Interaction of Dust-Acoustic Shock Waves in a Magnetized Dusty Plasma under the Influence of Polarization Force

Abstract

The interaction of dust-acoustic (DA) shock waves in a magnetized dusty plasma under the influence of nonextensively modified polarization force is investigated. The plasma model consists of negatively charged dust, Maxwellian electrons, nonextensive ions, and polarization force. In this investigation, we have derived the expression of polarization force in the presence of nonextensive ions and illustrated the head-on collision between two DA shock waves. The extended Poincaré–Lighthill–Kuo (PLK) method is employed to obtain the two-sided Korteweg–de Vries–Burgers (KdVB) equations and phase shifts of two shock waves. The trajectories and phase shifts of negative potential dust-acoustic shock waves after collision are examined. The combined effects of various physical parameters such as polarization force, nonextensivity of ions, viscosity of dust, and magnetic field strength on the phase shifts of DA shock waves have been studied. The present investigation might be useful to study the process of collision of nonlinear structures in space dusty plasma such as planetary rings where non-Maxwellian particles such as nonextensive ions, negatively charged dust, and electrons are present.

1. Introduction

Dusty plasma has fascinated the attention of many researchers for the study of different kinds of linear as well as nonlinear structures over the past many years. The presence of dust in an electron-ion plasma modifies the behaviour of nonlinear structures and generates new modes such as dust ion-acoustic waves [1] and dust-acoustic waves [2], which have been frequently studied theoretically and experimentally [3–6]. Dust-acoustic waves are low-frequency waves in which the inertia is provided by the dust mass and the restoring force is provided by the inertialess electrons and ions. Numerous investigations have reported the prevalent role of dust encountered in noctilucent clouds, planetary rings, comets, and plasma crystals [7–12] as well as in semiconductor manufacturing and ion implantation [13, 14]. Dust particulates in a dusty plasma are affected by various forces due to the large amount of dust charge on them, among which electrostatic force ${\mathbf{F}}_e=q_d\mathbf{E}$ ( $q_d=-Z_{d}e$ is dust charge, Z_d is the number of electrons residing on dust grain surface, and E is the electric field) has a significant influence on dust. As the bulk plasma is quasineutral, the electric field there is usually much weaker than the electric field in the sheath region. Hamaguchi Farouki [15, 16] reported that polarization force acts on the dust in the background of nonuniform plasma as a result of deformation of Debye sheath around the particulates. Spatial nonuniform dust density means that the dust charge is nonuniformly distributed in plasmas. The polarization force acts in the direction in which Debye length decreases and also acts in the direction opposite to the electrical force independent of the sign of dust charge [17]. Thus, the polarization force has a significant influence on the propagation characteristics of low-frequency linear DA waves. Various researchers have reported the effect of polarization force on different nonlinear structures (solitons and shocks) in the framework of Maxwellian and non-Maxwellian distribution in various plasma environments [17–25]. Sethi et al. [23] studied DA solitary and shock waves under the influence of superthermally modified polarization force in an unmagnetized dusty plasma comprising of negatively charged dust grains, superthermally distributed ions, and Maxwellian electrons. The superthermally modified polarization force was found to be greater for more superthermal particles. Singh et al. [24] illustrated that the superthermality of ions has a significant influence on polarization force which further altered the dynamics of DA cnoidal waves in dusty plasma. In the recent past, Singh and Saini [25] examined the impact of superthermal polarization force on the modulational instability of DA waves and transition of rogue wave triplets to super rogue waves in space dusty plasmas.

Wave-wave interaction is a very fascinating phenomenon apart from the wave propagation in plasmas that can modify the properties of waves. This unique phenomenon of wave-wave interaction can be described by the use of the extended PLK method and has been investigated by numerous authors [26–32]. The interaction between waves can take place as overtaking collision or as head-on collision. It is well acknowledged that solitons preserve their asymptotic form while undergoing collision. During the before and after collisions, the solitary waves retain their shape, and also they have a novel impact on phase shifts. In this investigation, we have focused our study on the head-on collision of two DA shock waves under the effect of polarization force and their trajectories after the collision. The investigations of the head-on collision of nonlinear shock waves under the influence of polarization force have been reported by numerous researchers in the framework of Maxwellian and non-Maxwellian distributions in different plasma systems [26, 27, 31]. Head-on collision of modulated dust-acoustic waves in strongly coupled dusty plasma is investigated by El-Labany et al. [26]. They considered negatively charged dust grains and Maxwellian electrons and ions and studied modulational instability and effect of polarization force on the phase shifts of dust-acoustic solitary waves. Head-on collision of dust-acoustic shock waves in strongly coupled dusty plasmas was investigated by El-Shamy and Al-Asbali [27]. In the recent past, Singh et al. [31] investigated the head-on collision in a dusty plasma among DA multisolitons with ions following the hybrid distribution under the effect of the polarization force. It was found that rarefactive DA multisolitons are formed and phase shifts are strongly influenced by the polarization force.

Although prolific literature on the interaction of DA waves has been reported, yet the combined effects of polarization force and nonextensivity of ions on the interaction of DA shock waves have not been investigated by researchers in dusty plasmas with application to planetary rings and comet tails. The manuscript is structured as follows. Section 2 presents the fluid model for a given plasma system. Derivations of two different types of KdV–Burgers (KdVB) equations and their corresponding phase shifts are displayed in Section 3. Numerical analysis and discussion are presented in Section 4. The last Section 5 is devoted to the conclusions.

2. The Fluid Equations

In statistical mechanics, there exist generally two types of systems such as extensive and nonextensive systems, depending on the range of interparticle forces. The extensive systems are considered for short-range interparticle forces and need Boltzmann–Gibbs (B-G) statistics. On the other hand, nonextensive systems hold for long-range interparticle forces, need generalized B-G statistics, and also include dissipative systems having nonvanishing thermodynamic currents. Tsallis [33] introduced q-nonextensive distribution which is used to model astrophysical regions (i.e., gravitational systems and stellar polytropes) [34, 35] where Maxwellian distribution might become inadequate. Tsallis q-entropy and the probabilistically independent postulate are known to be the basis of the nonextensive statistics. The principle of maximum q-entropy gives the power-law q-distribution which is called q-exponentials in nonextensive statistics. The nonextensivity is also known as pseudoadditivity or nonadditivity which gives both the q-entropy and the energy (for more details, see [36]). The parameter q measures the degree of nonextensivity of the system. The q-nonextensive number density of ions is given as follows [37]: (1) $$\begin{array}{}{n}_i=n_{i_0}{\left[1-\left(q-1\right)\frac{e\psi }{K_B{T}_i}\right]}^{\left(\right.q+1/2\left(q-1\right)\left.\right)}.\end{array}$$

We consider a magnetized dusty plasma comprising of inertial negatively charged dust, nonextensive ions, and Maxwellian electrons to study the head-on collision and phase shifts of two DA shock waves under the influence of nonextensively modified polarization force. Consider charge on negatively charged dust grain be $q_d=-Z_{d}e$ , and $n_{e_0}$ , $n_{i_0}$ , and $n_{d_0}$ are the unperturbed electron, ion, and dust number densities, respectively. Z_d is the dust grain charge. The charge neutrality condition at equilibrium yields $n_{e_0}+Z_d{n}_{d_0}=n_{i_0}$ which can be expressed as μ_e + 1 = μ_i, where ${\mu }_e=n_{e_0}/Z_d{n}_{d_0}=1/\rho -1$ and ${\mu }_i=n_{i_0}/Z_d{n}_{d_0}=\rho /\rho -1$ and $\rho =n_{i_0}/n_{e_0}$ . The electrons are considered Maxwellian with density $n_e=n_{e_0}\exp \left(e\psi |/|K_B{T}_e\right)$ . After Taylor expansion of the RHS of equation (1) for $e\psi /K_B{T}_i\ll 1$ , we have (2) $$\begin{array}{}{n}_i=n_{i_0}\left[1-h_1\frac{e\psi }{K_B{T}_i}+h_2{\left(\frac{e\psi }{K_B{T}_i}\right)}^2\right],\end{array}$$ where h₁ = q + 1/2 and h₂ = (q + 1)(3 − q)/8. The expression for polarization force F_pol is given by [15] (3) $$\begin{array}{}{\mathbf{F}}_{pol}=\frac{q_d^2}{2}\frac{\nabla {\lambda }_D}{{\lambda }_D^2}.\end{array}$$

We have now derived the polarization force (i.e., deformation of Debye sphere around the dust grains) acting on dust grains modified by nonextensive ions in a nonuniform dusty plasma comprising of nonextensive ions and Maxwellian electrons where the expression for modified Debye length λ_D for negative dust (i.e., $n_e{T}_i\ll n_i{T}_e$ ) is ${\lambda }_D\simeq \sqrt{K_B{T}_i/4\pi e^2{n}_i{h}_1}$ in the presence of nonextensive ions. The expression for nonextensively modified polarization force is obtained as (4) $$\begin{array}{}{\mathbf{F}}_{\text{pol}}=-Z_{d}eR{h}_0{\left(\frac{n_i}{n_{i_0}}\right)}^{1/2}\nabla \psi ,\end{array}$$ where $R=Z_d{e}^2/4K_B{T}_i{\lambda }_{D{i}_0}$ measures the polarization force effect, i.e., interaction of highly negatively charged dust with nonextensive ions with ${\lambda }_{D{i}_0}=\sqrt{K_B{T}_i/4\pi e^2{n}_{i_0}{h}_1}$ and $h_0=\left(h_1-\left(2h_2-\left(h_1^2|/|2\right)\left(e\psi |/|K_B{T}_i\right)\right)\right)$ . Thus, nonextensivity of ions (via q) notably alters the polarization force F_pol and also polarization parameter R. When q ⟶ 1, the polarization force expression equation (4) agrees well with the expression for the case of Maxwellian ions and electrons derived by Asaduzzaman et al. [18].

The dynamics of DA shocks waves in polarized dusty plasma is characterized by the following set of normalized fluid equations (continuity, momentum, and Poisson): (5) $$\begin{array}{}\frac{\partial n_d}{\partial t}+\nabla .\left(n_d{\mathbf{u}}_{\mathbf{d}}\right)=0,\end{array}$$ (6) $$\begin{array}{}\frac{\partial {\mathbf{u}}_{\mathbf{d}}}{\partial t}+\left({\mathbf{u}}_{\mathbf{d}}.\nabla \right){\mathbf{u}}_{\mathbf{d}}={\chi }_p\nabla \psi +{\eta }_l{\nabla }^2{\mathbf{u}}_{\mathbf{d}}-\Omega \left({\mathbf{u}}_{\mathbf{d}}\times \widehat{z}\right),\end{array}$$ (7) $$\begin{array}{}{\nabla }^2\psi =n_d+n_e-n_i,\end{array}$$ where $\nabla =\widehat{x}\left(\partial /\partial x\right)+\widehat{y}\left(\partial /\partial y\right)+\widehat{z}\left(\partial /\partial z\right)$ and ${\mathbf{u}}_{\mathbf{d}}=\widehat{x}{u}_{dx}+\widehat{y}{u}_{dy}+\widehat{z}{u}_{dz}$ . n_d, u_d, and ψ are the number density, velocity, and potential of dust fluid, σ = T_i/T_e, and ${\chi }_p=1-R\left(h_1-\left(2h_2+\left(h_1^2|/|2\right)\right)\right)$ . Density n_j (j = d, e, i) is normalized by unperturbed density of dust $n_{d_0}$ , dust velocity u_d by $\sqrt{Z_d{K}_B{T}_i/m_d}$ , ψ by K_BT_i/e, time t by ${\omega }_{pd}^{-1}$ , where ${\omega }_{pd}={\left(4\pi {\left(Z_{d}e\right)}^2{n}_{d_0}|/|m_d\right)}^{1/2}$ , and x by λ_Dd, where ${\lambda }_{Dd}={\left(K_B{T}_i|/|4\pi n_{d_0}{Z}_d{e}^2\right)}^{1/2}$ . ηl is the dust viscosity normalized by ${\omega }_{pd}{\lambda }_{Dd}^2$ . The magnetic field is assumed to be along the z-axis (i.e., $\mathbf{B}=B_0\widehat{\mathbf{z}}$ ), and the dust cyclotron frequency $\mathrm{\Omega }\left(=Z_{d}e{B}_0/m_d\right)$ is normalized by ω_pd.

3. Derivation of KdVB Equations and Phase Shifts

We employ an extended PLK perturbation method to investigate the collision between two DA shock waves in a magnetized dusty plasma. The perturbed quantities are described as follows: (8) $$\begin{array}{}Y=Y_0+{\displaystyle \sum _{r=1}^{\infty }{\mathrm{ϵ}}^{r+1}}{Y}_r,\hspace{1em}\Gamma ={\displaystyle \sum _{r=1}^{\infty }{\mathrm{ϵ}}^{r+2}}{\Gamma }_r,\end{array}$$ where $Y=\left(n_{\text{d}},u_{\text{d}z},\psi \right)$ , Y₀ = (1,0,0), and $\mathrm{\Gamma }=\left(u_{\text{d}x},u_{\text{d}y}\right)$ . The scaling variables x and t are stretched by the new coordinate system ξ, η, and τ in multiple scale variables: (9) $$\begin{array}{}\xi =\mathrm{ϵ}\left(l_{x}x+l_{y}y+l_{z}z-v_{0}t\right)+{\mathrm{ϵ}}^2{M}_0\left(\eta ,\tau \right)+\dots ,\end{array}$$ (10) $$\begin{array}{}\eta =\mathrm{ϵ}\left(l_{x}x+l_{y}y+l_{z}z+v_{0}t\right)+{\mathrm{ϵ}}^2{N}_0\left(\xi ,\tau \right)+\dots ,\end{array}$$ (11) $$\begin{array}{}\tau ={\mathrm{ϵ}}^{3}t,\end{array}$$ where ξ and η denote the opposite side trajectories of DA shock waves and ϵ is a small parameter determining the strength of nonlinearity. The functions M₀(η, τ) and N₀(ξ, τ) are determined later. l_x, l_y, and l_z represent the direction cosines along the x, y, and z directions, respectively, such that $l_x^2+l_y^2+l_z^2=1$ . The value of dust viscosity is considered small and its value is set as ${\eta }_l=ϵ{\eta }_{l_0}$ , where ${\eta }_{l_0}$ is O(1). Making use of equations (8)–(11) into (5)–(7), at the lowest orders of ϵ, we obtain the following first-order equations: (12) $$\begin{array}{}{v}_0\left(-\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right)n_{{\text{d}}_1}+l_z\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right)u_{\text{d}{z}_1}=0,\end{array}$$ (13) $$\begin{array}{}-v_0\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right)u_{{\text{d}}_1}+\left(1-R{h}_1\right)\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right){\psi }_1=0,\end{array}$$ (14) $$\begin{array}{}\left(1-R{h}_1\right)l_x\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right){\psi }_1-\Omega u_{\text{d}{y}_1}=0,\end{array}$$ (15) $$\begin{array}{}\left(1-R{h}_1\right)l_y\left(\frac{\partial }{\partial \xi }+\frac{\partial }{\partial \eta }\right){\psi }_1+\Omega u_{\text{d}{x}_1}=0,\end{array}$$ (16) $$\begin{array}{}{n}_{d1}=-a_1{\psi }_1,\end{array}$$ where a₁ = μ_eσ + μ_ih₁. On simplifying equations (12)–(16), the relations of different first-order physical quantities are obtained as follows: (17) $$\begin{array}{}{\psi }_1={\psi }_{1\xi }\left(\xi ,\tau \right)+{\psi }_{1\eta }\left(\eta ,\tau \right),\end{array}$$ (18) $$\begin{array}{}{n}_{{\text{d}}_1}=-\frac{\left(1-R{h}_1\right)l_z^2}{v_0^2}\left({\psi }_{1\xi }\left(\xi ,\tau \right)+{\psi }_{1\eta }\left(\eta ,\tau \right)\right),\end{array}$$ (19) $$\begin{array}{}{u}_{\text{d}{x}_1}=-\frac{\left(1-R{h}_1\right)l_y}{{\omega }_{cd}}\left(\frac{{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \xi }+\frac{{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \eta }\right),\end{array}$$ (20) $$\begin{array}{}{u}_{\text{d}{y}_1}=\frac{\left(1-R{h}_1\right)l_x}{{\omega }_{cd}}\left(\frac{{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \xi }+\frac{{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \eta }\right),\end{array}$$ (21) $$\begin{array}{}{u}_{\text{d}{z}_1}=-\frac{\left(1-R{h}_1\right)l_z}{v_0}\left({\psi }_{1\xi }\left(\xi ,\tau \right)-{\psi }_{1\eta }\left(\eta ,\tau \right)\right),\end{array}$$ (22) $$\begin{array}{}{n}_{{\text{d}}_1}=-a_1\left({\psi }_{1\xi }\left(\xi ,\tau \right)+{\psi }_{1\eta }\left(\eta ,\tau \right)\right).\end{array}$$

From equation (17), one predicts that there are two DA shock waves in which ${\psi }_{1\xi }\left(\xi ,\tau \right)$ is propagating towards the right and ${\psi }_{1\eta }\left(\eta ,\tau \right)$ is propagating towards the left. On solving equations (17)–(22), the expression for phase velocity in a magnetized dusty plasma under the effect of nonextensively modified polarization force is obtained as $v_0=l_z\sqrt{\left(1-R{h}_1\right)/a_1}$ .

Finally, for the next higher orders of ϵ, we obtain (23) $$\begin{array}{c}-4l_z\frac{{\partial }^2{u}_{\text{d}{z}_3}}{\partial \xi \partial \eta }={\displaystyle \int \left[\frac{\partial {\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \tau }+A{\psi }_{1\xi }\left(\xi ,\tau \right)\frac{\partial {\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial \xi }+B\frac{{\partial }^3{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial {\xi }^3}-C\frac{{\partial }^2{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial {\xi }^2}\right]}\text{d}\eta +{\displaystyle \int \left[\frac{\partial {\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \tau }-A{\psi }_{1\eta }\left(\eta ,\tau \right)\frac{\partial {\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial \eta }-B\frac{{\partial }^3{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial {\eta }^3}+C\frac{{\partial }^2{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial {\eta }^2}\right]}\text{d}\xi \\ +{\displaystyle \int {\displaystyle \int \left[2v_0\frac{\partial M_0}{\partial \eta }+D{\psi }_{1\eta }\left(\eta ,\tau \right)\right]}}\frac{{\partial }^2{\psi }_{1\xi }\left(\xi ,\tau \right)}{\partial {\xi }^2}\text{d}\xi \text{d}\eta \\ +{\displaystyle \int {\displaystyle \int \left[2v_0\frac{\partial N_0}{\partial \xi }+D{\psi }_{1\xi }\left(\xi ,\tau \right)\right]}}\frac{{\partial }^2{\psi }_{1\eta }\left(\eta ,\tau \right)}{\partial {\eta }^2}\text{d}\xi \text{d}\eta .\end{array}$$

From equation (23), it is noticed that the first (second) term of the right side is proportional to η (ξ), and thus, the integrating function is independent of η (ξ). These two secular terms in the above equation must be removed to avoid spurious resonance. Hence, we obtain the following nonlinear equations: (24) $$\begin{array}{}\frac{\partial {\varphi }_{\xi }}{\partial \tau }+A{\varphi }_{\xi }\frac{\partial {\varphi }_{\xi }}{\partial \xi }+B\frac{{\partial }^3{\varphi }_{\xi }}{\partial {\xi }^3}-C\frac{{\partial }^2{\varphi }_{\xi }}{\partial {\xi }^2}=0,\end{array}$$ (25) $$\begin{array}{}\frac{\partial {\varphi }_{\eta }}{\partial \tau }-A{\varphi }_{\eta }\frac{\partial {\varphi }_{\eta }}{\partial \eta }-B\frac{{\partial }^3{\varphi }_{\eta }}{\partial {\eta }^3}+C\frac{{\partial }^2{\varphi }_{\eta }}{\partial {\eta }^2}=0.\end{array}$$

In the above equations, we have considered ${\psi }_{1\xi }={\varphi }_{\xi }$ and ${\psi }_{1\xi }={\varphi }_{\eta }$ for simplicity. The nonlinear, dispersion, and dissipation coefficients, respectively, are given as (26) $$\begin{array}{}A=\frac{-3l_z^2\left(1-R{h}_1\right)}{2v_0}+\frac{v_{0}b}{a_1}+\frac{v_0\left(R\left(2h_2+\left(h_1^2|/|2\right)\right)\right)}{2\left(1-R{h}_1\right)},\end{array}$$ (27) $$\begin{array}{}B=\frac{\left(1-l_z^2\right)\left(1-R{h}_1\right)v_0}{2a_1{\Omega }^2}+\frac{v_0}{2a_1},\end{array}$$ (28) $$\begin{array}{}C=\frac{{\eta }_{l_0}}{2},\end{array}$$ where $b={\mu }_e{\sigma }^2/2-{\mu }_i{h}_2$ . Equations (24) and (25) are the KdV–Burgers equations of two shock waves travelling towards each other in the reference frame of ξ and η.

3.1. Solution of KdVB Equation and Phase Shifts

The steady-state solutions of equations (24) and (25) are obtained as follows [27]: (29) $$\begin{array}{}{\varphi }_{\xi }={\varphi }_m{\left[1-\text{tanh}\left[w\left(\xi -\left(\frac{6C^2}{25B}\right)\tau \right)\right]\right]}^2,\end{array}$$ (30) $$\begin{array}{}{\varphi }_{\eta }={\varphi }_m{\left[1+\text{tanh}\left[w\left(\eta +\left(\frac{6C^2}{25B}\right)\tau \right)\right]\right]}^2.\end{array}$$

The electrostatic potential for weak head-on collision is given as (31) $$\begin{array}{}\Phi ={\varphi }_m{\left[1-\text{tanh}\left[w\left(\xi -\left(\frac{6C^2}{25B}\right)\tau \right)\right]\right]}^2+{\varphi }_m{\left[1+\text{tanh}\left[w\left(\eta +\left(\frac{6C^2}{25B}\right)\tau \right)\right]\right]}^2,\end{array}$$ where ${\varphi }_m=3C^2/25AB$ is the maximum amplitude and w = 10B/C is the width of DA shock waves.

In the next higher order, the third and fourth terms on the right-hand side of equation (23) become secular and we obtain [27] (32) $$\begin{array}{}2v_0\frac{\partial M_0}{\partial \eta }=-D{\varphi }_{\eta },\end{array}$$ (33) $$\begin{array}{}2v_0\frac{\partial N_0}{\partial \xi }=-D{\varphi }_{\xi },\end{array}$$ where (34) $$\begin{array}{}D=\frac{v_{0}b}{a_1}+\frac{\left(1-R{h}_1\right)l_z^2}{2v_0}+\frac{v_0\left(R\left(2h_2+\left(h_1^2|/|2\right)\right)\right)}{2\left(1-R{h}_1\right)}.\end{array}$$

Equations (32) and (33) are solved using appropriate initial and boundary conditions to determine the phase shifts as (35) $$\begin{array}{}\Delta M_0\left(\eta ,\tau \right)\approx -{\mathrm{ϵ}}^2\frac{3{\eta }_{l0}}{50v_0}\frac{CD}{AB},\end{array}$$ (36) $$\begin{array}{}\Delta N_0\left(\xi ,\tau \right)\approx {\mathrm{ϵ}}^2\frac{3{\eta }_{l0}}{50v_0}\frac{CD}{AB}.\end{array}$$

These phase shifts are functions of different coefficients that are dependent on various physical parameters. Therefore, any variation in physical parameters makes a significant change in the phase shifts.

4. Numerical Analysis and Discussion

In this investigation, the head-on collision of DA shock waves in a magnetized dusty plasma consisting of nonextensive ions and Maxwellian electrons using the extended PLK perturbation method is presented. The numerical values of the typical physical parameters of planetary rings [20, 24], $n_{i_0}=5\times {10}^7{\text{cm}}^{-3},n_{e_0}=4\times {10}^7{\text{cm}}^{-3},Z_d=3\times {10}^3,n_d={10}^7{\text{cm}}^{-3},T_e=50\text{eV},T_i=0.05\text{eV},\text{and}R=0-0.14$ , are used for analysis. The shock waves are formed as a result of balance of nonlinearity, dispersion, and dissipation. The KdVB equations have been derived in equations (24) and (25). Numerically, it is seen that the nonlinear coefficient A (equation (26)) is negative (i.e., A < 0) and the dispersion coefficient B (equation (27)) is positive (i.e., B > 0) for the chosen set of parameters. It implies that AB < 0; as a result, negative potential shock structures are formed. Since the various physical parameters such as nonextensivity of ions, polarization force, obliqueness, magnetic field strength, and other plasma parameters have a profound influence on the characteristics of DA shock waves, it is of paramount importance to carry out numerical analysis.

4.1.

Influence of Nonextensivity of Ions (via <i>q</i>) on the Polarization Force

Figure 1 depicts the variation of the polarization force parameter R with nonextensive parameter (q). Since the parameter R is a function of nonextensivity of ions (via q), any variation in q will change the value of R. It is observed that as the nonextensive parameter q is increased, the polarization force parameter R increases. Thus, nonextensivity of ions has a profound influence on the polarization force parameter R (i.e., polarization force).

Figure 1:

The variation of polarization force parameter (R) with nonextensivity of ions (via q).

[Figure omitted. See PDF]

4.2. The Variation of Profile of Negative Polarity DA Shock Waves

Figure 2(a) depicts the variation of the amplitude of negative polarity DA shock waves in R − q plane. It is found that the amplitude of negative polarity DA shock waves is increased with rise in both polarization force parameter (via R) and nonextensive parameter (via q). Figure 2(b) shows that the amplitude of negative polarity shock waves is also enhanced with an increase in both magnetic field strength (via Ω) and obliqueness (via l_z). It is remarked that the amplitude of DA shock waves is modified with the variation in plasma parameters due to the change in nonlinear as well as dispersive effects under the influence of polarization force in the presence of dissipation effect.

Figure 2:

The variation of rarefactive DA shocks profile with respect to ξ (a) for different values of polarization parameter (R) and nonextensivity parameter (q) and (b) for different values of viscosity ${\eta }_{l_0}$ and magnetic field strength whereas other parameters ρ = 1.3, l_z = 0.8, σ = 0.25, Ω = 0.8, and η_l = 0.35 are fixed.

[Figure omitted. See PDF]

4.3. Evolution and Phase Shift of Two Negative Polarity DA Shock Waves

The collision process of DA shock waves in a magnetized dusty plasma for different time intervals is depicted in Figure 3. The mesial line indicated in the figure separates the before and after collision process. The upper red curves illustrate the profiles of DA shocks before collision and lower blue curves for after collision. This whole process illustrates that two opposite directional propagating negative polarity DA shock waves come closer, interact, and then depart (for more clarity, see animation of collision in the supplementary section (available here)). Figures 4(a) and 4(b) depict the variation of phase shift occurred during the collision process between two negative polarity DA shock waves for different values of dust fluid viscosity ( ${\eta }_{l_0}$ ). It is recognised that the magnitude of phase shift escalates with rise in viscosity (clearly seen in color bar). Figures 5(a) and 5(b) depict the variation of phase shift occurred during the collision process between two negative polarity DA shock waves for different values of magnetic field strength (via Ω). It is found that the magnitude of phase shift enhances as the value of Ω increases (see side color bar). Figure 6(a) illustrates the variation of phase shift with nonextensive parameter (q) for different values of polarization parameter (R) in the presence of magnetic field. It is seen that magnitude of phase shift escalates with rise in nonextensive parameter (q) and suppresses with increase in polarization force (via R). Figure 6(b) shows the same variation in the absence of magnetic field. It is found that magnitude of phase shift is more (almost four times greater) than the case in the absence of magnetic field. It is remarked that magnetic field strength has a significant influence on the profile of DA shocks. It is concluded that the phase shift of DA shock waves is significantly affected under the influence of polarization force (via R), nonextensivity parameter (via q), strength of magnetic field (via Ω), and other plasma parameters.

Figure 3:

The collision of two DA shock waves at different time intervals for q = 0.33, R = 0.1, ρ = 1.3, l_z = 0.8, σ = 0.25, Ω = 0.8, and ${\eta }_l=0.2$ . (a) τ = 50, (b) τ = 40, (c) τ = 30, (d) τ = 20, (e) τ = 0, (f) τ = −50, (g) τ = −40, (h) τ = −30, and (i) τ = −20.

[Figure omitted. See PDF]

Figure 4:

Collision process of two DA shock waves for (a) ${\eta }_{l_0}=0.1$ and (b) ${\eta }_{l_0}=0.2$ and ρ = 1.3, l_z = 0.8, σ = 0.25, Ω = 0.8, R = 0.11, and q = 0.33 are fixed.

[Figure omitted. See PDF]

Figure 5:

Collision process of two DA shock waves for (a) Ω = 0.2 and (b) Ω = 0.3 and ρ = 1.3, l_z = 0.8, σ = 0.25, R = 0.11, ${\eta }_l=0.2$ , and q = 0.3 are fixed.

[Figure omitted. See PDF]

Figure 6:

The variation of phase shift of two DA shock waves vs. q for different values of R (a) with magnetic field and (b) without magnetic field and ρ = 1.3, l_z = 0.8, σ = 0.25, and ${\eta }_l=0.2$ are fixed.

[Figure omitted. See PDF]

5. Conclusions

In the present investigation, we have studied the effects of nonextensively modified polarization force on the head-on collision between two DA shock waves in a magnetized dusty plasma composed of negatively charged dust, Maxwellian electrons, and q-nonextensive ions. The extended PLK method is employed to derive the two opposite directional KdVB equations and the expressions for phase shifts after collision. The q-nonextensive ions yield to escalate the polarization force parameter (R). Only negative polarity DA shock waves are formed under the influence of polarization force. The amplitude of DA shock waves and their phase shifts are significantly modified by the physical parameters such as nonextensivity of ions (via q), modified polarization force parameter (via R), viscosity (via ${\eta }_{l_0}$ ), obliqueness (via l_z), and magnetic field strength (via Ω). The time evolution of rarefactive DA shock waves under the influence of polarization force has been illustrated. The physical parameters have an emphatic effect on the characteristics of DA shock waves and their phase shifts occurred due to head-on collision among DA shock waves. It is important to conclude that the results of our present investigation might be useful for understanding the nonlinear features of disturbances in space/astrophysical dusty plasma systems (e.g., Saturn’s rings, interstellar clouds, and cometary environment) where negatively charged dust and nonextensive ions are present.

Data Availability

The derived data generated in this research will be shared on reasonable request to the corresponding author.

Disclosure

This paper was presented as an invited talk at the 12th International Conference on Plasma Science and Applications, held from 11 to 14 November 2019, Lucknow, India.

Conflicts of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as potential conflicts of interest.

Acknowledgments

This work was supported by DRS-II (SAP) no. F 530/17/DRS-II/2015(SAP-I) UGC, New Delhi.

References

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[2] Rao N. N., Shukla P. K., and Yu M. Y., “ Dust-acoustic waves in dusty plasmas,” Planetary and Space Science, vol. 38, no. 4, p. 543, 1990. 10.1016/0032-0633(90)90147-I

[3] Amin M. R., Morfill G. E., and Shukla P. K., “ Modulational instability of dust-acoustic and dust-ion-acoustic waves,” Physical Review E, vol. 58, no. 5, p. 6517, 1998. 10.1103/PhysRevE.58.6517

[4] Mamun A. A. and Shukla P. K., “ Electrostatic solitary and shock structures in dusty plasmas,” Physica Scripta, vol. T98, p. 107, 2002. 10.1238/Physica.Topical.098a00107

[5] Shukla P. K., “ Nonlinear waves and structures in dusty plasmas,” Physics of Plasmas, vol. 10, no. 5, p. 1619, 2003. 10.1063/1.1557071

[6] Shukla P. K. and Mamun A. A., Introduction to Dusty Plasma Physics, Institute of Physics Publishing, Bristol, England, UK, 2002. 10.1887/075030653X

[7] Bouchoule A., Plain A., Blondeau L. P., and Laure C., “ Particle generation and behavior in a silane-argon low pressure discharge under continuous or pulsed radio frequency excitation,” Journal of Applied Physics, vol. 70, 1991. 10.1063/1.349484

[8] Horanyi M. and Mendis D. A., “ The dynamics of charged dust in the tail of comet Giacobini-Zinner,” Journal of Geophysical Research, vol. 91, no. A1, p. 355, 1986. 10.1029/JA091iA01p00355

[9] Horanyi M. and Mendis D. A., “ The effects of electrostatic charging on the dust distribution at Halleyʼs Comet,” The Astrophysical Journal, vol. 307, p. 800, 1986. 10.1086/164466

[10] Mendis D. A. and Rosenberg M., “ Cosmic dusty plasma,” Annual Review of Astronomy and Astrophysics, vol. 32, no. 1, p. 419, 1994. 10.1146/annurev.aa.32.090194.002223

[11] Northrop T. G., “ Dusty plasmas,” Physica Scripta, vol. 45, no. 5, p. 475, 1992. 10.1088/0031-8949/45/5/011

[12] Verheest F., “ Waves and instabilities in dusty space plasmas,” Space Science Reviews, vol. 77, p. 267, 1996. 10.1007/BF00226225

[13] Adhikary N. C., Bailung H., Pal A. R., Chutia J., and Nakamura Y., “ Observation of sheath modification in laboratory dusty plasma,” Physics of Plasmas, vol. 14, no. 10, Article ID 103705, 2007. 10.1063/1.2798046

[14] Samarian A. A., James B. W., Vladimirov S. V., and Cramer N. F., “ Self-excited vertical oscillations in an rf-discharge dusty plasma,” Physical Review E, vol. 64, Article ID 025402, 2001. 10.1103/PhysRevE.64.025402

[15] Hamaguchi S. and Farouki R. T., “ Polarization force on a charged particulate in a nonuniform plasma,” Physical Review E, vol. 49, p. 4430, 1986. 10.1103/PhysRevE.49.4430

[16] Hamaguchi S. and Farouki R. T., “ Plasma particulate interactions in nonuniform plasmas with finite flows,” Physics of Plasmas, vol. 1, p. 21, 1994. 10.1063/1.870608

[17] Khrapak S. A., Ivlev A. V., Yaroshenko V. V., and Morfill G. E., “ Influence of a polarization force on dust acoustic waves,” Physical Review Letters, vol. 102, Article ID 245004, 2009. 10.1103/PhysRevLett.102.245004

[18] Asaduzzaman M., Mamun A. A., and Ashrafi K. S., “ Dust-acoustic waves in nonuniform dusty plasma in presence of polarization force,” Physics of Plasmas, vol. 18, no. 11, Article ID 113704, 2011. 10.1063/1.3657432

[19] Ashrafi K. S., Mamun A. A., and Shukla P. K., “ Time-dependent cylindrical and spherical dust-acoustic solitary and shock waves in a strongly coupled dusty plasma in the presence of polarization force,” Europhysics Letters, vol. 92, no. 1, Article ID 15004, 2010. 10.1209/0295-5075/92/15004

[20] Bandyopadhyay P., Prasad G., Sen A., and Kaw P. K., “ Experimental study of nonlinear dust acoustic solitary waves in a dusty plasma,” Physical Review Letters, vol. 101, Article ID 065006, 2008. 10.1103/PhysRevLett.101.065006

[21] Bandyopadhyay P., Konopka U., Khrapak S. A., Morfill G. E., and Sen A., “ Effect of polarization force on the propagation of dust acoustic solitary waves,” New Journal of Physics, vol. 12, no. 7, Article ID 073002, 2010. 10.1088/1367-2630/12/7/073002

[22] Mamun A. A., Ashrafi K. S., and Shukla P. K., “ Effects of polarization force and effective dust temperature on dust-acoustic solitary and shock waves in a strongly coupled dusty plasma,” Physical Review E, vol. 82, Article ID 026405, 2010. 10.1103/PhysRevE.82.026405

[23] Sethi P., Singh K., and Saini N. S., “ Effect of superthermal polarization force on dust acoustic nonlinear structures,” Zeitschrift für Naturforschung A, vol. 73, no. 9, p. 795, 2018. 10.1515/zna-2018-0079

[24] Singh K., Ghai Y., Kaur N., and Saini N. S., “ Effect of polarization force on dust acoustic cnoidal waves in dusty plasma,” European Physical Journal D, vol. 72, p. 160, 2018. 10.1140/epjd/e2018-90228-2

[25] Singh K. and Saini N. S., “ The evolution of rogue wave triplets and super rogue waves in superthermal polarized space dusty plasma,” Physics of Plasmas, vol. 26, no. 11, Article ID 113702, 2019. 10.1063/1.5119894

[26] El-Labany S. K., El-Taibany W. F., El-Shamy E. F., El-Depsy A., and Zedan N. A., “ Head-on-collision of modulated dust acoustic waves in strongly coupled dusty plasma,” Physics of Plasmas, vol. 19, no. 10, Article ID 103708, 2012. 10.1063/1.4762847

[27] El-Shamy E. F. and Al-Asbali A. M., “ Head-on collision of dust-acoustic shock waves in strongly coupled dusty plasmas,” Physics of Plasmas, vol. 21, no. 9, Article ID 093701, 2014. 10.1063/1.4894478

[28] Ghosh U. N., Chatterjee P., and Roychoudhury R., “ The effect of q-distributed electrons on the head-on collision of ion acoustic solitary waves,” Physics of Plasmas, vol. 19, no. 1, Article ID 012113, 2012. 10.1063/1.3675603

[29] Han J.-N., Yang X.-X., Tian D.-X., and Duan W.-S., “ Head-on collision of ion-acoustic solitary waves in a weakly relativistic electron-positron-ion plasma,” Physics Letters A, vol. 372, no. 27-28, p. 4817, 2008. 10.1016/j.physleta.2008.05.018

[30] Saini N. S. and Singh K., “ Head-on collision of two dust ion acoustic solitary waves in a weakly relativistic multicomponent superthermal plasma,” Physics of Plasmas, vol. 23, no. 10, Article ID 103701, 2016.

[31] Singh K., Sethi P., and Saini N. S., “ Effect of polarization force on head-on collision between multi-solitons in dusty plasma,” Physics of Plasmas, vol. 25, no. 3, Article ID 033705, 2018. 10.1063/1.5020194

[32] Xue J. K., “ Head-on collision of dust-acoustic solitary waves,” Physical Review E, vol. 69, Article ID 016403, 2004. 10.1103/PhysRevE.69.016403

[33] Tsallis C., “ Possible generalization of Boltzmann-Gibbs statistics,” Journal of Statistical Physics, vol. 52, no. 1-2, p. 479, 1988. 10.1007/BF01016429

[34] Lima J. A. S., Silva R. Jr., and Santos J., “ Plasma oscillations and nonextensive statistics,” Physical Review E, vol. 61, no. 3, p. 3260, 2000. 10.1103/PhysRevE.61.3260

[35] Plastino A. R. and Plastino A., “ Stellar polytropes and Tsallis' entropy,” Physics Letters A, vol. 174, no. 5-6, p. 384, 1993. 10.1016/0375-9601(93)90195-6

[36] Shalini, Saini N. S., and Misra A. P., “ Modulation of ion-acoustic waves in a nonextensive plasma with two-temperature electrons,” Physics of Plasmas, vol. 22, no. 9, Article ID 092124, 2015. 10.1063/1.4931074

[37] Roy K., Saha T., Chatterjee P., and Tribeche M., “ Large amplitude double-layers in a dusty plasma with aq-nonextensive electron velocity distribution and two-temperature isothermal ions,” Physics of Plasmas, vol. 19, no. 4, Article ID 042113, 2012.

References

[1] Shukla P. K. and Silin V. P., “Dust ion-acoustic wave,” Physica Scripta, vol. 45, no. 5, p. 508, 1992.10.1088/0031-8949/45/5/015 [2] Rao N. N., Shukla P. K., and Yu M. Y., “Dust-acoustic waves in dusty plasmas,” Planetary and Space Science, vol. 38, no. 4, p. 543, 1990.10.1016/0032-0633(90)90147-I [3] Amin M. R., Morfill G. E., and Shukla P. K., “Modulational instability of dust-acoustic and dust-ion-acoustic waves,” Physical Review E, vol. 58, no. 5, p. 6517, 1998.10.1103/PhysRevE.58.6517 [4] Mamun A. A. and Shukla P. K., “Electrostatic solitary and shock structures in dusty plasmas,” Physica Scripta, vol. T98, p. 107, 2002.10.1238/Physica.Topical.098a00107 [5] Shukla P. K., “Nonlinear waves and structures in dusty plasmas,” Physics of Plasmas, vol. 10, no. 5, p. 1619, 2003.10.1063/1.1557071 [6] Shukla P. K. and Mamun A. A., Introduction to Dusty Plasma Physics, Institute of Physics Publishing, Bristol, England, UK, 2002.10.1887/075030653X [7] Bouchoule A., Plain A., Blondeau L. P., and Laure C., “Particle generation and behavior in a silane-argon low pressure discharge under continuous or pulsed radio frequency excitation,” Journal of Applied Physics, vol. 70, 1991.10.1063/1.349484 [8] Horanyi M. and Mendis D. A., “The dynamics of charged dust in the tail of comet Giacobini-Zinner,” Journal of Geophysical Research, vol. 91, no. A1, p. 355, 1986.10.1029/JA091iA01p00355 [9] Horanyi M. and Mendis D. A., “The effects of electrostatic charging on the dust distribution at Halleyʼs Comet,” The Astrophysical Journal, vol. 307, p. 800, 1986.10.1086/164466 [10] Mendis D. A. and Rosenberg M., “Cosmic dusty plasma,” Annual Review of Astronomy and Astrophysics, vol. 32, no. 1, p. 419, 1994.10.1146/annurev.aa.32.090194.002223 [11] Northrop T. G., “Dusty plasmas,” Physica Scripta, vol. 45, no. 5, p. 475, 1992.10.1088/0031-8949/45/5/011 [12] Verheest F., “Waves and instabilities in dusty space plasmas,” Space Science Reviews, vol. 77, p. 267, 1996.10.1007/BF00226225 [13] Adhikary N. C., Bailung H., Pal A. R., Chutia J., and Nakamura Y., “Observation of sheath modification in laboratory dusty plasma,” Physics of Plasmas, vol. 14, no. 10, Article ID 103705, 2007.10.1063/1.2798046 [14] Samarian A. A., James B. W., Vladimirov S. V., and Cramer N. F., “Self-excited vertical oscillations in an rf-discharge dusty plasma,” Physical Review E, vol. 64, Article ID 025402,2001.10.1103/PhysRevE.64.025402 [15] Hamaguchi S. and Farouki R. T., “Polarization force on a charged particulate in a nonuniform plasma,” Physical Review E, vol. 49, p. 4430, 1986.10.1103/PhysRevE.49.4430 [16] Hamaguchi S. and Farouki R. T., “Plasma particulate interactions in nonuniform plasmas with finite flows,” Physics of Plasmas, vol. 1, p. 21, 1994.10.1063/1.870608 [17] Khrapak S. A., Ivlev A. V., Yaroshenko V. V., and Morfill G. E., “Influence of a polarization force on dust acoustic waves,” Physical Review Letters, vol. 102, Article ID 245004, 2009.10.1103/PhysRevLett.102.245004 [18] Asaduzzaman M., Mamun A. A., and Ashrafi K. S., “Dust-acoustic waves in nonuniform dusty plasma in presence of polarization force,” Physics of Plasmas, vol. 18, no. 11, Article ID 113704, 2011.10.1063/1.3657432 [19] Ashrafi K. S., Mamun A. A., and Shukla P. K., “Time-dependent cylindrical and spherical dust-acoustic solitary and shock waves in a strongly coupled dusty plasma in the presence of polarization force,” Europhysics Letters, vol. 92, no. 1, Article ID 15004, 2010.10.1209/0295-5075/92/15004 [20] Bandyopadhyay P., Prasad G., Sen A., and Kaw P. K., “Experimental study of nonlinear dust acoustic solitary waves in a dusty plasma,” Physical Review Letters, vol. 101, Article ID 065006, 2008.10.1103/PhysRevLett.101.065006 [21] Bandyopadhyay P., Konopka U., Khrapak S. A., Morfill G. E., and Sen A., “Effect of polarization force on the propagation of dust acoustic solitary waves,” New Journal of Physics, vol. 12, no. 7, Article ID 073002, 2010.10.1088/1367-2630/12/7/073002 [22] Mamun A. A., Ashrafi K. S., and Shukla P. K., “Effects of polarization force and effective dust temperature on dust-acoustic solitary and shock waves in a strongly coupled dusty plasma,” Physical Review E, vol. 82, Article ID 026405, 2010.10.1103/PhysRevE.82.026405 [23] Sethi P., Singh K., and Saini N. S., “Effect of superthermal polarization force on dust acoustic nonlinear structures,” Zeitschrift für Naturforschung A, vol. 73, no. 9, p. 795, 2018.10.1515/zna-2018-0079 [24] Singh K., Ghai Y., Kaur N., and Saini N. S., “Effect of polarization force on dust acoustic cnoidal waves in dusty plasma,” European Physical Journal D, vol. 72, p. 160, 2018.10.1140/epjd/e2018-90228-2 [25] Singh K. and Saini N. S., “The evolution of rogue wave triplets and super rogue waves in superthermal polarized space dusty plasma,” Physics of Plasmas, vol. 26, no. 11, Article ID 113702, 2019.10.1063/1.5119894 [26] El-Labany S. K., El-Taibany W. F., El-Shamy E. F., El-Depsy A., and Zedan N. A., “Head-on-collision of modulated dust acoustic waves in strongly coupled dusty plasma,” Physics of Plasmas, vol. 19, no. 10, Article ID 103708, 2012.10.1063/1.4762847 [27] El-Shamy E. F. and Al-Asbali A. M., “Head-on collision of dust-acoustic shock waves in strongly coupled dusty plasmas,” Physics of Plasmas, vol. 21, no. 9, Article ID 093701, 2014.10.1063/1.4894478 [28] Ghosh U. N., Chatterjee P., and Roychoudhury R., “The effect of q-distributed electrons on the head-on collision of ion acoustic solitary waves,” Physics of Plasmas, vol. 19, no. 1, Article ID 012113, 2012.10.1063/1.3675603 [29] Han J.-N., Yang X.-X., Tian D.-X., and Duan W.-S., “Head-on collision of ion-acoustic solitary waves in a weakly relativistic electron-positron-ion plasma,” Physics Letters A, vol. 372, no. 27-28, p. 4817, 2008.10.1016/j.physleta.2008.05.018 [30] Saini N. S. and Singh K., “Head-on collision of two dust ion acoustic solitary waves in a weakly relativistic multicomponent superthermal plasma,” Physics of Plasmas, vol. 23, no. 10, Article ID 103701, 2016. [31] Singh K., Sethi P., and Saini N. S., “Effect of polarization force on head-on collision between multi-solitons in dusty plasma,” Physics of Plasmas, vol. 25, no. 3, Article ID 033705, 2018.10.1063/1.5020194 [32] Xue J. K., “Head-on collision of dust-acoustic solitary waves,” Physical Review E, vol. 69, Article ID 016403, 2004.10.1103/PhysRevE.69.016403 [33] Tsallis C., “Possible generalization of Boltzmann-Gibbs statistics,” Journal of Statistical Physics, vol. 52, no. 1-2, p. 479, 1988.10.1007/BF01016429 [34] Lima J. A. S., Silva R.Jr., and Santos J., “Plasma oscillations and nonextensive statistics,” Physical Review E, vol. 61, no. 3, p. 3260, 2000.10.1103/PhysRevE.61.3260 [35] Plastino A. R. and Plastino A., “Stellar polytropes and Tsallis' entropy,” Physics Letters A, vol. 174, no. 5-6, p. 384, 1993.10.1016/0375-9601(93)90195-6 [36] Shalini, Saini N. S., and Misra A. P., “Modulation of ion-acoustic waves in a nonextensive plasma with two-temperature electrons,” Physics of Plasmas, vol. 22, no. 9, Article ID 092124, 2015.10.1063/1.4931074 [37] Roy K., Saha T., Chatterjee P., and Tribeche M., “Large amplitude double-layers in a dusty plasma with aq-nonextensive electron velocity distribution and two-temperature isothermal ions,” Physics of Plasmas, vol. 19, no. 4, Article ID 042113, 2012.