Astrophysics and Space Science

, Volume 339, Issue 2, pp 261–267

Planar and nonplanar ion acoustic shock waves with nonthermal electrons and positrons


  • Prasanta Chatterjee
    • Department of MathematicsSiksha Bhavana Visva Bharati University
  • Deb Kumar Ghosh
    • Department of MathematicsSiksha Bhavana Visva Bharati University
    • Department of MathematicsWest Bengal State University
Original article

DOI: 10.1007/s10509-012-1011-z

Cite this article as:
Chatterjee, P., Ghosh, D.K. & Sahu, B. Astrophys Space Sci (2012) 339: 261. doi:10.1007/s10509-012-1011-z


The nonlinear propagation of ion acoustic shock waves (IASWs) are studied in an unmagnetized plasma consisting of nonthermal electrons, nonthermal positrons, and singly charged adiabatically hot positive ions, whose dynamics is governed by the two dimensional nonplanar Kadomstev-Petviashvili-Burgers (KPB) equation. The shock solution of the KPB equations is obtained numerically. The effects of several parameters and ion kinematic viscosities on the properties of ion acoustic shock waves are discussed in planar and nonplanar geometry. It is shown that the ion acoustic shock wave propagating in cylindrical/spherical geometry with transverse perturbation will be deformed as time goes on. Also, it is seen that the strength and the steepness of the IASWs increases with increasing β, the nonthermal parameter.


Ion acoustic wavesShock waves

1 Introduction

The study of propagation of solitary waves in electron-positron-ion (EPI) plasma is of considerable importance for several reasons. Such plasmas are expected to exist in the early universe (Misner et al. 1973; Rees 1983). They are also found in the galactic nuclei (Mille and Witta 1987; Begelman et al. 1984), in the polar regions of neutron stars (Michel 1982, 1991), in the inner regions of the accretion disks surrounding black holes (Rees 1971), at the center of our galaxy (Burns 1983), in pulsar magnetospheres and in the plasmas in intense laser fields (Liang et al. 1998; Berezhiani et al. 1992). Recently, positrons have also been produced in tokamaks due to collisions of runaway electrons with plasma ions or thermal electrons (Helander and Ward 2003), which have also been observed in the joint European Torus (Gill 1993) and JT-60U (Yoshino et al. 1999). Most of the astrophysical plasmas contain ions in addition to electrons and positrons. So three component EPI plasmas can exist in nature, and it is important to study nonlinear structures such as solitons, shocks, etc., in such plasmas. It has been observed that the nonlinear waves in plasmas having positrons behave differently than usual plasmas containing electrons and ions. Therefore, a study of the propagation of solitary waves in EPI plasmas is a subject of considerable interest. Greaves et al. (1994) have reported that advances in the positron trapping technique have led to room-temperature plasmas of 107 positrons with lifetime of 103 s. Because of long lifetime of the positrons most of the astrophysical (Tandberg-Hansen and Emshie 1988; Piran 1999) and laboratory plasmas (Surko et al. 1986; Tinkle et al. 1994; Greaves and Surko 1995) become an admixture of electrons, positrons, and ions. Therefore, the study of EPI plasmas is important to understand the behavior of both astrophysical and laboratory plasmas. Popel et al. (1995) showed that the presence of positrons in electron ion plasmas significantly reduces the amplitude of ion acoustic solitary waves.

A great deal of research has been made to study the electron-positron and electron-positron-ion plasmas during the last three decades (Verheest et al. 1995; Nejoh 1996; Salahuddin et al. 2002; Mushtaq and Shah 2005; Mahmood and Akhtar 2008; Shukla et al. 2004; Pakzad 2011). Recently, based on Maxwellian assumption, many authors have been studied the propagation of ion acoustic waves in electron-positron-ion plasma (Sabry et al. 2009; El-Awady et al. 2010; Akbari-Moghanjoughi 2010; Shah et al. 2010). However, in these studies electrons were assumed to be isothermal. But in a number of heliospheric environments the plasma contains the nonthermally distributed ions or electrons (Verheest 2000, 2009; Shukla and Mamun 2002; Verheest and Pillay 2008; Mamun et al. 1996). Nonthermal ions have been observed in and around the Earth’s bowshock and foreshock (Asbridge et al. 1968; Feldman et al. 1983). Therefore non-isothermality plays an important role in determining the nature of nonlinear waves. Cairns et al. (1995) used nonthermal distribution of electrons to study the ion-acoustic solitary structures obserbed by the Freja and Viking satellites (Boström 1992; Dovner et al. 1994). It was noticed that inclusion of nonthermal electrons predicts negatively charged potential structures. Non-thermal electrons and positrons are predicted to exist in the expansion phenomenon of laser induced plasmas (Doumaz and Djebli 2010). Ali et al. (2007) investigated linear and nonlinear ion-acoustic waves in an unmagnetized electron-positron-ion quantum plasma. Moslem et al. (2010a, 2010b) studied three dimensional cylindrical Kadomtsev-Petviashvili equation in a very dense electron-positron-ion plasma. Alinejad (2010) studied dust ion-acoustic solitary and shock waves in a dusty plasma with non-thermal electrons.

A medium with dispersive and significant dissipative properties, supports the existence of shock waves instead of solitons. There are many phenomena such as Landau damping, kinematic viscosity among the plasma constituents, as well as collisions between charged particles and neutrals which are responsible for the dissipation in the system. Andersen et al. (1967) reported the formation of shocks in Q machine experiments. The dissipative Burger term in the Korteweg-de Vries-Burger (KdVB) (in one dimension) and the Kadomtsev-Petviashvilli-Burgers (KPB) (in two-dimension) equations arises by considering the kinematic viscosity among the plasma constituents. Mamun (1997) and later Tang and Xue (2004), Pakzad (2009), Das et al. (2009) also considered the nonthermal electrons to study ion acoustic waves. Recently, Saeed and Shah (2010) studied nonlinear KdVB equation for ion acoustic shock waves in a weakly relativistic electron-positron-ion plasma with thermal ions. More recently, Akhtar and Hussain (2011) investigated ion acoustic shock waves in degenerate plasmas.

However, most of the theoretical studies were confined to the unbounded planar geometry, which may not be a realistic situation in laboratory devices and space. Recent theoretical studies for solitary/shock waves in nonplanar geometry (Mamun and Shukla 2001; Jukui and He 2003; Xue 2003a, 2003b, 2005; Masood and Rizvi 2010; Noaman-ul-Haq et al. 2010; Jehan et al. 2011) show that the properties of solitary/shock waves in bounded nonplanar cylindrical/spherical geometry are very different from that of the planar geometry. In this paper, we study the nonlinear propagation of ion acoustic shock waves (IASWs) in a plasma system comprising of nonthermal electrons, nonthermal positrons, and singly charged adiabatically hot positive ions. The organization of the paper is as follows. In Sect. 2 mathematical formulation and derivation of cylindrical/spherical KPB equations are given. In Sect. 3 the numerical solutions of shock waves are presented, while Sect. 4 is kept for conclusion.

2 Mathematical formulation and derivation of cylindrical/spherical KPB equation

We consider a homogeneous, collisionless, unmagnetized plasma consisting of nonthermal electrons, nonthermal positrons, and singly charged adiabatically hot positive ions. The phase velocity of IAWs is assumed to be much smaller than the electron and positron thermal velocities and larger than the ion thermal velocity (vthiω/kvthe;vthp), where the inertia is provided by the ion mass and the restoring force is provided by the thermal pressures of electrons and positrons. It should be noted that typically Te=Tp. The basic system of equations in cylindrical and spherical geometry in such a plasma model is governed by
$$ \frac{\partial n_i}{\partial t}+\nabla.(n_iv_i)=0$$
$$ \frac{\partial v_i}{\partial t}+(v_i.\nabla)v_i=\frac{e}{m_i}E-\frac{1}{n_im_i}\nabla p_i+\mu \nabla^2v_i.$$
Where ni is the ion number density, vi is the ion fluid velocity, mi is the ion mass, e the magnitude of the electron charge, E the electric field, μ the kinematic viscosity of electron-positron-ion plasma, and pi, the pressure of the adiabatically hot ions, is given by the following thermodynamic equation of state:
$$ p_i=p_{i0}\biggl(\frac{n_i}{n_{i0}}\biggr)^\gamma.$$

Here pi0=ni0Ti0 is the ion pressure at equilibrium, ni0 is the unperturbed ion density, Ti0 is the ion equilibrium temperature, and the adiabatic constant is defined as \(\gamma=\frac{(N+2)}{N}\), where N represents the number of degrees of freedom of the ions. Since the kinematic viscosity represents the diffusion in momentum, therefore, we include this contribution from ions and neglect the contribution of electrons and positrons.

The electrons and positrons are assumed to obey non-thermal distribution on the ion acoustic time scale and are given by the following expressions (Cairns et al. 1995):
and the system of equations is closed with the help of Poisson’s equation,
$$ \nabla.E=-4\pi e(n_i+n_p-n_e),$$
where np and ne are the number densities, while Tp and Te are temperatures of positrons and the electrons (in the energy units), respectively. The electric field E=−∇ϕ, such that ϕ represents the electrostatic wave potential. At equilibrium, we have ni0+np0=ne0, where np0 and ne0 are the unperturbed positron and electron number densities, respectively.
The normalized form of continuity, momentum, and Poisson’s equations in nonplanar cylindrical and spherical geometries are as follows:
where, ν=0, 1, 2 for planar, cylindrical and spherical geometries, respectively. Note that the expression for ∇2 is different for cylindrical and spherical geometries and therefore caution needs to be exercised while opening it in the said geometries. Here, the ion fluid velocity vi is normalized by the ion acoustic speed \(C_{s}=\sqrt{\frac{T_{e}}{m_{i}}}\), ion density ni is normalized by electron equilibrium density ne0, and ϕ, the electrostatic wave potential, is normalized by \(\frac{T_{e}}{e}\). The space and time variables are in the units of electron Debye length \(\lambda_{De}=(\frac{T_{e}}{4\pi n_{e0}e^{2}})^{\frac{1}{2}}\) and \(\frac{\lambda_{De}}{C_{s}}\), respectively. We have also defined \(\alpha=\frac{n_{p0}}{n_{e0}}\), \(\sigma=\frac{T_{i0}}{T_{e}}\), \(\beta=\frac {4\alpha_{1}}{1+3\alpha_{1}}\), where α1 determines the population of nonthermal electrons and \(\eta=\frac{\mu}{(\lambda_{De}C_{s})}\).
In order to investigate the IASWs in unmagnetized e-p-i plasma, we stretch the independent variables as (Washimi and Taniuti 1966; Xue 2003a, 2003b; (Ali and Shukla 2006); Moslem et al. 2010a, 2010b)
$$\xi=\epsilon^\frac{1}{2}(r-v_0t),\qquad\chi=\epsilon^{-\frac{1}{2}} \theta,\qquad\tau=\epsilon^\frac{3}{2} t, $$
where ϵ is a small expansion parameter and v0 is wave phase velocity normalized to Cs.
Employing the reductive perturbation technique, we expand the perturbed quantities ni, ui, vi, and ϕ about their equilibrium values in powers of ϵ, such that
The value of η is assumed to be small, so that we may let
$$ \eta=\epsilon^\frac{1}{2} \eta_0$$
where η0 is O(1).
Substituting (9)–(11) in (5)–(8) and collecting different powers of ϵ, we obtain the following equations to the lowest order in ϵ:
$$n_i^{(1)}=\frac{(1-\alpha)\phi^{(1)}}{v_0^2(1-\frac{\gamma \sigma}{v_0^2})},\qquad u_i^{(1)}=\frac{\phi^{(1)}}{v_0(1-\frac{\gamma \sigma}{v_0^2})}.$$
Solving for v0, we get
$$ v_0=\sqrt{\frac{(1-\alpha)+\gamma\sigma(1-\beta+\alpha-\alpha\beta)}{(1-\beta+\alpha-\alpha\beta)}}$$
The next higher order equations in ϵ are given by
Now, eliminating \(n_{i}^{(2)}\), \(u_{i}^{(2)}\), and ϕ(2) from (13)–(16), we obtain the following Kadomstev-Petviashvili-Burgers (KPB) equation:
where Ξ=0, 1 for planar and nonplanar geometries, respectively. A and C are the coefficients of nonlinearity and dissipation, respectively, whereas B and D are the coefficients of dispersion. ν=0 and Ξ=0 correspond to the KPB equation in planar geometry, whereas ν=1,2 and Ξ=1 correspond to the KPB equations in the cylindrical and spherical geometries, respectively.

3 Numerical results and discussions

There are several methods to solve the nonlinear partial differential equations, for instance, inverse scattering method (Ablowitz and Clarkson 1991), Hirota bilinear formalism (Hirota 1971), Backlund transformation (Miura 1978), tanh method (Malfliet 1992, 2004), etc. However, when the partial differential equation in a system is governed by the combined effect of dispersion and dissipation, the most convenient method to solve the nonlinear partial differential equation is tanh method (Malfliet 1992, 2004). The initial profile that we have used in our numerical analysis is
Equation (22) is a special shock wave solution for (17) obtained by solving the planar KPB equation using the tanh method (Malfliet 1992, 2004). In the above solution, mainly the factor \(\frac{C}{10B}\) determines the steepness of the shock. It is clear that the nonlinear coefficient A does not affect the shock steepness, whereas the weak transverse dispersion coefficient D affects neither the shock height nor its steepness. It only plays a role in shifting the shock from its initial position with the passage of time. When the geometrical effect is taken into account, an exact analytical solution of (17) is not possible. Therefore, it is necessary to plot numerical solution of (17) for a better understanding the nature of the shock wave. The practical applications of the nonplanar geometries are laser-induced implosion, shock tube, star formation, supernova explosions, etc. The numerical results of (17) for in-going shock waves are shown in Figs. 16. The initial condition that we have used in our numerical results is the form of the stationary solution (22) at τ=−20. In Fig. 1 we have plotted the shock wave profile for planar and nonplanar (cylindrical/spherical) geometries at τ=−5. It is seen that strength of the shock is maximum for the spherical geometry, intermediate for cylindrical geometry, while it is minimum for the planar geometry. Again, the steepness of the shock front follows the same trend as that of strength of the shock. Also, it is mentioned that the increase in steepness tells us that the change in parameters downstream of the shock is drastic while the amplitude of the shock determines its strength. The structures of the shock profile in nonplanar geometries differs significantly from the planar geometry due to the presence of ν/τ term in (17). If we increase |τ| to very large values, the nonplanar geometries would approach the planar geometry.
Fig. 1

Plot of ϕ(1) against ξ for different values of ν for the solution of (17), where χ=−50, σ=0.01, α=0.1, β=0.5, γ=3, τ=−5, η0=0.4

Figures 2 and 3 show the time evolution of shock structures in cylindrical and spherical geometry, respectively. It is observed that the EPI plasma admits compressive shocks. It is also clear that as time increases (i.e., for smaller |τ|), the height and steepness of the shock wave in cylindrical and spherical geometry increases (due to the decrease of nonlinearity coefficient A) significantly. In spherical geometries, the shock propagate faster in comparison to cylindrical geometries owing to the fact that shocks diverge at large value of potentials in the case of spherical geometry. The increase is more pronounced in the spherical geometry by comparison with the cylindrical geometry meaning that the density compression can be more effectively obtained in a spherical geometry. Figure 4 displays the cylindrical and spherical shock wave structures at different β, the nonthermal parameter. This shows that the shock height and width increases as the value of nonthermal parameter β is increased. Actually, this happens due to the reason that it decreases both the predominant and weak dispersive coefficients and the nonlinearity coefficient. Kinematic viscosities of ions play important role in numerical investigation of KPB equations. The initial speed of nonlinear structure depends on the value of dissipation coefficients. In Fig. 5 the effect of increasing η0 in both cylindrical and spherical geometries are investigated. It is clear that as the total value of viscosity increases (i.e., increasing in dissipation of the system), the strength of shocks increases in e-p-i plasmas and the effect is more pronounced in spherical by comparison with the cylindrical geometry. Physically, increasing the kinematic viscosity is equivalent to increasing the dissipation in the system and, consequently, the observed increase in the shock strength. This tells us that not only the strength of the spherical shock is greater than the cylindrical shock for all the chosen values of kinematic viscosity, but also the change in the shock strength owing to the 1/r2 dependence in the spherical case as opposed to 1/r dependence in the cylindrical case. Figure 6 represents the shock wave structures at different α, the positron concentration, in cylindrical and spherical geometry. It is found that the height and steepness of the shock profile decreases with increasing α. It happens on the basis of the driving force of the ion acoustic wave, as the driving force for the ion acoustic wave is provided by ions inertia. Actually, increase in positron concentration (depopulation of ions) causes decrease in the driving force, which is provided by the ion inertia, and consequently shock wave enervates. Figure 7 explores the influence of the ion temperature σ on shock structure in the considered nonthermal plasma system. It is noticed that shock strength decreases with increasing σ. Physically, the increase in ion temperature can be depicted as increase in ion thermal velocities as a result of which the convection thrives and at the same time dispersion curtails in the system, consequently the shock amplitude is decreased.
Fig. 2

The profiles of shock wave for several values of τ in cylindrical geometry, where the other parameters are same as Fig. 1
Fig. 3

The profiles of shock wave for several values of τ in spherical geometry, where the other parameters are same as Fig. 1
Fig. 4

Cylindrical and spherical shock structure evolved at τ=−10 for several β, where the other parameters are same as Fig. 1
Fig. 5

Cylindrical and spherical shock structure evolved at τ=−10 for several η0, where the other parameters are same as Fig. 1
Fig. 6

Cylindrical and spherical shock structure evolved at τ=−10 for several α, where the other parameters are same as Fig. 1
Fig. 7

Cylindrical and spherical shock structure evolved at τ=−10 for several σ, where the other parameters are same as Fig. 1

4 Conclusion

We have investigated the planar and nonplanar IASWs in an unmagnetized plasma consisting of nonthermal electrons, nonthermal positrons, and singly charged adiabatically hot positive ions. The dynamics of such wave is described by the two dimensional nonplanar KPB equations. The numerical results reveal that a shock wave can exist in a bounded nonplanar geometry under the transverse perturbation, but the ion acoustic shock wave propagating in cylindrical/spherical geometry with transverse perturbation will be deformed as time goes on. It is found that the shock strength and propagation speed of shocks is maximum for the spherical, intermediate for cylindrical, and minimum for the planar geometry. It is seen that the increasing kinematic viscosity enhances the shock strength. We have also shown that the presence of the nonthermally distributed electrons and positrons can change the dynamical evolution of the nonplanar shock wave structures from the one-dimensional planar geometry. Our results may help to understand the salient features of IASWs in space as well as laboratory plasmas.

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© Springer Science+Business Media B.V. 2012