetosheath region (Pickettet al., 2005). However, most of these studies are limitedto solitary EAWs or confined to one-dimensional geometrywhich may not be a realistic situation in laboratory orspace plasmas, since the observed waves are certainly notbounded in one-dimension. Examples of nonplanar geom-etries of practical interest are capsule implosion, shocktube, star formation, supernovae explosion. . .etc. The aimof the present paper is therefore to investigate EA shockwaves in nonplanar cylindrical and spherical geometriesto show how nonplanar EA shock waves may differ fromtheir one-dimensional geometry counterpart. It may beuseful to note that recently nonplanar nonlinear waveshave received a good deal of interest (Mamun and Shukla,2001; Mamun and Shukla, 2002; Khan et al., 2008; Sarmaand Bujarbarua, 1984; Sabry et al., 2009; Mamun andShukla, 2009; ul Haq et al., 2010; Liu et al., 2010; Sahu,2010; Mamun and Shukla, 2010; Eslami et al., 2011; Daset al., 1989). The plan of the paper is as follows. Nonplanarmodified KdVB equation is derived in section 2. In section3 the numerical results and discussion are given whilesection 4 is kept for conclusion.

2. Basic equations and derivation of KdVB equations in

nonplanar geometry

We consider a homogeneous, collisionless, unmagne-tized three component plasma consisting of cold electrons,immobile ions and Boltzmann distributed hot electrons andstudy the nonlinear propagation of electron acoustic shockwaves. The nonlinear dynamics of EA oscillations is gov-erned by the following dimensionless equations

@nc

@tþ 1

rm

@

@rðrmncucÞ ¼ 0; ð1Þ

@uc

@tþ uc

@uc

@r¼ a

@/@rþ g

1

rm

@

@rrm @uc

@r

� �� muc

r2

� �; ð2Þ

1

rm

@

@rrm @/@r

� �¼ nh þ

1

anc � 1þ 1

a

� �; ð3Þ

where nh ¼ expð/Þ; m ¼ 0, for one dimensional geometryand m ¼ 1; 2 for cylindrical and spherical geometry, respec-tively. In the above equations, nc(nh) is the cold (hot) elec-tron number density normalized to its equilibrium value nc0

(nh0), uc is the cold electron velocity normalized toCe ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT h=am

p, and / is the electrostatic wave potential

to kBT h=e; a ¼ nh0=nc0 > 1, m is the electron mass, T h isthe temperature of hot electron, e is the magnitude of theelectron charge and kB is the Boltzmann constant, g is theelectron kinematic viscosity. Time and space variables arenormalized respectively to the inverse of cold electronplasma frequency x�1

pc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim=4pnc0e2

pand the hot electron


Debye length kDh ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT h=4pnh0e2

p, respectively. We have

assumed that ve � c, where ve is the electron thermal velocity

and c is the velocity of light. In equilibrium, we havenc0 þ nh0 ¼ ni0.

We now introduce the stretched coordinates

n ¼ �1=2ðr� v0tÞ; s ¼ �3=2t ð4Þ

and expand nc , uc and / in a power series of � as

nc ¼ 1þ �nð1Þc þ �2nð2Þc þ � � � ; ð5Þuc ¼ �uð1Þc þ �2uð2Þc þ � � � ; ð6Þ

/ ¼ �/ð1Þ þ �2/ð2Þ þ � � � ; : ð7Þ

In many experimental situations the value of g is small, sowe may set g ¼ �1=2g0. g0 is O (1). Substituting (5)–(7) into(1)–(3), we obtain from the lowest order in �; nð1Þc ¼ �a/ð1Þ,uð1Þc ¼ � a

v0/ð1Þ and v2

0 ¼ 1.To next higher order in �, we obtain a set of equations,

@nð1Þc

@s� v0

@nð2Þc

@nþ @

@nðuð2Þc þ nð1Þc uð1Þc Þ þ

mv0s

uð1Þc ¼ 0; ð8Þ

@uð1Þc

@s� v0

@uð2Þc

@nþ uð1Þc

@uð1Þc

@n¼ a

@/ð2Þ

@nþ g0

@2uð1Þc

@n2; ð9Þ

@2/ð1Þ

@n2¼ 1

anð2Þc þ /ð2Þ þ 1

2/ð1Þ

2: ð10Þ

Combining Eqs. (8)–(10), we deduce cylindrical andspherical modified KdVB equation for electron acousticwaves

@/ð1Þ

@sþ m

2s/ð1Þ þ A/ð1Þ

@/ð1Þ

@nþ B

@3/ð1Þ

@n3� C

@2/ð1Þ

@n2

¼ 0; ð11Þ

where A ¼ � v0

21þ 3að Þ;B ¼ v0

2and C ¼ g0

2.

In the above equations A, B, and C are the coefficientsof nonlinearity, dispersion, and dissipation, respectively.The Burger term (C) arises due to the effect of kinematicviscosity. The Burger term implies the possibility of theexistence of a shock-like solution. If the dissipation termis negligible compared to the dispersion terms, then soliton-ic structure arises by balancing the effects of dispersive andnonlinear terms. On the other hand, if the couplingbecomes very strong the shock waves will appear. The nat-ure of these shock profiles depends on the relative valuesbetween the dispersive and dissipative coefficients B andC, respectively. It is also seen from Eq. (11) that the non-planar geometrical effect is significant when s! 0 andweaker for larger value of jsj.

3. Numerical results and discussion

There are several methods to solve the nonlinear partialdifferential equations, for instance, inverse scatteringmethod (Ablowitz and Clarkson, 1991), Hirota bilinear




Fig. 2. Cylindrical shock profile for different values of kinematic viscosityg0, where the other parameter are same as in Fig. 1.

Fig. 3. Spherical shock profile for different values of kinematic viscosityg0, where the other parameter are same as in Fig. 1.

B. Sahu, M. Tribeche / Advances in Space Research xxx (2013) xxx–xxx 3

formalism (Hirota, 1971), Backlund transformation(Miura, 1978), tanh method (Roychoudhury et al., 1999),etc. However, when the partial differential equation in asystem is governed by the combined effect of dispersionand dissipation, the most convenient method to solve thenonlinear partial differential equation is tanh method(Roychoudhury et al., 1999). Therefore, using the tanhmethod, the travelling solution of the KdVB Eq. (11) form ¼ 0 turns out to be

/ðn; sÞ ¼ a0 þ a1 tanhfaðn� V sÞg þ a2tanh2faðn� V sÞg;ð12Þ

where

a0 ¼1

AðVþ 12Ba2Þ; a1 ¼ �

12Ca5A

; a2 ¼ �12Ba2

A;

a ¼ � C10B

and V is the shock wave velocity. When the geometricaleffect is taken into account, an exact analytical solutionof Eq. (11) is not possible. Therefore, it is necessary to plotnumerical solution of Eq. (11) for a better understanding ofthe nature of the shock wave. The initial profile that wehave used in all our numerical results is the stationary solu-tion (12). In Fig. 1 we plot the numerical solution of Eq.(11) for the shock wave structure evolved at s ¼ �3 in dif-ferent geometries. It is seen that the shock height and shapechanges in different geometries. It is also seen that a kinkwave structure is formed in cylindrical/spherical geome-tries. It should be noted that the strength and steepnessof the shock waves in nonplanar geometries differ moder-ately from the planar geometry due to the presence of m=sterm in Eq. (11). If the value of jsjincreases, the nonplanargeometries would approach the planar geometry. In Fig. 2and Fig. 3 we plot the numerical solution of Eq. (11) fordifferent values of g0 in cylindrical (m ¼ 1) and spheri-cal(m ¼ 2) geometry, respectively. These figures show howoscillatory profile of shock is reduced to monotonicshock structure. It is clear that the transformation from

Fig. 1. Nonlinear shock waves are shown for different values m, wherea ¼ 3:0; g0 ¼ 0:5; V ¼ 1 and s ¼ �3.

Fig. 4. Variation in /ð1Þ with respect to n at different values of s for m ¼ 1,where g0 ¼ 0:4, and the other parameter are same as in Fig. 1.


monotonic shock to oscillatory shock is due to variationin g0. In fact, when dissipation term is negligible compared




Fig. 5. Variation in /ð1Þ with respect to n at different values of s for m ¼ 2,where g0 ¼ 0:4, and the other parameter are same as in Fig. 1.

Fig. 6. Cylindrical shock profile for different values a, where the otherparameter are same as in Fig. 4.

Fig. 7. Spherical shock profile for different values a, where the otherparameter are same as in Fig. 4.

4 B. Sahu, M. Tribeche / Advances in Space Research xxx (2013) xxx–xxx

to the nonlinearity and dispersion terms, then oscillatorystructure will appear by balancing the effects of dispersiveand nonlinear terms. On the other hand, if the kinematic


viscosity becomes very strong then monotonic shocks willappear by balancing the effects of dissipative and dispersiveterms. So the presence of the Burgers’ term exhibits anydisturbance from developing into oscillatory structuresand leads to the formation of a shock wave. Fig. 4 andFig. 5 show the shock wave structure for different valuesof s in cylindrical and spherical geometry respectively. Itis clear that as the value of jsj increases, the amplitude ofthe shock structures decreases and the solution looks likethose for one dimensional KdVB solutions. In Fig. 6 andFig. 7 we display the cylindrical and spherical shock profilefor different a. It is found that the number of peaks andtroughs of the shock structures increases as the value of aincreases.

4. Conclusion

To conclude, we have addressed the problem of cylindri-cal and spherical EA shock waves in an unmagnetized dis-sipative plasma consisting of cold electrons, immobile ionsand Boltzmann distributed hot electrons. A KdVB-likeequation has been derived based on the reductive perturba-tion method. It has been found that the propagation char-acteristics of the cylindrical and spherical EA shock wavessignificantly differ from those of one-dimensional EA shockwaves. Our results may help to understand the salient fea-tures of multi-dimensional EA waves that may occur in dis-sipative space and/or laboratory plasmas such as capsuleimplosion, shock tube, star formation, supernovaeexplosion. . .etc.

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