Hall instability in electron magnetohydrodynamics is interpreted as the shear-Hall instability driven jointly by helicoidal oscillations and shear in the electron current velocity. This explanation suggests an antiparallel orientation of the background magnetic field and vorticity of the current velocity as the necessary condition for Hall instability. The condition is tested and generally confirmed by numerical computations in plane slab geometry. Unstable eigenmodes are localized in the spatial regions of the antiparallel field and vorticity. Computations of the tearing-type mode of the instability are complemented by (and generally agree with) asymptotic analytical estimations for large Hall numbers. The stabilizing effect of perfect conductor boundary conditions is found and explained. For large Hall numbers, the growth rates approach the power-law dependence $\sigma \propto B^\alpha \eta ^{1-\alpha }$ on the magnetic field ( $B$ ) and diffusivity ( $\eta$ ). Almost all computations give the power index $\alpha = 3/4$ with one exception of the tearing-type mode with vacuum boundary conditions for which case $\alpha = 2/3$ .

中文翻译：

---

霍尔不稳定性：撕裂模式的起源、性质和渐近理论

电子磁流体动力学中的霍尔不稳定性被解释为由螺旋振荡和电子电流速度中的剪切共同驱动的剪切霍尔不稳定性。这种解释表明背景磁场的反平行方向和电流速度的涡度是霍尔不稳定性的必要条件。该条件通过平面板几何中的数值计算进行测试和确认。不稳定的本征模位于反平行场和涡度的空间区域。对大霍尔数的渐近分析估计对不稳定性撕裂型模式的计算进行了补充（并且通常同意）。找到并解释了完美导体边界条件的稳定效应。对于大霍尔数， $\sigma \propto B^\alpha \eta ^{1-\alpha }$ 在磁场（ $B$ ) 和扩散率 ( $\eta$ ）。几乎所有的计算都给出了功率指数 $\alpha = 3/4$ 除了具有真空边界条件的撕裂型模式之外，在这种情况下 $\alpha = 2/3$ .