2.1. Governing equations

The fluid equations are used for both ions and electrons, i.e. when kinetic effects, such as the Bernstein modes (Bernstein Reference Bernstein1958), can be neglected.

Conservation of mass can be constructed by taking the zeroth moment of the kinetic equation. Here, ionisation and recombination are neglected. Thus, the continuity equation can be written as

(2.1)\begin{equation} \frac{\partial n_s}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}(n_s\boldsymbol{u}_s) = 0, \end{equation}

where $n_s$ is the number density and $\boldsymbol {u}_s$ is the bulk velocity for species $s$.

The equation for the fluid momentum can be formulated by taking the first moment of the kinetic equation, which can be written using conservative or primitive variables. Assuming that the plasma is collisionless and the distribution function is close to an isotropic Maxwellian distribution function, the conservation of momentum can be written as

(2.2)\begin{equation} \frac{\partial (m_sn_s\boldsymbol{u}_s)}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot}(m_sn_s\boldsymbol{u}_s\boldsymbol{u_s}) ={-}\boldsymbol{\nabla} p_s + q_s n_s(\boldsymbol{E} + \boldsymbol{u}_s\times\boldsymbol{B}), \end{equation}

where $m_s$ is the mass, $p_s$ is the pressure, $q_s$ is the charge, $\boldsymbol {E}$ is the electric field and $\boldsymbol {B}$ is the magnetic field. Using the source-less continuity equation, as shown in (2.1), the momentum equation can also be given, using the primitive variables, by

(2.3)\begin{equation} \frac{\partial \boldsymbol{u}_s}{\partial t} + (\boldsymbol{u}_s\boldsymbol{\cdot} \boldsymbol{\nabla})\boldsymbol{u}_s ={-}\frac{\boldsymbol{\nabla} p_s}{m_s n_s} + \frac{q_s}{m_s} (\boldsymbol{E} + \boldsymbol{u_s}\times \boldsymbol{B}). \end{equation}

Note that the pressure is a scalar term, which is valid when the velocity distribution function (VDF) is close to an isotropic Maxwellian distribution function, that is, the temperatures in three directions are equal. Under this condition, the pressure can be written using the ideal gas law: $p_s = n_s k_B T_s$, where $k_B$ is the Boltzmann constant and $T_s$ is the temperature for species $s$.

In the present model, the ideal gas law is assumed for the electron fluid model. It is to be noted that standard drift models account for gyroviscosity effects, which arise due to the non-Maxwellian distribution and lead to cancellation of the diamagnetic drift in the inertia term in the momentum equation (Ramos Reference Ramos2005; Schnack et al. Reference Schnack, Barnes, Brennan, Hegna, Held, Kim, Kruger, Pankin and Sovinec2006). Although the gyroviscosity effects may play an important role in low-temperature, partially magnetised plasmas (Smolyakov et al. Reference Smolyakov, Chapurin, Frias, Koshkarov, Romadanov, Tang, Umansky, Raitses, Kaganovich and Lakhin2016), the validity of the drift models and the necessity of including gyroviscosity effects in low-temperature magnetised plasmas needs to be investigated. In the state-of-the-art computational models for low-temperature plasmas, simplified fluid models such as the drift-diffusion model are known to represent the physical processes (Kushner Reference Kushner2009; Hara Reference Hara2019). In this paper, a five-moment model that neglects the gyroviscosity effects is used, based on recent numerical simulations that show low-frequency rotating spokes (Mansour & Hara Reference Mansour and Hara2022). The inclusion of the gyroviscosity effects is reserved for future work.

2.2. Linear perturbation analysis for partially magnetised plasmas

We consider the growth of the instabilities in $y$ direction, that is, the direction in which the electrons drift. Under the linear perturbation analysis, a plasma property $Q$ can be described as a sum of the steady-state quantity and a linear perturbation, such that

(2.4)\begin{equation} Q = Q_0 + Q_1 \exp(-{\rm i}\omega t + {\rm i} k_y y), \end{equation}

where $Q_0$ and $Q_1$ are the equilibrium (steady-state) and first-order perturbation terms of a plasma property $Q$, respectively, $\omega$ is the frequency, $t$ is time and $k_y$ is the wave number in the $y$ direction. Here, $\omega = \omega _r + {\rm i}\gamma$, where $\omega _r$ is the real frequency and $\gamma$ is the imaginary part which corresponds to the growth rate.

2.3. Zeroth-order (equilibrium) equations for magnetised electrons

Inserting (2.4) into the governing equations and considering the zeroth-order terms leads to the equilibrium equations. For magnetised electrons, using (2.3) and considering the equilibrium bulk velocity $\boldsymbol {u}_{e0} = (u_{e0x}, u_{e0y}, u_{e0z})^\intercal$, where subscripts $x$, $y$ and $z$ denote the direction, the steady-state momentum equation in $x$ and $y$ directions can be written as

(2.5)\begin{gather} 0 ={-}\frac{k_B T_e}{m_e n_0}\frac{\partial n_0}{\partial x} -\frac{e}{m_e} (E_0 + u_{e0y}B_0) , \end{gather}

(2.6)\begin{gather}0 = \frac{e}{m_e} u_{e0x}B_0 , \end{gather}

where $e$ is the elementary charge, $T_e$ is the electron temperature, $m_e$ is the electron mass and $n_0$ is the equilibrium density, assuming quasineutrality for the equilibrium condition.

Recall that the equilibrium density gradient and electric field are considered to exist locally only in $x$ direction. Although (2.6) results in $u_{e0x}=0$, (2.5) yields the well-known drifts:

(2.7)\begin{equation} u_{e0y} ={-}\frac{E_0}{B_0} - \frac{k_B T_e}{e n_0} \frac{ n_0'}{B_0}, \end{equation}

where $n_0'={\rm d}n_0/{{\rm d} x}$ is the plasma density gradient. The first term in (2.7) is the $\boldsymbol {E} \times \boldsymbol {B}$ drift and the second term is the diamagnetic drift, which can be written as $u_E$ and $u_*$, respectively. The diamagnetic drift does not come from the single-particle trajectory analysis but appears as an equilibrium drift from the fluid theory, whereas the $\boldsymbol {E} \times \boldsymbol {B}$ drift can be derived from single particle trajectories. Nonetheless, the diamagnetic drift is a steady-state bulk velocity that can propagate in the same or opposite direction of the $\boldsymbol {E} \times \boldsymbol {B}$ drift.

As the density gradient is only considered in $x$ direction and using $u_{e0x} = 0$ obtained from (2.6), the steady-state conservation of mass can be written as

(2.8)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}_{e0} = 0. \end{equation}

This relation shows that $u_{e0y}$ is constant in $y$ direction, i.e. homogeneous, which is consistent with the configuration considered.

2.4. First-order (linear perturbation) equations for magnetised electrons

Let us consider the perturbation terms of the electron bulk velocity to be $\boldsymbol {u}_{e1} = (u_{e1x}, u_{e1y}, 0)^\intercal$ for magnetised electrons. Using (2.8), the first-order conservation of mass can be derived from (2.1) as

(2.9)\begin{equation} \frac{ \partial n_{e1}}{\partial t} + n_0\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{u}_{e1} + \boldsymbol{u}_{e0}\boldsymbol{\cdot}\boldsymbol{\nabla} n_{e1} + \boldsymbol{u}_{e1}\boldsymbol{\cdot} \boldsymbol{\nabla} n_0 = 0. \end{equation}

Using the linear perturbation shown in (2.4), the perturbed electron density can be written as

(2.10)\begin{equation} n_{e1} = \frac{n_0k_yu_{e1y} -{\rm i} u_{e1x} n_0'}{\tilde{\omega}}, \end{equation}

where $\tilde {\omega }= \omega - k_yu_{e0y}$. Note that $u_{e0y}$ is given in (2.7).

The first-order momentum equation can be derived from (2.2). Note that pressure term leads to

(2.11)\begin{equation} \frac{k_B T_e \boldsymbol{\nabla}(n_0+n_{e1})}{m_e(n_0+n_{e1})} =\frac{k_B T_e}{m_e}\left[ \frac{\boldsymbol{\nabla} n_0}{n_0} \left( 1 - \frac{n_{e1}}{n_0} \right) + \frac{\boldsymbol{\nabla} n_{e1}}{n_0} \right], \end{equation}

which results in two contributions to the linearised momentum equation. Thus, using (2.11), the first-order momentum equation for magnetised electrons can be given by

(2.12)\begin{equation} -{\rm i}\tilde{\omega} \boldsymbol{u}_{e1} = v_{{\rm th}}^2 k_n \frac{n_{e1}}{n_0} \hat{\boldsymbol{x}} - v_{{\rm th}}^2 \frac{\boldsymbol{\nabla} n_{e1}}{n_0} - \frac{e}{m_e}(\boldsymbol{E}_1 + \boldsymbol{u}_{e1}\times\boldsymbol{B}_0), \end{equation}

where $v_{{\rm th}} = (k_B T_e/m_e)^{1/2}$ is the electron thermal speed and $k_n = n_0' / n_0$ is the inverse of the density gradient length scale, which is also defined similarly by Sakawa et al. (Reference Sakawa, Joshi, Kaw, Chen and Jain1993) and Smolyakov et al. (Reference Smolyakov, Chapurin, Frias, Koshkarov, Romadanov, Tang, Umansky, Raitses, Kaganovich and Lakhin2016). Taking the perturbation terms in $y$ direction, e.g. $\phi _1 \exp (-{\rm i}\omega t +{\rm i} k_y y)$ for the electric field using the electrostatic assumption: $\boldsymbol {E}=-\boldsymbol {\nabla } \phi$, (2.12) can be written for $x$ and $y$ directions as

(2.13)\begin{equation} \begin{bmatrix} -{\rm i}\tilde{\omega} & \omega_{{\rm ce}}\\ -\omega_{{\rm ce}} & -{\rm i}\tilde{\omega} \end{bmatrix} \begin{bmatrix} u_{e1x}\\ u_{e1y} \end{bmatrix} =\begin{bmatrix} v_{{\rm th}}^2 k_n \frac{n_{e1}}{n_0}\\ -{\rm i} v_{{\rm th}}^2 k_y \frac{n_{e1}}{n_0} +{\rm i} \frac{e}{m_e}k_y\phi_1 \end{bmatrix}, \end{equation}

where $\omega _{{\rm ce}} = eB_0/m_e$ is the electron gyrofrequency. Solving for $u_{e1x}$ and $u_{e1y}$ in (2.13) gives

(2.14)\begin{gather} u_{e1x} = \frac{{\rm i}}{\tilde{\omega}^2-\omega_{{\rm ce}}^2} \left[ \tilde{\omega} k_n v_{{\rm th}}^2 \frac{n_{e1}}{n_0} - \omega_{{\rm ce}}k_y\left(v_{{\rm th}}^2 \frac{n_{e1}}{n_0} - \frac{e}{m_e} \phi_1 \right) \right], \end{gather}

(2.15)\begin{gather}u_{e1y} = \frac{1}{\tilde{\omega}^2-\omega_{{\rm ce}}^2} \left[- \omega_{{\rm ce}}k_n v_{{\rm th}}^2 \frac{n_1}{n_0} + \tilde{\omega}k_y\left( v_{{\rm th}}^2 \frac{n_{e1}}{n_0} - \frac{e}{m_e}\phi_1 \right) \right]. \end{gather}

Equations (2.14) and (2.15) are similarly derived in the Rayleigh–Taylor instability analysis (Chen Reference Chen1984). If cold electrons are assumed (i.e. $v_{{\rm th}} = 0$) and one considers the low-frequency approximation (i.e. $\tilde {\omega }^2 \ll \omega _{{\rm ce}}^2$), (2.14) and (2.15) reduce to $u_{e1x} = E_{1y}/B_0$ and $u_{e1y}= {\rm i}\tilde {\omega } u_{e1x}/\omega _{{\rm ce}}$, which are equivalent to the perturbed $\boldsymbol {E} \times \boldsymbol {B}$ and polarisation drifts, respectively. However, it can be seen that the determinant of the matrix in (2.13) becomes negative if $\tilde {\omega }^2 \gg \omega _{{\rm ce}}^2$. In this case, $u_{e1x}$ moves in the opposite direction of the perturbed $\boldsymbol {E} \times \boldsymbol {B}$ drift, which seems to be non-physical (see Appendix A for some discussions). Therefore, assuming $\tilde {\omega }^2 \ll \omega _{{\rm ce}}^2$ and using (2.14) and (2.15), (2.10) can be written as

(2.16)\begin{equation} \frac{n_{e1}}{n_0} = \frac{e\phi_1 }{m_e} \frac{ k_y^2 \tilde{\omega}- k_nk_y \omega_{{\rm ce}}}{ \omega_{{\rm ce}}^2 \tilde{\omega} + \left( k_y^2 + k_n^2\right)v_{{\rm th}}^2 \tilde{\omega} - 2k_nk_y \omega_{{\rm ce}}v_{{\rm th}}^2}. \end{equation}

Note that this equation is similar to (10) in Sakawa et al. (Reference Sakawa, Joshi, Kaw, Chen and Jain1993), except for the coefficient of the last term in the denominator.

2.5. Dispersion function for unmagnetised ions

Let us derive the zeroth-order equations for unmagnetised ions. If restricting the argument to the cross-field direction ($x$ and $y$), that is, neglecting the plasma dynamics in $z$ direction (along the magnetic field), the steady-state continuity equation gives, $n_0 u_{i0x} ={\rm const}.$, where $u_{i0x}$ is the equilibrium ion bulk velocity in $x$ direction. Assuming that the ion bulk velocity is negligible in $y$ direction (i.e. $u_{i0y} = 0$), the linear perturbation of the ion number density, $n_{i1}$, can be derived from the linearised conservation of mass as

(2.17)\begin{equation} n_{i1} = \frac{n_0k_yu_{i1y} -{\rm i} u_{i1x} n_0'}{ \omega - {\rm i} k_n u_{i0x}}, \end{equation}

where ${\boldsymbol {u}}_{i1} = (u_{i1x}, u_{i1y},0)^\intercal$ is the linear perturbation of the ion bulk velocities.

Assuming cold ions ($T_i= 0$, where $T_i$ is the ion temperature) and only considering perturbation in $y$ direction, as shown in (2.4), the linearised momentum equation for unmagnetised ions can be written as

(2.18)\begin{gather} u_{1x} = 0, \end{gather}

(2.19)\begin{gather}u_{1y} = \frac{k_y}{{\omega}} \frac{e}{m_i}\phi_1, \end{gather}

where $m_i$ is the ion mass. If one further assumes that the effects of $k_n u_{i0x}$ to be small, (2.17) can be reduced to

(2.20)\begin{equation} \frac{n_{i1}}{n_0} = \frac{e k_y^2}{m_i \omega^2}\phi_1. \end{equation}

This is consistent with the dispersion function used for unmagnetised ions in Sakawa et al. (Reference Sakawa, Joshi, Kaw, Chen and Jain1993).

3.1. Fluid dispersion relation

Assuming quasineutrality and perturbations occur in the azimuthal direction, which are similar assumptions employed for the Rayleigh–Taylor instability theory (Chen Reference Chen1984), and using (2.16) and (2.20), the dispersion relation for partially magnetised plasmas can be derived as

(3.1)\begin{equation} \frac{m_e k_y}{m_i \omega^2 } = \frac{ k_y \tilde{\omega}- k_n \omega_{{\rm ce}}}{ [\omega_{{\rm ce}}^2 + \left( k_y^2 + k_n^2\right)v_{{\rm th}}^2] \tilde{\omega} - 2k_nk_y \omega_{{\rm ce}}v_{{\rm th}}^2}, \end{equation}

where $\tilde {\omega } = \omega - k_y u_{e0y}$ is used throughout the derivation. Here, this drift-shifted frequency can be written as $\tilde {\omega } = \omega - \omega _E - \omega _*,$ where $\omega _E = k_y u_E$, $\omega _* = k_y u_*$, $u_E = - E_0 / B_0$ and $u_* = - k_n k_B T_e / (e B_0)$, as can be seen from (2.7). It can be seen that (3.1) yields a third-order equation for $\omega$, from which the damping and linear instability growth can be evaluated.

3.2. Instability criteria in the large-wavelength limit

In the limit of a large-wavelength mode, a simplified form of (3.1) can be obtained by neglecting the $k_y \tilde {\omega }$ term in the numerator of (3.1) compared with $k_n \omega _{{\rm ce}}$, that is, $k_y \tilde {\omega } \ll k_n \omega _{{\rm ce}}$. In addition, $u_* \omega _{{\rm ce}} = - k_n v_{{\rm th}}^2$ gives $k_n^2 v_{{\rm th}}^2 = \tilde {u}_*^2 \omega _{{\rm ce}}^2$, where $\tilde {u}_*=u_*/v_{{\rm th}}$. Therefore, in this limit, (3.1) can be written as

(3.2)\begin{equation} 0 ={-} \frac{k_n m_i }{m_e \omega_{{\rm ce}}} \omega^2 - k_y(1 + \tilde{u}_*^2) \omega + \tilde{u}_*^2 k_y^2 (u_E+u_*) + k_y^2 (u_E-u_*). \end{equation}

For the solution to have an unstable mode, that is, a root with a positive growth rate, the discriminant of (3.2) must be negative. The instability condition can therefore be given by

(3.3)\begin{equation} \frac{m_e }{m_i}(1+\tilde{u}_*^2)^2-4\tilde{u}_* \left[ (1+\tilde{u}_*^2) \tilde{u}_E - (1-\tilde{u}_*^2) \tilde{u}_* \right] < 0, \end{equation}

where $\tilde {u}_E = u_E / v_{{\rm th}}$. By rewriting (3.3) considering $m_e \ll m_i$, the condition for the partially magnetised plasma to have an unstable mode in the limit of large wavelength can be written as

(3.4)\begin{equation} \tilde{u}_* F(\tilde{u}_*, \tilde{u}_E) >0, \end{equation}

where

(3.5)\begin{equation} F(\tilde{u}_*, \tilde{u}_E) = \tilde{u}_*^3 + \tilde{u}_E \tilde{u}_*^2 - \tilde{u}_* + \tilde{u}_E . \end{equation}

As can be seen from (3.4), the two instability criteria can be obtained as (i) $\tilde {u}_*>0$ and $F(\tilde {u}_*, \tilde {u}_E) >0$ and (ii) $\tilde {u}_*<0$ and $F(\tilde {u}_*, \tilde {u}_E) <0$.

It can be seen that (3.3), in the limit of $|\tilde {u}_*| \rightarrow 0$ and considering $m_e \ll m_i$, reduces to $u_E u_* > 0$, which is the instability criterion for the MSHI (Sakawa et al. Reference Sakawa, Joshi, Kaw, Chen and Jain1993). See Appendix B for further discussion about the comparison of the present dispersion relation and MSHI.

Figure 1 shows the shape of (3.5) for two values of $\tilde {u}_E$, that is, $\tilde {u}_E = -0.1$ and $-0.6$, to illustrate the range of unstable and stable roots in the limit of large wavelength and low frequency. Here, negative $u_E$ values are chosen, as the $\boldsymbol {E} \times \boldsymbol {B}$ drift occurs in $-y$ direction when considering $E_0$ in $x$ direction and $B_0$ in $z$ direction. Unlike the MSHI that predicts one region for unstable modes, that is, $\tilde {u}_* \tilde {u}_E > 0$, two unstable regions can be seen in figure 1(a), whereas three unstable regions can be seen in figure 1(b).

Figure 1.

Cubic function, $F(\tilde {u}_*, \tilde {u}_E)$, shown in (3.5) for (a) $u_E = -0.6 v_{{\rm th}}$ and (b) $u_E = -0.1 v_{{\rm th}}$, resulting in two and three instability regions, respectively. Instabilities occur according to (3.4) in the regions where the product of the $u_*$ and the function $F(\tilde {u}_*, \tilde {u}_E)$ is positive. The red dashed line denotes $F=0$ and the unstable regions are indicated in blue.

Figure 2 shows the unstable and stable regions of the gradient drift instability in the large-wavelength limit, as a function of the diamagnetic drift and $\boldsymbol {E} \times \boldsymbol {B}$ drift. The unstable roots are obtained according to the instability condition shown in (3.4), and the results are consistent with figure 1, showing the two and three unstable regions depending on $u_*$ and $u_E$. A few observations about the instability criteria can be made as follows.

1. (i) When $|\tilde {u}_E| > 0.3$, there are two regions that satisfy the instability condition shown in (3.4). One is $u_E u_*>0$, leading to a condition that can be written as $\boldsymbol {E}_0 \boldsymbol {\cdot }\boldsymbol {\nabla } n_0 >0$, which is similar to the MSHI. In this regime, both $\boldsymbol {E} \times \boldsymbol {B}$ and diamagnetic drifts are in the same direction, leading to instability. The other region that results in instability can be found at $u_* > v_{{\rm th}}$ while $u_E < 0$, as shown in figures 1(a) and 2. It is interesting to note that $\boldsymbol {E}_0 \boldsymbol {\cdot }\boldsymbol {\nabla } n_0<0$ in this second region, which is a different instability criterion compared to the MSHI. This indicates that a strong diamagnetic drift (compared with the magnitude of the $\boldsymbol {E} \times \boldsymbol {B}$ drift and the electron thermal speed) can drive an instability at large wavelength whereas the diamagnetic drift is in the opposite direction to the $\boldsymbol {E} \times \boldsymbol {B}$ drift.

2. (ii) When $0< |\tilde {u}_E| \lessapprox 0.3$, there are three regions that result in linear instability. As can be seen from figure 1(b), $F$ becomes positive in a certain range within $u_*<0$, leading to a stable root because $u_* F < 0$. The appearance of a stable mode in the $\boldsymbol {E}_0 \boldsymbol {\cdot } \boldsymbol {\nabla } n_0 >0$ region can also be seen from figure 2. It can be shown from (3.5) that $F\approx (\tilde {u}_*-\tilde {u}_E)(\tilde {u}_*+ \tilde {u}_E - 1 ) (\tilde {u}_*+ \tilde {u}_E + 1 )$ when assuming $|\tilde {u}_E| \ll 1$. Hence, the three instability regions can be observed approximately at (i) $\tilde {u}_* < -1 - \tilde {u}_E$, (ii) $\tilde {u}_E<\tilde {u}_* < 0$ and (iii) $\tilde {u}_* > 1-\tilde {u}_E$ for $\tilde {u}_E<0$.

3. (iii) When $\tilde {u}_E=0$, there are two regions where the unstable roots exist. Equation (3.5) reduces to $F = \tilde {u}_* (\tilde {u}_* - 1 ) (\tilde {u}_* + 1 )$ when $\tilde {u}_E=0$. Hence, the instability regions can be observed at (i) $\tilde {u}_* < -1$, and (ii) $\tilde {u}_* > 1$, which can be seen from figure 2. The solutions indicate that a low-frequency, large-wavelength instability can occur with a strong diamagnetic drift in the absence of any electric field.

Figure 2.

Unstable and stable regions of the gradient drift instability in the large-wavelength limit, as a function of $u_E$ and $u_*$. The blue region indicates where the unstable modes exist, whereas unstable modes do not exist in the stable region, shown in white.

Figure 3 shows the schematic of the instability criteria for the low-frequency rotating spokes. The first condition $u_E u_*>0$ is similar to the MSHI, in which it is predicted that the $\boldsymbol {E} \times \boldsymbol {B}$ and diamagnetic drifts must be in the same direction. Another instability condition observed from the derivations shown in this section (i.e. (3.4) and (3.5)) illustrates that the partially magnetised plasma can be unstable when the diamagnetic drift is sufficiently larger than the electron thermal speed, even if the electric field and the density gradient exist in the opposite direction.

Figure 3.

Instability criteria for low-frequency rotating spokes. (a) The $\boldsymbol {E} \times \boldsymbol {B}$ drift and diamagnetic drift are in the same direction. (b) The $\boldsymbol {E} \times \boldsymbol {B}$ drift and diamagnetic drift are in the opposite direction and the diamagnetic drift must be larger than the electron thermal speed.

3.3. Direction of the wave propagation

When the instability condition is met in (3.3), the real part of the solution in (3.2) can be obtained as

(3.6)\begin{equation} \omega_r = \frac{(1+\tilde{u}_*^2) c_s }{2u_*} k_y c_s, \end{equation}

where $c_s=(k_B T_e /m_i)^{1/2}$ is the ion acoustic speed. The phase velocity of the wave can be obtained from $v_\phi = \omega _r / k_y$. It can therefore be seen from (3.6) that the wave propagates in the direction of the diamagnetic drift, which may lead to the rotating spokes to propagate in the direction of $\boldsymbol {E} \times \boldsymbol {B}$ drift or $-\boldsymbol {E} \times \boldsymbol {B}$ drift, depending on the instability criteria discussed in § 3.2.

3.4. Resonance

It can be seen from (2.14) and (2.15) that the linear perturbation of the electron bulk velocities shows resonance when $\tilde {\omega }^2 \approx \omega _{{\rm ce}}^2$. In other words, $\omega \approx \pm \omega _{{\rm ce}} + k_y u_{e0y}$, where $u_{e0y}$ is the sum of the $\boldsymbol {E} \times \boldsymbol {B}$ drift and diamagnetic drift, given in (2.7). If the real frequency is of the order of ion plasma frequency, $\omega _r \ll \omega _{{\rm ce}}$, as shown later in § 4, a resonance condition for the dispersion relation can be considered to be

(3.7)\begin{equation} k_y ={\mp} \frac{\omega_{{\rm ce}}}{u_{e0y}}. \end{equation}

It can be seen that the smaller (larger) the electron drift, the larger (smaller) $k_y$ at which resonance occurs. Equation (3.7) indicates that the electron drift is in resonance with the electron gyromotion, which is akin to the electrostatic two-stream instability where the electron drift is in resonance with the electron plasma frequency.

Equation (3.7) can also be written as $k_y \lambda _D = \tilde {u}_{e0y}^{-1}{\omega _{{\rm ce}}}/{\omega _{{\rm pe}}}$ or $k_y r_L = \tilde {u}_{e0y}^{-1}$, where $\omega _{{\rm pe}}=(e^2n_0 / m_e \epsilon _0)^{1/2}$ is the electron plasma frequency, $\lambda _D = (\epsilon _0 k_B T_e / e^2 n_0)^{1/2}$ is the Debye length, $\epsilon _0$ is the vacuum permittivity, $\tilde {u}_{e0y}=u_{e0y}/v_{{\rm th}}$ and $r_L = v_{{\rm th}}/\omega _{{\rm ce}}$ is the electron Larmor radius. Although the kinetic effects, such as the electron Bernstein mode leading to electron cyclotron drift instability (Cavalier et al. Reference Cavalier, Lemoine, Bonhomme, Tsikata, Honoré and Grésillon2013; Hara & Tsikata Reference Hara and Tsikata2020), may play an important role when $k_y r_L=O(1)$, the results in the present fluid theory are applicable for the smaller $k_y$ range, e.g. $k_y r_L<1$. Investigation of the coupling of the present fluid theory and the kinetic dispersion relation (Chang & Callen Reference Chang and Callen1992) is reserved for future work.

5.1. Penning discharge

The Penning discharge, as shown in figure 6(a), operates using an outer cylinder as an anode with a cathode placed along the centreline, in addition to an applied axial magnetic field (Hoh Reference Hoh1963; Simon Reference Simon1963). The plasma density is typically largest near the cathode, generating a radially inward plasma density gradient, $\boldsymbol {\nabla }_r n_0<0$. At the same time, the applied electric field is also radially inward, $E_r<0$. Penning discharge typically naturally satisfies the condition $\boldsymbol {E}_0 \boldsymbol {\cdot }\boldsymbol {\nabla } n_0 >0$, that is, the $\boldsymbol {E} \times \boldsymbol {B}$ and diamagnetic drifts are in the same direction. Therefore, as shown in figure 3(a), the Penning-type discharge satisfies (3.4), generating a gradient drift instability at low frequency.

5.2. Cylindrical magnetron

The cylindrical magnetron, shown in figure 6(b), is used to study critical ionisation velocity (CIV) phenomena (Brenning et al. Reference Brenning, Lundin, Minea, Costin and Vitelaru2013). In addition, such a cross-field configuration is used in high-power microwave sources. For its use as a microwave generator, ion formation is to be avoided because the working principle is that the electrons from the cathode are trapped (i.e. insulated) by the applied or induced magnetic field to generate high-power microwaves (Benford et al. Reference Benford, Swegle and Schamiloglu2015). However, as the devices increase in power and become more compact, plasma generation is unavoidable as the current density in the system increases (Hadas et al. Reference Hadas, Sayapin, Krasik, Bernshtam and Schnitzer2008). Although low-frequency plasma oscillations are not well-studied due to the short pulse operation, it can be seen from figure 6(b) that the partially magnetised plasma is unstable in such configurations, similar to the Penning discharge.

5.3. Planar magnetron discharge

Figure 6(c) shows a schematic of the planar magnetron, used for plasma-assisted sputtering and deposition (Waits Reference Waits1978). An axial electric field and a radial magnetic field are applied close to the cathode surface. The crossed electric and magnetic fields generate electron drifts in the azimuthal direction.

Recent experimental observations in high-power impulse magnetron sputtering (HiPIMS) show that the rotating spoke direction may be a function of the current (Anders & Yang Reference Anders and Yang2017) and can be reversed during a certain operation (Hecimovic et al. Reference Hecimovic, Maszl, von der Gathen, Böke and von Keudell2016). Similar reversal of the rotating spoke propagation is found in a micro-magnetron discharge (Ito et al. Reference Ito, Young and Cappelli2015; Marcovati et al. Reference Marcovati, Ito and Cappelli2020). The global axial plasma profile, such as the location and amplitude of the density and potential gradients, can affect the characteristics of the gradient drift instability, leading to rotating spokes.

In addition, magnetron sputtering devices have a large magnetic field inhomogeneity in the region where the rotating spokes occur, leading to the possibility of gradient drift instabilities due to magnetic field gradients (Esipchuk & Tilinin Reference Esipchuk and Tilinin1976; Tilinin Reference Tilinin1977; Frias et al. Reference Frias, Smolyakov, Kaganovich and Raitses2013) playing an important role in the formation of rotating spokes.

5.4. HET

The HET operates using a radial applied magnetic field and axial electric field, as shown in figure 6(d). In the plume of a HET, the $\boldsymbol {E} \times \boldsymbol {B}$ drift is much larger than the diamagnetic drift and the diamagnetic drift is in the opposite direction to the $\boldsymbol {E} \times \boldsymbol {B}$ drift, which is shown as region 2 in figure 6(d). Consequently, using the instability criteria discussed in (3.4), the partially magnetised plasma in the HET plume is stable against the Penning-type instability. The simulation results by Kawashima et al. (Reference Kawashima, Hara and Komurasaki2018) were compared with a class of gradient drift instabilities in the presence of a magnetic field gradient (Esipchuk & Tilinin Reference Esipchuk and Tilinin1976; Tilinin Reference Tilinin1977; Frias et al. Reference Frias, Smolyakov, Kaganovich and Raitses2013) and showed that the gradient drift instability due to the magnetic field gradient grows in the plume region, which leads to the excitation of rotating spokes.

However, if one considers the plasma inside the channel, shown as region 1 in figure 6(d), the cross-field electron transport is primarily driven by the pressure gradient near the anode. It can therefore be considered that the magnitude of the diamagnetic drift is larger than that of the $\boldsymbol {E} \times \boldsymbol {B}$ drift, while the two drifts may be in the opposite direction. Thus, this condition may lead to the gradient drift instability presented in this paper to be unstable. The spoke rotation will be in the direction of the diamagnetic drift driven by the plasma density gradient inside the channel, which turns out to be in the same direction as the $\boldsymbol {E} \times \boldsymbol {B}$ drift due to the applied electric field (set up by the anode and cathode) in the system.

In summary, there are two scenarios in which azimuthally rotating spokes can occur in HETs and in the planar magnetron discharge. One is due to the instabilities that are caused by the magnetic field gradient. The other possibility is the gradient drift instability, discussed in the present paper, which is caused by the diamagnetic drift in the absence of magnetic field gradients.