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Hall diffusion and the magnetorotational instability in protoplanetary discs

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DOI: 10.1111/j.1365-2966.2011.20022.x
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The destabilising effect of Hall diffusion in a Keplerian disc allows the MRI to occur for much lower ionisation levels than would otherwise be possible. However, simulations suggest that the consequences for the saturated state are not as significant as suggested by the linear instability. Close inspection reveals that that 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 MHD turbulence in protoplanetary discs. We conduct a local stability analysis for a vertical, weak magnetic field subject to axisymmetric perturbations with a vertical wave vector. The diffusivity dependence is presented using analytic expressions and contours in the eta_H - eta_P plane for the maximum growth rate and corresponding wave number, the upper cut-off for unstable wave numbers, and the loci that divide the plane into regions of different characteristic behaviour. In the highly-diffusive limit the magnetic field decouples from the fluid perturbations and the diffusive MRI reduces to a diffusive plane-parallel shear instability with effective shear rate 1.5 Omega. We give analytic expressions for the growth rate and wave number of the most unstable mode. 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 1 au, including x-ray and cosmic-ray ionisation and a population of 1 micron 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. Existing estimates of the depth of magnetically active layers in protoplanetary discs are likely to be wildly inaccurate. [Abridged]

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