On the linear stability of weakly ionized, magnetized planar shear flows

We investigate the effects of ambipolar diffusion and the Hall effect on the stability of weakly ionized, magnetized planar shear flows. Employing a local approach similar to the shearing-sheet approximation, we solve for the evolution of linear perturbations in both streamwise-symmetric and non-streamwise-symmetric geometries using Wentzel-Kramers-Brillouin techniques and/or numerical methods. We find that instability arises from the combination of shear and non-ideal magnetohydrodynamic processes and is a result of the ability of these processes to influence the free energy path between the perturbations and the shear. They turn what would be simple linear-in-time growth due to current and vortex stretching from shear into exponentially growing instabilities. Our results aid in understanding previous work on the behaviour of weakly ionized accretion discs. In particular, the recent finding that the Hall effect and ambipolar-diffusion destabilize both positive and negative angular velocity gradients acquires a natural explanation in the more general context of this paper. We construct a simple toy model for these instabilities based upon transformation operators (shears, rotations and projections) that captures both their qualitative and, in certain cases, exact quantitative behaviour.