Hall diffusion and the magnetorotational instability in protoplanetary discs

The destabilizing effect of Hall diffusion in a weakly ionized Keplerian disc allows the magnetorotational instability (MRI) to occur for much lower ionization levels than would otherwise be possible. However, simulations incorporating Hall and Ohm diffusion give the impression that the consequences of this for the non-linear saturated state are not as significant as suggested by the linear instability. Close inspection reveals that this is not actually the case as the simulations have not yet probed the Hall-dominated regime. Here we revisit the effect of Hall diffusion on the MRI and the implications for the extent of magnetohydrodynamic (MHD) turbulence in protoplanetary discs, where Hall diffusion dominates over a large range of radii. We conduct a local, linear analysis of the instability for a vertical, weak magnetic field subject to axisymmetric perturbations with a purely vertical wave vector. In contrast to previous analyses, we express the departure from ideal MHD in terms of Hall and Pedersen diffusivities η _H and η _P, which provide transparent notation that is directly connected to the induction equation. This allows us to present a crisp overview of the dependence of the instability on magnetic diffusivity. We present analytic expressions and contours in the η _H-η _P plane for the maximum growth rate and corresponding wavenumber, the upper cut-off for unstable wavenumbers and the loci that divide the plane into regions of different characteristic behaviour. We find that for where v _A is the Alfvén speeds and Ω is the Keplerian frequency, Hall diffusion suppresses the MRI irrespective of the value of η _P. In the highly diffusive limit, the magnetic field decouples from the fluid perturbations and simply diffuses in the background Keplerian shear flow. The diffusive MRI reduces to a diffusive plane-parallel shear instability with effective shear rate (3/2)Ω. We give simple analytic expressions for the growth rate and wavenumber of the most unstable mode. We review the varied and confusing parametrizations of magnetic diffusion in discs that have appeared in the literature, and confirm that simulations examining the saturation of the instability under Hall-Ohm diffusion are consistent with the linear analysis and have yet to probe the 'deep' Hall regime characteristic of protoplanetary discs where Hall diffusion is expected to overcome resistive damping. Finally, we illustrate the critical effect of Hall diffusion on the extent of dead zones in protoplanetary discs by applying a local stability criterion to a simple model of the minimum-mass solar nebula at 1au, including X-ray and cosmic ray ionization and a population of 1-m grains. Hall diffusion increases or decreases the MRI-active column density by an order of magnitude or more, depending on whether B is parallel or antiparallel to the rotation axis, respectively. We conclude that existing estimates of the depth of magnetically active layers in protoplanetary discs based on damping by Ohm diffusion are likely to be wildly inaccurate.