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 weaklyionized accretion disks, including the effect of the Hall term and ohmicdissipation, is investigated using local three-dimensional MHD simulations andvarious initial magnetic field geometries. When the magnetic Reynolds number,Re_M \equiv v_A^2 / \eta \Omega (where v_A is the Alfven speed, \eta themagnetic diffusivity, and \Omega the angular frequency), is initially largerthan a critical value Re_{M, crit}, the MRI evolves into MHD turbulence inwhich angular momentum is transported efficiently by the Maxwell stress. IfRe_M < Re_{M, crit}, however, ohmic dissipation suppresses the MRI, and thestress is reduced by several orders of magnitude. The critical value is in therange of 1 - 30 depending on the initial field configuration. The Hall effectdoes not modify the critical magnetic Reynolds number by much, but enhances thesaturation level of the Maxwell stress by a factor of a few. We show that thesaturation level of the MRI is characterized by v_{Az}^2 / \eta \Omega, wherev_{Az} is the Alfven speed in the nonlinear regime along the vertical componentof the field. The condition for turbulence and significant transport is givenby v_{Az}^2 / \eta \Omega \gtrsim 1, and this critical value is independent ofthe 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 estimatedobservationally, and the magnetic Reynolds number v_A^2 / \eta \Omega is largerthan about 30, this would imply the MRI is operating in the disk.