Magnetorotational instability in protoplanetary discs

We investigate the linear growth and vertical structure of the magnetorotational instability (MRI) in weakly ionized, stratified accretion discs. The magnetic field is initially vertical and dust grains are assumed to have settled towards the mid-plane, so charges are carried by electrons and ions only. Solutions are obtained at representative radial locations from the central protostar for different choices of the initial magnetic field strength, sources of ionization, disc structure and configuration of the conductivity tensor. The MRI is active over a wide range of magnetic field strengths and fluid conditions in low-conductivity discs. Moreover, no evidence was found of a low-limit field strength below which unstable modes do not exist. For the minimum-mass solar nebula model, incorporating cosmic ray ionization, perturbations grow at 1 au for B less than or similar to 8 G. For a significant subset of these strengths (200 mG less than or similar to B less than or similar to 5 G), the maximum growth rate is of the order of the ideal magnetohydrodynamic (MHD) rate (0.75 Omega). Hall conductivity modifies the structure and growth rate of global unstable modes at 1 au for all magnetic field strengths that support MRI. As a result, at this radius, modes obtained with a full conductivity tensor grow faster and are active over a more extended cross-section of the disc than perturbations in the ambipolar diffusion limit. For relatively strong fields (e.g. B greater than or similar to 200 mG), ambipolar diffusion alters the envelope shapes of the unstable modes, which peak at an intermediate height, instead of being mostly flat as modes in the Hall limit are in this region of parameter space. Similarly, when cosmic rays are assumed to be excluded from the disc by the winds emitted by the magnetically active protostar, unstable modes grow at this radius for B less than or similar to 2 G. For strong fields, perturbations exhibit a kink at the height where X-ray ionization becomes active. Finally, for R= 5 au (10 au), unstable modes exist for B less than or similar to 800 mG (B less than or similar to 250 mG) and the maximum growth rate is close to the ideal-MHD rate for 20 less than or similar to B less than or similar to 500 mG (2 less than or similar to B less than or similar to 50 mG). Similarly, perturbations incorporating Hall conductivity have a higher wavenumber and grow faster than solutions in the ambipolar diffusion limit for B less than or similar to 100 mG (B less than or similar to 10 mG). Unstable modes grow even at the mid-plane for B greater than or similar to 100 mG (B similar to 1 mG), but for weaker fields, a small dead region exists. This study shows that, despite the low magnetic coupling, the magnetic field is dynamically important for a large range of fluid conditions and field strengths in protostellar discs. An example of such magnetic activity is the generation of MRI unstable modes, which are supported at 1 au for field strengths up to a few gauss. Hall diffusion largely determines the structure and growth rate of these perturbations for all studied radii. At radii of order 1 au, in particular, it is crucial to incorporate the full conductivity tensor in the analysis of this instability and more generally in studies of the dynamics of astrophysical discs.