The effect of the Hall term on the nonlinear evolution of the magnetorotational instability. I. Local axisymmetric simulations

The effect of the Hall term on the evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks is investigated using local axisymmetric simulations. First, we show that the Hall term has important effects on the MRI when the temperature and density in the disk is below a few thousand K and between 10¹³ and 10¹⁸ cm^-3, respectively. Such conditions can occur in the quiescent phase of dwarf nova disks, or in the inner part (inside 10-100 AU) of protoplanetary disks. When the Hall term is important, the properties of the MRI are dependent on the direction of the magnetic field with respect to the angular velocity vector Ω. If the disk is threaded by a uniform vertical field oriented in the same sense as Ω, the axisymmetric evolution of the MRI is an exponentially growing two-channel flow without saturation, When the field is oppositely directed to Ω, however, small scale fluctuations prevent the nonlinear growth of the channel flow and the MRI evolves into MHD turbulence. These results are anticipated from the characteristics of the linear dispersion relation. In axisymmetry on a field with zero-net flux, the evolution of the MRI is independent of the size of the Hall term relative to the inductive term. The evolution in this case is determined mostly by the effect of ohmic dissipation.