Gradient-drift instability applied to Hall thrusters

The stability of gradient-drift waves in a Hall-type plasma thruster is investigated within the framework of two-fluid ideal magnetohydrodynamics. The analysis is based on the dispersion relation, which includes the effects of equilibrium electron current, finite ion flow velocity, electron inertia, electron temperature, magnetic field and plasma density gradients, and also the Debye length effects. The features of unstable modes are calculated along the thruster channel. Three spatially separated areas of instability are revealed: (i) the near-anode region with long-wavelength azimuthal oscillations, (ii) the main part of the acceleration channel with short-wavelength axial modes destabilized by macroscopic ion flow, and (iii) the plume region characterized by short-wavelength oblique waves.


Introduction
The operation of a Hall-type plasma thruster exhibits a wide range of oscillations [1]. There are usually three typical frequency bands: ∼1-10 kHz, ∼100kHz, and ∼1-10 MHz. The low frequency band is traditionally associated with the axial breathing modes [2,3] and azimuthal rotating spokes [4]. The middle frequency band corresponds to the transient-time oscillations [5]. The high frequency band includes different types of plasma waves and is related both to the high-frequency branch of transient-time oscillations and Rayleightype instabilities [1,6,7]. The observed fluctuations may be responsible for turbulence and anomalous electron transport across the external magnetic field [8,9].
The gradient-drift instability is inherent in partially magnetized plasmas immersed in crossed external electric and magnetic fields [10]. In this paper, we study the stability of gradient-drift modes along the thruster channel of classical plasma thruster SPT-100. Our approach is based on the solution of the dispersion relation obtained recently in [11] within the framework of the advanced ideal two-fluid model. It includes the effects of the equilibrium E×B electron current, finite ions velocity, electron inertia, magnetic field and plasma density gradients, and the Debye length effects.
The paper is organized as follows. In section 2 the dispersion law for the gradient drift modes under consideration is introduced. In section 3 the distribution of the employed plasma parameters along the thruster channel is presented. Section 4 is devoted to the numerical solution of the considered dispersion relation. Studies of finite electron temperature effects and the effects of equilibrium ion flow are performed consistently. A comparison of the obtained features of instability with simple analytical expressions from [12] is given. In section 5 we briefly summarize and discuss the results of the paper.

Dispersion relation for gradient-drift waves
For the description of a Hall-type plasma thruster geometry we consider a simplified slab model in Cartesian coordinates {x, y, z}-see figure 1. The z-coordinate corresponds to the radial direction of the predominant magnetic field B=Be z . The x-coordinate corresponds to the axial direction along the thruster channel, i.e., it is the direction of the external electric field = E E e x and of the accelerated ions velocity = v v e ; x i i in general all equilibrium plasma parameters depend on x. The y coordinate is in the azimuthal direction coinciding with the direction of stationary electron flow = u u e y e e , which is a superposition of[ ] E B -drift velocity in crossed electric and magnetic fields, , and gradient drift velocity in inhomogeneous magnetic field, . Hereafter, the CGS units are used; c is the speed of light, e is the elementary charge, T e is the temperature of electrons.
To study the stability of gradient-drift waves we use the dispersion relation, which follows from the more general one 6 derived in [11]: is the electron cyclotron frequency; ( ) m e,i are the masses of electron and ion, respectively; parameters κ n and κ B describe the gradients of plasma density, n, and of magnetic field magnitude along the thruster . The perturbations are characterized by the frequency ω, and by the transverse (with respect to equilibrium magnetic field) wave-vector = + k k k x y 2 2 . Due to the geometry used, we call the perturbations with k x ?k y axial perturbations and perturbations with k y ?k x azimuthal ones. Below a right-handed coordinate system is used, so that Dispersion relation (1) describes electrostatic plasma perturbations in the frequency range between the ion and electron cyclotron frequencies, w w w   Bi Be , propagating strictly perpendicularly to the external magnetic field. The first two terms in equation (1) are related to electron inertia and the Debye length effects. The third term corresponds to the response of ions and includes the ion flow velocity, v i . The last term is due to the electron response. It incorporates the equilibrium electron flow, u e , and the gradients of plasma density and magnetic field magnitude. In the equation (1) the effects of temperature perturbations and of equilibrium temperature gradient are not included and the cold ion approximation, w k v Ti (v Ti is the ion thermal velocity), is used.

Axial profiles of equilibrium plasma parameters
In the classical coaxial Hall thruster the magnetic field increases from the anode region and reaches its maximum at the channel exit. Then it decreases in the plume region. To approximate this behavior, we use the expression from [13]: Here B m is a maximal value of the magnetic field at position x=d corresponding to the exit plane; x=0 indicates the position of the anode; ν 1 is a coefficient characterizing the rate of change of function B(x). According to the data from different thrusters studies (see, e.g., [13][14][15]) the plasma density profile often has the same shape as the magnetic field. It reaches its maximum inside the acceleration channel and rapidly decreases towards the channel exit. Thus, for the plasma density profile we can use the following approximation: where n m is a maximal value of the plasma density at some point < l d; 1 ν 2 is a coefficient characterizing the rate of change of function n(x). Then the parameters κ B and κ n linearly depend on x: Here d is the length of the acceleration channel. 6 The dispersion relation of [11] also includes the finite electron Larmor radius (FLR) effects. The negligibility of FLR effects for typical parameters of the Hall plasma thruster is shown in [12].
For further analysis we set ν 1 =2.5, ν 2 =4 and = l d 0.62 1 . For these values the profiles (2) and (3) are in good qualitative agreement with the smoothed profiles in the SPT-100 thruster presented in [14]. The considered profiles of and k ( ) x n are shown in figure 2. To describe the equilibrium electric field we introduce the electrostatic potential Φ in the form: where Φ m is the potential on the anode; l 2 corresponds to some point inside the acceleration channel (l 2 <d), where the electric field reaches its maximal value; 2 is a normalizing factor; ν 3 is a coefficient characterizing the rate of change of function Φ(x) and a 2 is a parameter which allows us to set a minimal value of the potential behind the channel exit. Then the profile of axial electric field and a 2 =0.1 expression (5) gives almost the same shape for the electric potential as in [13,14]. The profiles of Φ(x) and E(x) are shown in figure 3.
For the analysis of instability the following absolute values of plasma parameters are used below: B m =180 G, = · n 0.5 10 m 12 cm −3 and Φ m =270 V, and xenon plasma is considered.

Instability behavior in the Hall thruster
In what follows we solve equation (1) numerically with the plasma profiles presented in the previous section and study the features of gradient-drift instability along the thruster channel. If the instability occurs there are many unstable modes simultaneously coexisting in the considered position along the channel. Unless noted otherwise, at fixed axial position we present the features of the most unstable mode of the spectrum-the mode with the largest growth rate.
Three models are considered in succession: the model with cold electrons and stationary ions (u e =V E , v i =0), the model with hot electrons and stationary ions ( In [11] it was analytically shown that the general necessary instability condition for gradient-drift waves with the dispersion law (1) is In a uniform magnetic field (κ B =0) and neglecting equilibrium ion flow (v i =0) condition (7) reduces to the wellknown instability criterion by Simon and Hoh [16,17]: In the general case the gradient-drift instability is due to a combination of electronÉ B and magnetic drift flows and the ion flow parallel to the electric field (the instability drive) and the inhomogeneity of plasma density and magnetic field magnitude in the direction perpendicular to the magnetic field (the instability trigger).
Typical operational regimes of Hall plasma thrusters are characterized by high values of electron current (u e up to 10 8 cm s −1 ). In [12] it is shown that when equilibrium ion flow is neglected the most unstable modes are purely azimuthal ones and in the limit of strong instability drive ( Be Bi 1 2 is the lower-hybrid frequency) their main features can be described by the following expressions: Here * k y , w * r and * g are the wave number, frequency and growth rate of the most unstable mode in the spectrum, correspondingly. The sign of wave phase velocity in the azimuthal direction coincides with the sign of equilibrium electron rotation velocity, u e . For further analysis we fix the sign of k y assuming that k y >0. Below, expressions (8) are compared with the results of the direct numerical calculation.

Plasma with cold electrons
Under the assumption of cold electrons T e →0 the instability drive is only due to electronÉ B-drift, u e =V E , which, for the profiles considered, does not change the sign along the axis. The necessary instability condition (7) in this case takes the simplest form

Plasma with hot electrons
Taking into account the finite electron temperature effects, the instability drive also includes the magnetic drift term, , and the necessary instability condition takes the form: To study the instability behavior we use the temperature profile obtained numerically in [14] and approximate it by a simple analytical expression: A comparison of the instability features in the near-anode region for models with cold and hot electrons is shown in figure 8. The differences between the instability features are as follows: the finite electron temperature effects result in the increase of both the growth rate and frequency-on average by ∼109kHz and ∼48kHz, correspondingly. The most unstable mode is located at  x 0.21 and has the growth rate * g p  2 830 kHz and the frequency * - f 505 kHz. The wavelength of unstable modes changes the most. On average the wavenumber decreases by ∼1.4rad cm −1 and for the most unstable mode it equals *  k 1.78 rad y cm −1 . The decrease in the wavenumber with the increase in instability drive is predicted by analytical formula (8).
The features of instability in the plume region at > x x b out are presented in figure 9. The most unstable mode is characterized by the following features: x=1.47, * g p  2 605 kHz, *  f 396 kHz, and *  k 46 rad y cm −1 . The instability in the plume region is characterized by the shorter wavelengths compared to the near-anode instability. The shorter wavelengths of the most unstable modes in the plume region are due to the weaker instability drive (u e =0 at x out b ) leading to the increase of * k y .
B on x/d. The green color shows the stability region in the model with hot electrons, and the red color shows the unstable regions.

Influence of the equilibrium ion flow
To study the influence of finite ion flow velocity we use an assumption of the ballistic ion acceleration. Then the dependence of v i on the x-coordinate is given by the relation: The obtained profile of v i is shown in figure 3. The numerical solution of equation (1) shows that in the near-anode region ( < x x b in ) the finite ion flow velocity does not affect the instability behavior due to the smallness of v i in this area-see figure 10. The most unstable modes in this region remain the purely azimuthal ones (see figure 10(c), which shows the growth rate of instability in the k x -k y -plane at x=0.21) and their features are the same as described in section 4.2.
It has been shown above that the main part of the acceleration channel (     signs. This is demonstrated in figure 11(c), showing the contour plots of the growth rate in the k x -k y plane at x=0.5.

Summary and discussion
The full picture of gradient-drift instability in the Hall plasma thruster is summarized in figure 13. The acceleration channel can be divided into three spatially-separated areas characterized by different types of unstable oscillations: (i) the near-anode region ( < x x b in ) with purely azimuthal long-wavelength oscillations, (ii) the main part of the acceleration channel ) with axial short-wavelength modes, (iii) the plume region ( > x d) with oblique propagating waves. In the near-anode region the long-wavelength perturbations * - k 0.01 2.1 rad y cm −1 with frequencies in the range * - ( -) f 295 505 kHz propagating in E×B-direction are excited. The mechanism of instability of these oscillations is related to classical azimuthal gradient-drift instability [18], which occurs from the inhomogeneity of plasma density and magnetic field in accordance with the criterion . The features of instability in the near-anode region are well described by analytical expressions (8).
Unstable modes in the main part of the acceleration channel are in the high-frequency range *  f --( ) 0.31 3.39 MHz. They propagate from anode to cathode and have short wavelengths, * -- ( ) k 9.36 15.1 x rad cm −1 . Axial modes are destabilized by the equilibrium ion flow, however, the mechanism of their excitation distinctly differs from the axial current flow instability [19] existing even at k y =0.
In the plume region ( > x d) unstable waves propagate both in x and y directions. The wave rotates in the E×B direction and moves away from the thruster. The unstable modes are characterized by the wide ranges of frequencies --*  ( ) f 0.01 12.16 MHz and wavelengths, *  k x --( ) 0.2 44.3 rad cm −1 and -*  k 0.7 26.1 y rad cm −1 . They are also strongly affected by the equilibrium ion flow.
The effects of finite electron temperature and ion flow velocity considered in the paper appear to be crucial for the description of plasma stability in the Hall thruster. It is shown that the commonly used approximation of cold electrons and stationary ions [1,6,18,20,21] works well only in the nearanode region of the plasma thruster. The electromagnetic effects considered in [18,20,21] [18]. For the considered profiles the electromagnetic effects are important only for the perturbations with  k 0.2 1.8 2 rad 2 cm −2 . Such unstable modes exist only in the very narrow region near x in b -see figure 10(b) -therefore, the electromagnetic effects are not of significant interest for the presented study. The results of calculations performed taking into account the finite electron Larmor radius effects [11,22] differ by less than 10% from the presented results. This small difference is explained by the large value of electron current (instability drive) in typical operational regimes of the Hall plasma thrusters. The non-ideal effects such as the collisions between electrons and neutral atoms and ionization processes can also be important in Hall thrusters. Such effects are outside of the scope of this paper. The analysis of their influence on fluid-like plasma instabilities can be found, e.g., in [23][24][25][26][27].