Hall instability: origin, properties and asymptotic theory for its tearing mode

Abstract

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 $\alpha = 2/3$ .