The Effect of the Hall Term on the Nonlinear Evolution of the Magnetorotational Instability. II. Saturation Level and Critical Magnetic Reynolds Number

The nonlinear evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks, including the effect of the Hall term and ohmic dissipation, is investigated using local three-dimensional MHD simulations and various initial magnetic field geometries. When the magnetic Reynolds number, Re_M ≡ v/ηΩ (where v_A is the Alfvén speed, η is the magnetic diffusivity, and Ω is the angular frequency), is initially larger than a critical value Re_M,crit, the MRI evolves into MHD turbulence in which angular momentum is transported efficiently by the Maxwell stress. If Re_M < Re_M,crit, however, ohmic dissipation suppresses the MRI, and the stress is reduced by several orders of magnitude. The critical value is in the range of 1-30 depending on the initial field configuration. The Hall effect does not modify the critical magnetic Reynolds number by much but enhances the saturation level of the Maxwell stress by a factor of a few. We show that the saturation level of the MRI is characterized by v/ηΩ, where v_Az is the Alfvén speed in the nonlinear regime along the vertical component of the field. The condition for turbulence and significant transport is given by v/ηΩ ≳ 1, and this critical value is independent of the strength and geometry of the magnetic field or the size of the Hall term. If the magnetic field strength in an accretion disk can be estimated observationally and the magnetic Reynolds number v/ηΩ is larger than about 30, this would imply that the MRI is operating in the disk.