Density gradient driven instability in an E→×B→ plasma system having temperature gradients

Abstract

A theoretical model is developed to investigate the density gradient driven instability in an $\overrightarrow{E}\times \overrightarrow{B}$ plasma system with the consideration of finite temperature of both the electrons and the ions where these species are magnetized. The dependence of temperature of the electrons and ions, charge of the ions, plasma background density, and strength of magnetic field on the variation of normalized growth rate has been investigated and portrayed for different channel lengths. The ion temperature gradient that generally exists in the considered plasma systems is also included to investigate its impact on the growth rate of the instability.

KEYWORDS:

• Density gradient driven instability
• channel length
• $\overrightarrow{E}\times \overrightarrow{B}$ plasma
• temperature gradient
• magnetic field

1. Introduction

From the last more than two decades, Hall thrusters have been widely adopted among several propulsion devices because of their higher thrust to power ratio and lower input power. Hall thrusters find diverse applications in commercial space missions and also in telecommunication satellites [1]. On account of electrons’ azimuthal drift, they are termed as closed drift thrusters. Depending on the framework and material of the discharged channel, thrusters are of usually two types: Anode Layer Thruster and Magnetic Layer Thruster. The latter is comprised of one expanded acceleration channel, whereas the former is composed of two-stage channels for acceleration and ionization separately, which are separated by an electrode [2, 3]. Because of the existence of radial magnetic field and axial electric field, Hall thrusters provide large thrust density and a high impulse. This in turn results in the acceleration of propellants to an immense velocity. Both acceleration and ionization of the propellant are achieved inside the insulated discharge region (cylindrical chamber), called a channel, in the existence of radial magnetic field and axial electric field. The erosion of the channel walls may occur due to the bombardment of the ions and the electrons. Consequently, the lifetime and efficiency of Hall thrusters may be affected significantly [3, 4]. Moreover, a disruption in the communication through satellites may occur because of the electrostatic charging, produced due to the existence of extensive exhaust beam divergence [5]. Therefore, several imperative parameters like optimum thrust to power ratio, minimum erosion rate, lower beam divergence, etc. are required to be analyzed for establishing a hassle-free and an effective thruster’s operation. Distinct varieties of the instabilities are originated from the oscillations developed in the chamber.

Several varieties of discharge instabilities originated in Hall thrusters, within the domain of 10³–10⁹ Hz, have been discussed by Choueiri [2]. The growth of Rayleigh instability in a Hall thruster channel in the presence of dust has been investigated by Malik et al. [6]. The transport and characteristics effects of the electron drift instability in Hall-effect thrusters were explored by Lafleur et al. [7]. High-frequency instability and phenomena of electron transport in a Hall thruster have been investigated by Lazurenko et al. [8]. With the negligence of thermal motion of the plasma species, Litvak and Fisch [9] have analytically investigated the gradient-driven Rayleigh-type instabilities in Hall thrusters. The numerical study of low-frequency oscillations in Stationary Plasma Thruster (SPT) have been carried out by Chable and Rogier [10].

In the azimuthal direction, an electric field is generated because of the difference in electron and ion drift velocities in the Hall thrusters. In the same direction, it leads to the generation of the electrostatic waves. In the plasma, such waves could absorb free energy from the density gradient, built-up and lead to instability. Therefore, with the extensive interpretation of physical phenomena accountable for such instabilities, we can have a better analysis of efficiency of Hall thrusters. Hence, with the development of these devices, the concern of investigation of instabilities has always been the centre of attention for several researchers [11–14]. Barral and Ahedo [15] have been developed a low-frequency approach of breathing oscillations. Barral and Peradzynski [16] have investigated the low-frequency oscillations under the impact of ionization. In the strong magnetic field regimes, Meezan et al. [17] have investigated the impact of fluctuations to electron transport. In a coaxial $\overrightarrow{E}\times \overrightarrow{B}$ discharge plasma, an approach to determine the azimuthal drift current has been devised by Thomas et al. [18]. By dealing with the finite temperature of electrons, a theoretical model has been presented by Kapulkin and Guelman [19] to analyse the low-frequency instabilities around anode regime. On the other hand, Sedighi et al. [20, 21] have investigated relevant instabilities in fluid-conveying C-BN hybrid – or hetero-nanotube under a magneto-thermal environment. Makinde et al. [22] have investigated the role of transverse magnetic field on the development of steady flow regime for an incompressible fluid in the boundary layer limit of a semi-infinite vertical plate. Khurana et al. [23] have used linear theory based on normal mode to study the magneto-convection influenced by a gyrotactic behaviour of algal suspensions along with rotation in the nanoliquid layer.

In this paper, we have developed a theoretical model for analyzing density gradient driven instabilities in an $\overrightarrow{E}\times \overrightarrow{B}$ plasma where a gradient exists in the ion and electron temperature. This is inspired due to the fact that under the action of cross field, the plasma species can have different drifts and collisions because of which there are fair chances of energy exchange and alteration of their distribution. This can lead to the gradient in temperature of the ions and electrons. In the proposed system two types of gradient-driven instabilities are found to evolve, out of which one grows with a very small rate and hence, is not evaluated in detail. The growth rate is investigated through the solution of the dispersion equation. The magnitude of normalized growth rate is determined and depicted as a function of the magnetic field, charge on ions, ion temperature gradient, channel length, background plasma density and temperature of the ions and electrons. The finding of this work shall play an imperative role in the technologies like electric propulsion [24] and magnetized plasma sources used in plasma material interaction / surface processing [25–28].

2. Mathematical formulation

In the present work, we have considered both the ions and the electrons to be magnetized under the impact of the external magnetic field ($B_0$), applied in the z-direction. We have assumed non-zero temperatures of both the ions and the electrons; thus, a non-zero pressure-gradient term has been considered in the respective fluid equations. The two-dimensional adiabatic approximation is adopted to investigate the given system, i.e. space coordinates are chosen to be varied in the x – and y – directions.

In the present model, ${\text{m}}_{\text{j}}$ is the mass, ${\text{T}}_{\text{j}}$ is the temperature and ${\text{n}}_{\text{j}}$ is the density of the ions ($j=i$) and the electrons ($j=e$). $-e$ and $Ze$, respectively, are the charges of the electrons and the ions. V and U depict the velocities of the ions and the electrons, respectively. The oscillating and un-perturbed parts of densities are represented by $\left(n_{i1},n_{e1}\right)$ and $\left(n_{i0},n_{e0}\right)$ respectively, whereas those of velocities are represented by $\left(V_{x1},U_{x1},V_{y1},U_{y1}\right)$ and $\left(V_{x0},U_{x0},V_{y0},U_{y0}\right)$, respectively. The electric field has its oscillating value as ${\overrightarrow{E}}_1$ and the associated potential is ${\phi }_1$. The oscillating quantities are taken to vary as e^{i(ωt − ky)}, where ω is the frequency of oscillations and $k$ is the wavenumber associated with the oscillations of wavelength $\lambda$.

Continuity equations and equations of motion for both the ion and electron fluids, in the linearized form, are written as

2.1. For ions

Continuity equation (1) $\begin{array}{rl}& \frac{\mathrm{\partial }{\text{n}}_{\text{i}1}}{\mathrm{\partial }\text{t}}+{\text{n}}_{\text{i}0}\frac{\mathrm{\partial }{\text{V}}_{\text{x}1}}{\mathrm{\partial }\text{x}}+{\text{V}}_{\text{x}1}\frac{\mathrm{\partial }{\text{n}}_{\text{i}0}}{\mathrm{\partial }\text{x}}\\ & +{\text{n}}_{\text{i}0}\frac{\mathrm{\partial }{\text{V}}_{\text{y}1}}{\mathrm{\partial }\text{y}}+{\text{V}}_{\text{y}0}\frac{\mathrm{\partial }{\text{n}}_{\text{i}1}}{\mathrm{\partial }\text{y}}=0\end{array}$(1)

The x-component of the equation of motion (2) $\begin{array}{rl}& \frac{\mathrm{\partial }{\text{V}}_{\text{x}1}}{\mathrm{\partial }\text{t}}+{\text{V}}_{\text{y}0}\frac{\mathrm{\partial }{\text{V}}_{\text{x}1}}{\mathrm{\partial }\text{y}}+\frac{\text{Ze}}{{\text{m}}_{\text{i}}}\frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }\text{x}}-\frac{\text{Ze}}{{\text{m}}_{\text{i}}}\left({\text{V}}_{\text{y}0}+{\text{V}}_{\text{y}1}\right){\text{B}}_0\\ & +\frac{2{\text{T}}_{\text{i}}}{{\text{m}}_{\text{i}}{\text{n}}_{\text{i}}}\left(\frac{\mathrm{\partial }{\text{n}}_{\text{i}0}}{\mathrm{\partial }\text{x}}+\frac{\mathrm{\partial }{\text{n}}_{\text{i}1}}{\mathrm{\partial }\text{x}}\right)+\frac{2}{{\text{m}}_{\text{i}}}\frac{\mathrm{\partial }{\text{T}}_{\text{i}}}{\mathrm{\partial }\text{x}}=0\end{array}$(2)

The y-component of the equation of motion (3) $\begin{array}{rl}& \frac{\mathrm{\partial }{\text{V}}_{\text{y}1}}{\mathrm{\partial }\text{t}}+{\text{V}}_{\text{x}1}\frac{\mathrm{\partial }{\text{V}}_{\text{y}0}}{\mathrm{\partial }\text{x}}+{\text{V}}_{\text{y}0}\frac{\mathrm{\partial }{\text{V}}_{\text{y}1}}{\mathrm{\partial }\text{y}}+\frac{\text{Ze}}{{\text{m}}_{\text{i}}}\frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }\text{y}}+\frac{\text{Ze}}{{\text{m}}_{\text{i}}}{\text{V}}_{\text{x}1}{\text{B}}_0\\ & +\frac{2{\text{T}}_{\text{i}}}{{\text{m}}_{\text{i}}{\text{n}}_{\text{i}}}\frac{\mathrm{\partial }{\text{n}}_{\text{i}1}}{\mathrm{\partial }\text{y}}=0\end{array}$(3)

2.2. For electrons

Continuity equation (4) $\begin{array}{rl}& \frac{\mathrm{\partial }{\text{n}}_{\text{e}1}}{\mathrm{\partial }\text{t}}+{\text{n}}_{\text{e}0}\frac{\mathrm{\partial }{\text{U}}_{\text{x}1}}{\mathrm{\partial }\text{x}}+{\text{n}}_{\text{e}0}\frac{\mathrm{\partial }{\text{U}}_{\text{y}1}}{\mathrm{\partial }\text{y}}\\ & +{\text{U}}_{\text{x}1}\frac{\mathrm{\partial }{\text{n}}_{\text{e}0}}{\mathrm{\partial }\text{x}}+{\text{U}}_{\text{y}0}\frac{\mathrm{\partial }{\text{n}}_{\text{e}1}}{\mathrm{\partial }\text{y}}=0\end{array}$(4)

The x-component of equation of motion (5) $\begin{array}{rl}& \frac{\mathrm{\partial }{\text{U}}_{\text{x}1}}{\mathrm{\partial }\text{t}}+{\text{U}}_{\text{y}0}\frac{\mathrm{\partial }{\text{U}}_{\text{x}1}}{\mathrm{\partial }\text{y}}-\frac{\text{e}}{{\text{m}}_{\text{e}}}\frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }\text{x}}+\frac{\text{e}}{{\text{m}}_{\text{e}}}\left({\text{U}}_{\text{y}0}+{\text{U}}_{\text{y}1}\right){\text{B}}_0\\ & +\frac{2{\text{T}}_{\text{e}}}{{\text{m}}_{\text{e}}{\text{n}}_{\text{e}}}\left(\frac{\mathrm{\partial }{\text{n}}_{\text{e}0}}{\mathrm{\partial }\text{x}}+\frac{\mathrm{\partial }{\text{n}}_{\text{e}1}}{\mathrm{\partial }\text{x}}\right)+\frac{2}{{\text{m}}_{\text{e}}}\frac{\mathrm{\partial }{\text{T}}_{\text{e}}}{\mathrm{\partial }\text{x}}=0\end{array}$(5)

The y-component of the equation of motion (6) $\begin{array}{rl}& \frac{\mathrm{\partial }{\text{U}}_{\text{y}1}}{\mathrm{\partial }\text{t}}+{\text{U}}_{\text{x}1}\frac{\mathrm{\partial }{\text{U}}_{\text{y}0}}{\mathrm{\partial }\text{x}}+{\text{U}}_{\text{y}0}\frac{\mathrm{\partial }{\text{U}}_{\text{y}1}}{\mathrm{\partial }\text{y}}-\frac{\text{e}}{{\text{m}}_{\text{e}}}\frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }\text{y}}\\ & -\frac{\text{e}}{{\text{m}}_{\text{e}}}{\text{U}}_{\text{x}1}{\text{B}}_0+\frac{2{\text{T}}_{\text{e}}}{{\text{m}}_{\text{e}}{\text{n}}_{\text{e}}}\frac{\mathrm{\partial }{\text{n}}_{\text{e}1}}{\mathrm{\partial }\text{y}}=0\end{array}$(6)

Here, ${\mathrm{\Omega }}_e=\left(\text{e}{\text{B}}_0/{\text{m}}_{\text{e}}\right)$ and ${\mathrm{\Omega }}_i=\left(\text{Ze}{\text{B}}_0/{\text{m}}_{\text{i}}\right)$, respectively, are the electron-cyclotron and ion-cyclotron frequencies. This can be noted that the ion and electron temperature gradients are also retained in the respective equation of motions, since the plasma species in the considered plasma systems carry finite temperatures and the temperature also shows a spatial variation. The temperature is taken to have a gradient only in the axial direction, i.e. x-axis. The temperature gradient has also been seen in other fluidic motions [29, 30]. The ions and electrons have $\overrightarrow{E}\times \overrightarrow{B}$ drifts in the y-direction.

The above equations (1)–(6) are solved for the oscillating quantities and relations are obtained for the perturbed densities $n_{e1}$and $n_{i1}$ in terms of the oscillating potential ${\phi }_1$. We define $A_1$ = $i\omega -ik{\text{U}}_{\text{y}0}$, $B_1=1+\left(2T_e{k}^2/A_1{m}_e\omega \right)$,  ${\omega }_{\text{e}}=(n_{e0}{e}^2/m_e{ϵ}_0{)}^{\left(1/2\right)}$, $A^{\mathrm{\prime }}=i\omega -ik{\text{V}}_{\text{y}0}$, $B^{\mathrm{\prime }}=1+\left(2T_i{k}^2/A^{\mathrm{\prime }}{m}_i\omega \right)$ and ${\omega }_{\text{i}}=(n_{i0}{e}^2/m_i{ϵ}_0{)}^{\left(1/2\right)}$, and put the expressions of $n_{e1}$and $n_{i1}$ in the following Poisson’s equation (7) ${ϵ}_0{\mathrm{\nabla }}^2{\phi }_1=-Ze{\text{n}}_{\text{i}1}+\text{e}{\text{n}}_{\text{e}1}$(7)

Hence, we obtain (8) $\begin{array}{rl}& \left(\frac{{\mathrm{\partial }}^2{\phi }_1}{\mathrm{\partial }{\text{x}}^2}-{\text{k}}^2{\phi }_1\right)+\frac{{\mathrm{\partial }}^2{\phi }_1}{\mathrm{\partial }{\text{x}}^2}[\frac{{\text{n}}_{\text{e}0}{\text{e}}^2}{X_1{A}_1{B}_1{ϵ}_0{\text{m}}_{\text{e}}{\omega }_{\text{e}}}\\ & -\frac{{\text{Z}}^2{\text{e}}^2{\text{n}}_{\text{i}0}}{S_1{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\omega }_{\text{i}}{ϵ}_0}]+{\text{k}}^2{\phi }_1\left(\frac{{\text{Z}}^2{\text{e}}^2{\text{n}}_{\text{i}0}{\omega }_{\text{i}}}{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}^2{ϵ}_0}\right)\\ & +{{\phi }_1}_{}\left[\frac{\text{Zeik}{P}_1{\omega }_{\text{i}}}{Q_1{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}^2}-\frac{\text{ik}{T}_1{\omega }_{\text{e}}\text{e}}{U_1{\text{m}}_{\text{e}}{\mathrm{\Omega }}_{\text{e}}^2}+\frac{V_1{\text{e}}^2{\text{B}}_0\text{ik}}{W_1{X}_1{\text{m}}_{\text{e}}^2{\omega }_{\text{e}}}\right]\\ & +[\phantom{\frac{2\text{ik}{\text{n}}_{\text{i}0}\text{Ze}}{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}^2{ϵ}_0}\frac{\mathrm{\partial }{\text{T}}_{\text{i}}}{\mathrm{\partial }\text{x}}\frac{\mathrm{\partial }{\text{V}}_{\text{y}0}}{\mathrm{\partial }\text{x}}}-\frac{2{\text{T}}_{\text{i}}{P}_1}{Q_1{\text{m}}_{\text{i}}{\text{n}}_{\text{i}0}{\mathrm{\Omega }}_{\text{i}}}\frac{\mathrm{\partial }{\text{n}}_{\text{i}0}}{\mathrm{\partial }\text{x}}-\frac{2P_1}{Q_1{\mathrm{\Omega }}_i{\text{m}}_{\text{i}}}\frac{\mathrm{\partial }{\text{T}}_{\text{i}}}{\mathrm{\partial }\text{x}}\\ & -\frac{2{\text{T}}_{\text{i}}{P}_1\text{Ze}{\text{B}}_0}{Q_1{\text{m}}_{\text{i}}^2{\mathrm{\Omega }}_{\text{i}}^2{\text{n}}_{\text{i}0}}\frac{\mathrm{\partial }{\text{n}}_{\text{i}0}}{\mathrm{\partial }\text{x}}-\frac{2P_1\text{Ze}{\text{B}}_0}{Q_1{\text{m}}_{\text{i}}^2{\mathrm{\Omega }}_{\text{i}}^2}\frac{\mathrm{\partial }{\text{T}}_{\text{i}}}{\mathrm{\partial }\text{x}}\\ & +\frac{2{\text{T}}_{\text{i}}\text{Zeik}}{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}{ϵ}_0}\frac{\mathrm{\partial }{\text{n}}_{\text{i}0}}{\mathrm{\partial }\text{x}}+\frac{2\text{ik}{\text{n}}_{\text{i}0}\text{Ze}}{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}{ϵ}_0}\frac{\mathrm{\partial }{\text{T}}_{\text{i}}}{\mathrm{\partial }\text{x}}\\ & +\frac{2\text{ik}{\text{Z}}^2{\text{e}}^2{\text{T}}_{\text{i}}{\text{B}}_0}{{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}\mathrm{m}}_{\text{i}}^2{\mathrm{\Omega }}_{\text{i}}^2{ϵ}_0}\frac{\mathrm{\partial }{\text{n}}_{\text{i}0}}{\mathrm{\partial }\text{x}}+\frac{2\text{ik}{\text{Z}}^2{\text{e}}^2{\text{n}}_{\text{i}0}{\text{B}}_0}{{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}\mathrm{m}}_{\text{i}}^2{\mathrm{\Omega }}_{\text{i}}^2{ϵ}_0}\frac{\mathrm{\partial }{\text{T}}_{\text{i}}}{\mathrm{\partial }\text{x}}\\ & +\frac{\left(2T_1{T}_{\mathrm{e}}{\text{ikeB}}_0\right)}{\left(W_1{X}_1\right(m_{\mathrm{e}}{\omega }_{\mathrm{e}}{)}^2)}-\frac{\left(2P_1{T}_{\mathrm{i}}\text{k}{\omega }^2\right)}{\left(Q_1{\text{m}}_{\mathrm{i}}{\mathrm{\Omega }}_{\mathrm{i}}^2{\omega }_{\mathrm{i}}\right)}\\ & +\frac{2\text{ik}{\text{n}}_{\text{i}0}\text{Ze}}{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}^2{ϵ}_0}\frac{\mathrm{\partial }{\text{T}}_{\text{i}}}{\mathrm{\partial }\text{x}}\frac{\mathrm{\partial }{\text{V}}_{\text{y}0}}{\mathrm{\partial }\text{x}}]=0\end{array}$(8)

Here $P_1=\left(2{\text{Z}}^2{\text{e}}^2\text{ik}{\text{n}}_{\text{i}0}{\text{B}}_0/{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}{ϵ}_0\right)+\left({\text{Z}}^3{\text{e}}^3\text{ik}{\text{n}}_{\text{i}0}{B}_0^2/{{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}\mathrm{m}}_{\text{i}}^2{\mathrm{\Omega }}_{\text{i}}^2{ϵ}_0\right)+\left({\text{Z}}^2{\text{e}}^2\text{ik}{\text{n}}_{\text{i}0}{\text{B}}_0/{\mathrm{A}}^{\mathrm{\prime }}{\mathrm{B}}^{\mathrm{\prime }}{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}^2{ϵ}_0\right)\left(\mathrm{\partial }{\text{V}}_{\text{y}0}/\mathrm{\partial }\text{x}\right)$, $Q_1=1+\left(2\text{Ze}{\text{B}}_0/{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}\right)+\left({\text{Z}}^2{\text{e}}^2{\text{B}}_0^2/{\text{m}}_{\text{i}}^2{\mathrm{\Omega }}_{\text{i}}^2\right)+\left(\text{Ze}{\text{B}}_0/{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}^2\right)\left(\mathrm{\partial }{\text{V}}_{\text{y}0}/\mathrm{\partial }\text{x}\right)$, $S_1=1+\left({\mathrm{\Omega }}_i^2/{\omega }_i^2\right)+\left(2Ze{B}_0{\mathrm{\Omega }}_i/m_i{\omega }_i^2\right)+2\left({\mathrm{\Omega }}_i/{\omega }_i^2\right)\left(\mathrm{\partial }{\text{V}}_{\text{y}0}/\mathrm{\partial }\text{x}\right)+\left({\text{Z}}^2{\text{e}}^2{\text{B}}_0^2/{\text{m}}_{\text{i}}^2{\omega }_{\text{i}}^2\right)$, $T_1=\left(ik{n}_{e0}{e}^2{B}_0/A_1{B}_1{m}_e{\mathrm{\Omega }}_e^2{ϵ}_0\right)\left(\mathrm{\partial }{U}_{y0}/\mathrm{\partial }x\right)+\left(2ik{n}_{e0}{e}^2{B}_0/A_1{B}_1{m}_e{\mathrm{\Omega }}_e{ϵ}_0\right)+\left(ik{n}_{e0}{e}^3{B}_o^2/A_1{B}_1{m}_e^2{\mathrm{\Omega }}_e^2{ϵ}_0\right)$, $U_1=1-\left(e{B}_0/m_e{\mathrm{\Omega }}_e^2\right)\left(\mathrm{\partial }{U}_{y0}/\mathrm{\partial }x\right)+\left(2e{B}_0/m_e{\mathrm{\Omega }}_e\right)$, $W_1=1+\left(4{\mathrm{\Omega }}_e^2/{\omega }_e^2\right)-\left(2{\mathrm{\Omega }}_e/{\omega }_e^2\right)\left(\mathrm{\partial }{U}_{y0}/\mathrm{\partial }x\right)$ and $X_1=1+\left(4{\mathrm{\Omega }}_e^2/{\omega }_e^2\right)-\left(e{B}_0/m_e{\omega }_e^2\right)\left(\mathrm{\partial }{U}_{y0}/\mathrm{\partial }x\right)$.

In the acceleration channel, the electron density, the ion density, the electron drift velocity and ion drift velocity have been assumed to follow (9) $A_0=A_{00}\exp \left[-10{\left(\frac{x}{d}\right)}^2\right]$(9) Here, the peak values of ${\text{n}}_{\text{e}0},{\text{n}}_{\text{i}0},{\text{U}}_0$ and ${\text{V}}_0$ are represented by $A_{00}$. Using the above velocities and densities profiles, the un-perturbed part of Equation (8) is solved to get the growth rate of the instabilities. After some mathematical calculations and arranging the terms, the following dispersion equation is obtained (10) $G_1{\omega }^6+G_2{\omega }^5+G_3{\omega }^4+G_4{\omega }^3+G_5{\omega }^2+G_6\omega +G_7=0$(10) where $G_1=QLiF$, $G_2=B^{\mathrm{\prime }\mathrm{\prime }}iLF+QL{X}^{\mathrm{\prime }}+QKiF$, $G_3=A^{\mathrm{\prime }\mathrm{\prime }}LFi+B^{\mathrm{\prime }\mathrm{\prime }}L{X}^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }}KFi+QL{Y}^{\mathrm{\prime }}+QK{X}^{\mathrm{\prime }}$, $G_4=A^{\mathrm{\prime }\mathrm{\prime }}L{X}^{\mathrm{\prime }}+A^{\mathrm{\prime }\mathrm{\prime }}KFi+B^{\mathrm{\prime }\mathrm{\prime }}L{Y}^{\mathrm{\prime }}+QL{Z}^{\mathrm{\prime }}+Q{Y}^{\mathrm{\prime }}K+PI{C}^{\mathrm{\prime }\mathrm{\prime }}Li$, $G_5=A^{\mathrm{\prime }\mathrm{\prime }}L{Y}^{\mathrm{\prime }}+A^{\mathrm{\prime }\mathrm{\prime }}K{X}^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }}L{Z}^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }}K{Y}^{\mathrm{\prime }}+QK{Z}^{\mathrm{\prime }}+PI{C}^{\mathrm{\prime }\mathrm{\prime }}Ki-BNLPIi+PID{C}^{\mathrm{\prime }\mathrm{\prime }}L$, $G_6=A^{\mathrm{\prime }\mathrm{\prime }}L{Z}^{\mathrm{\prime }}+A^{\mathrm{\prime }\mathrm{\prime }}K{Y}^{\mathrm{\prime }}+B^{\mathrm{\prime }\mathrm{\prime }}K{Z}^{\mathrm{\prime }}-BNKPIi+PID{C}^{\mathrm{\prime }\mathrm{\prime }}K-PIDBNL$ and $G_7=A^{\mathrm{\prime }\mathrm{\prime }}K{Z}^{\mathrm{\prime }}-PIDBNK$.

Here, $A^{\mathrm{\prime }\mathrm{\prime }}=MN-Q{N}^2$, $B^{\mathrm{\prime }\mathrm{\prime }}=Mi-2iNQ$, $C^{\mathrm{\prime }\mathrm{\prime }}=AN-Bi$, $X^{\mathrm{\prime }}=iG$, $Y^{\mathrm{\prime }}=iH+CG+EG$, $Z^{\mathrm{\prime }}=CH+EH$, $Q=\left(2T_{i}ik/Q_1{m}_i{\mathrm{\Omega }}_i^2\right)\left[\right(2Z^2{e}^{2}ik{n}_{i0}{B}_0/m_i{\mathrm{\Omega }}_i{ϵ}_0)+(Z^2{e}^{2}ik{n}_{i0}{B}_0/m_i{\mathrm{\Omega }}_i^2{ϵ}_0\left)\right(\mathrm{\partial }{V}_{y0}/\mathrm{\partial }x\left)\right]$, $P=\left(2T_{e}ke/m_e{ϵ}_0\right)\left(\mathrm{\partial }{n}_{e0}/\mathrm{\partial }x\right)$, $N=-ik{V}_{y0}$, $\begin{array}{rl}M& =-\frac{2T_i}{Q^{\mathrm{\prime }}{m}_i{n}_{i0}{\mathrm{\Omega }}_i}\frac{\mathrm{\partial }{n}_{i0}}{\mathrm{\partial }x}[\phantom{\frac{Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{V}_{y0}}{\mathrm{\partial }x}}\frac{2Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i{ϵ}_0}\\ & +\frac{Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{V}_{y0}}{\mathrm{\partial }x}]-\frac{2T_{i}Ze{B}_0}{Q^{\mathrm{\prime }}{m}_i^2{\mathrm{\Omega }}_i^2{n}_{i0}}[\frac{2Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i{ϵ}_0}\\ & +\frac{Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{V}_{y0}}{\mathrm{\partial }x}]+\frac{2T_{i}Zeik}{m_i{\mathrm{\Omega }}_i{ϵ}_0}\frac{\mathrm{\partial }{n}_{i0}}{\mathrm{\partial }x}\\ & +\frac{2ik{Z}^2{e}^2{T}_i{B}_0}{m_i^2{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{n}_{i0}}{\mathrm{\partial }x}-\frac{2}{Q^{\mathrm{\prime }}{m}_i}\frac{\mathrm{\partial }{T}_i}{\mathrm{\partial }x}[\frac{2Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i{ϵ}_0}\\ & +\frac{Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{V}_{y0}}{\mathrm{\partial }x}]-\frac{2Z_e{B}_0}{Q^{\mathrm{\prime }}{m}_i^2{\mathrm{\Omega }}_i^2}\frac{\mathrm{\partial }{T}_i}{\mathrm{\partial }x}\\ & \times [\frac{2Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i{ϵ}_0}+\frac{Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{V}_{y0}}{\mathrm{\partial }x}]\\ & +\frac{2ik{n}_{i0}Ze}{m_i{ϵ}_0}\frac{\mathrm{\partial }{T}_i}{\mathrm{\partial }x}+\frac{2Z^2{e}^{2}ik{n}_{i0}{B}_0}{m_i^2{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{T}_i}{\mathrm{\partial }x}\\ & +\frac{2ik{n}_{i0}Ze}{m_i{\mathrm{\Omega }}_i^2{ϵ}_0}\frac{\mathrm{\partial }{V}_{y0}}{\mathrm{\partial }x}\frac{\mathrm{\partial }{T}_i}{\mathrm{\partial }x},\end{array}$ $Q^{\mathrm{\prime }}=1+\left(2\text{Ze}{\text{B}}_0/{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}\right)+\left({\text{Z}}^2{\text{e}}^2{\text{B}}_0^2/{\text{m}}_{\text{i}}^2{\mathrm{\Omega }}_{\text{i}}^2\right)+\left(\text{Ze}{\text{B}}_0/{\text{m}}_{\text{i}}{\mathrm{\Omega }}_{\text{i}}^2\right)\left(\mathrm{\partial }{\text{V}}_{\text{y}0}/\mathrm{\partial }\text{x}\right)$, $L=i+\left(2T_e{k}^2/m_e{\mathrm{\Omega }}_e^2\right)$, $K=-\left(2T_e{k}^{2}ik{U}_{y0}/m_e{\mathrm{\Omega }}_e^2\right)-ik{U}_{y0}$, $J=2e{B}_0{\mathrm{\Omega }}_e{m}_e+2e^2{B}_0^2-2e{B}_0{m}_e\left(\mathrm{\partial }{U}_{y0}/\mathrm{\partial }x\right)$, $I=m_e{n}_{e0}{\mathrm{\Omega }}_e^2$, $H=-ik{U}_{y0}{m}_e{n}_{e0}{\mathrm{\Omega }}_e^{2}D-2T_e{k}^4{n}_{e0}{U}_{y0}^2$, $G=iD{m}_e{n}_{e0}{\mathrm{\Omega }}_e^2-4T_e{k}^3{n}_{e0}{U}_{y0}+k{U}_{y0}{m}_e{n}_{e0}{\mathrm{\Omega }}_e^2$, $F=-2T_e{k}^2{n}_{e0}-m_e{n}_{e0}{\mathrm{\Omega }}_e^2$, $E=(e^2{B}_0^2-e{B}_0{m}_e(\mathrm{\partial }{U}_{y0}/\mathrm{\partial }x\left)/m_e^2{\mathrm{\Omega }}_e\right)$, $D=-ik{V}_{y0}$, $C=-ik{V}_{y0}$, $B=ik{V}_{y0}+\left(2T_{i}i{k}^3{V}_{y0}/m_i{\mathrm{\Omega }}_i^2\right)$ and $A=i+\left(2T_{i}i{k}^2/m_i{\mathrm{\Omega }}_i^2\right)$.

3. Results and discussion

The above-defined dispersion equation, i.e. Equation (10) has been solved numerically to determine the growth rate of instabilities for various parameters like plasma background density, ion temperature, electron temperature, ion temperature gradient, magnetic field, charge on ions and channel length. In Figures 1–4, the growth rate of the instability has been normalized with the ion-plasma frequency (${\omega }_{\text{i}}$). The behaviour of the normalized growth rate as a function of the magnetic field (in Tesla) for different charge of the ions (Z) is investigated and portrayed in Figure 1 with (i) channel length (d) = 2.5 m and (ii) d = 5 m. The magnitude of normalized growth rate is found to diminish with increasing magnetic field and higher charge of the ions for both the cases of channel length. The reduction in the growth rate with the magnetic field is consistent to the observation of Pachauri et al. [31], who considered dust in the system; the result with higher charge also matches with their finding. Also, a slight reduction in the normalized growth rate is observed for the enlarged channel length from 2.5 m to 5 m. On the other hand, the difference in the magnitude of normalized growth rate for the different charge of the ions is significantly increased with increasing magnetic field for d = 2.5 m, whereas, there is not much enhancement for the case of d = 5 m.

Figure 1. Normalized growth rate as a function of magnetic field (in Tesla) for different charge on ions (Z) with (i) channel length (d) = 2.5 m and (ii) d = 5 m, when $\lambda$ = 5 cm, x = $\lambda /4$, $n_{e00}$ = $n_{i00}$ = ${10}^{18}$ $m^{-3}$, $T_i$ = 0.3 eV, $T_e$ = 1.5 eV, $m_i$ = 1.6. kg, $V_{y00}$ = ${10}^3$ m/s, $U_{y00}$ = ${10}^5$ m/s and $T_i/x$ = 1 eV/m.

Figure 2. Normalized growth rate as a function of ion temperature gradient (in eV/m) for different values of x with (i) channel length (d) = 2.5 m and (ii) d = 5 m, when $\lambda$ = 5 cm, $n_{e00}$ = $n_{i00}$ = ${10}^{18}$ $m^{-3}$, $T_i$ = 0.3 eV, $T_e$ = 1.5 eV, $m_i$ = 1.6. kg, $V_{y00}$ = ${10}^3$ m/s, $U_{y00}$ = ${10}^5$ m/s, Z = 1 and B = 1 T.

Figure 3. Normalized growth rate as a function of ion temperature (in eV) for different electron temperature $\left(T_e\right)$ in eV with (i) channel length (d) = 2.5 m and (ii) d = 5 m when $\lambda$ = 5 cm, x = $\lambda /4$, $n_{e00}$ = $n_{i00}$ = ${10}^{18}$ $m^{-3}$, $m_i$ = 1.6. kg, $V_{y00}$ = ${10}^3$ m/s, $U_{y00}$ = ${10}^5$ m/s, Z = 1, B = 1 T and $T_i/x$ = 1 eV/m.

Figure 4. Normalized growth rate as a function of background plasma density (in $m^{-3}$) for different channel lengths (d) when $\lambda$ = 5 cm, x = $\lambda /4$, $T_i$ = 0.3 eV, $T_e$ = 1.5 eV, $m_i$ = 1.6. kg, $V_{y00}$ = ${10}^3$ m/s, $U_{y00}$ = ${10}^5$ m/s, Z = 1, B = 1 T and $T_i/x$ = 1 eV/m.

An equal temperature gradient is taken for both the species and hence, the normalized growth rate is plotted only as a function of the ion temperature gradient (in eV/m) in Figure 2. This is done for different values of x, i.e. at different positions in the channel as (i) channel length d = 2.5 m and (ii) d = 5 m. The magnitude of the growth rate is found to enhance with an increment in the ion temperature gradient. This can be seen from the basic relation (9) that the gradient reduces with increasing values of x from $\left(\lambda /2\right)$ to $\lambda$. Since the cause for the present instability is the gradient in the temperature as well as in the density, this is expected that the instability grows with higher rate for $x=\left(\lambda /2\right)$ . This is also an important observation that the difference in the magnitude of the normalized growth rate is decreased with the increasing gradient. Also, a small reduction in normalized growth rate is seen for the increased channel length from 2.5 m to 5 m, which confirms the results obtained in Figure 1.

The behaviour of normalized growth rate as a function of ion temperature (in eV) for different electron temperature $\left(T_e\right)$ in eV is studied through Figure 3 with (i) channel length d = 2.5 m and (ii) d = 5 m. Larger growth rate is seen with the increased temperatures of both the ions and the electrons for both the cases of channel length. Also, a significant reduction in the magnitude of the normalized growth rate is recorded with an increased channel length from 2.5 m to 5 m.

Finally in Figure 4, we show the behaviour of normalized growth rate as a function of background plasma density (in $m^{-3}$) for different channel lengths. Here, the magnitude of normalized growth rate is found to increase with the increased background plasma density, whereas it is reduced with the increased channel length. This confirms the results obtained in Figures 1–3, which also portrayed the reduction in normalized growth rate with the increased channel length. This can also be seen that the growth rate of the instability almost saturates for the much higher density. An early saturation takes place for the case of d = 2.5 in comparison to the case of d = 5. This is due to the stronger density gradient present in the system in the case of d = 2.5 that the instability grows with higher growth rate and also saturates early.

The above results show that the instability driven by the density gradient is very sensitive to the spatial variation of the ions and electrons temperature also. The instability does not grow at the same rate throughout the channel rather it depends on the position where one observes the same. The plasma density initially raises the growth of the instability at faster rate, after which the rate saturates. These results were obtained based on the linear analysis of the problem. However, computers (simulation) can do a greater job and provide the real scenario of the instabilities. This is also true that it becomes difficult to demarcate various instabilities while one uses the computer approach, but the present calculations can easily establish the regions for each kind of instability. The present calculations showed two types of the instabilities in the said system, but we plotted the results only for one kind of instability. Actually, the magnitude of the second kind of instability was very small in comparison to the discussed instability. Also, there were some propagating modes as per the dispersion equation (10), but their velocity was very small. For these reasons, we focused only on one instability and left the discussion on the others. Those small magnitude modes and instabilities could be understood as only the noise in the system and are not expected to grow sufficiently.

As an application of the present model, we mention that the $\overrightarrow{E}\times \overrightarrow{B}$ plasma is used in Hall thrusters [6, 14, 32] and magnetic nozzle [24, 33] for prolonged space propulsion/mission. Also, inductively coupled plasma along with such a cross field arrangement has been used for etching [34–36], film deposition [37–39] etc. in semiconductor industries. However, in these applications, only the electrons are magnetized which enhance the plasma production and hence, ultimately control the etching aspect ratio and film quality. Since our calculations have shown that such a plasma is susceptible to the instability if there are gradients in the density and temperature of the plasma species, in view of this the current model would enable the experimentalists to conduct experiments smoothly as they can keep the system away from the instabilities by generating plasma with the required properties.

4. Conclusions

Density gradient driven instability in an $\overrightarrow{E}\times \overrightarrow{B}$ plasma system was explored numerically by assuming the finite temperature of both the plasma species, i.e. the ions and the electrons. All the calculations were performed by considering both the electrons and the ions to be magnetized. The magnitude of normalized growth rate was found to increase with an increment in the ion and the electron temperature and background plasma density, whereas it diminished with the increasing magnetic field strength and charge of the ions. The growth rate was also found to increase when the plasma has larger ion or electron temperature gradient. The dependence of plasma parameters on the growth rate for different channel length was also evaluated and a reduction in its magnitude was observed with the increased channel length.

Nomenclature

$B_0$	=	External magnetic field
$e$	=	Electronic charge
E1	=	Perturbed electric field
$k$	=	Wavenumber
λ	=	Wavelength of oscillations
$m_e$	=	Electron mass
$m_i$	=	Ion mass
${\text{n}}_{\text{e}}$	=	Electron density
${\text{n}}_{\text{e}0}$	=	Background electron density
${\text{n}}_{\text{e}1}$	=	Perturbed electron density
${\text{n}}_{\text{i}}$	=	Ion density
${\text{n}}_{\text{i}0}$	=	Background ion density
${\text{n}}_{\text{i}1}$	=	Perturbed ion density
ω	=	Frequency of oscillations
ωe	=	Electron plasma frequency
Ωe	=	Electron cyclotron frequency
ωi	=	Ion plasma frequency
Ωi	=	Ion cyclotron frequency
${\phi }_1$	=	Perturbed electric potential
${\text{T}}_{\text{e}}$	=	Electron temperature
${\text{T}}_{\text{i}}$	=	Ion temperature
${\text{U}}_0$	=	Electron velocity
${\text{U}}_{\text{x}0}$	=	x-component of electron velocity
${\text{U}}_{\text{y}0}$	=	y-component of electron velocity
${\text{V}}_0$	=	Ion velocity
${\text{V}}_{\text{x}0}$	=	x-component of ion velocity
${\text{V}}_{\text{y}0}$	=	y-component of ion velocity
$Ze$	=	Ion charge

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Government of India and Government of France (IFCPAR/CEFIPRA: Indo-French Centre for the Promotion of Advanced Research) are thankfully acknowledged for providing the financial support through Indo-French project (Proj-5204-3/RP03052).