Saturation and Thermalization of the Magnetorotational Instability: Recurrent Channel Flows and Reconnections

The nonlinear evolution and the saturation mechanism of the magnetorotational instability (MRI) are investigated using three-dimensional resistive MHD simulations. A local shearing box is used for our numerical analysis and the simulations begin with a purely vertical magnetic field. We find that the magnetic stress in the nonlinear stage of the MRI is strongly fluctuating. The time evolution shows the quasi-periodic recurrence of spike-shape variations typically for a few orbits which correspond to the rapid amplification of the magnetic field by the nonlinear growth of a two-channel solution followed by the decay through magnetic reconnections. The increase rate of the total energy in the shearing box system is analytically related to the volume-averaged torque in the system. We find that at the saturated state this energy gain of the system is balanced with the increase of the thermal energy mostly due to the joule heating. The spike-shape time evolution is a general feature of the nonlinear evolution of the MRI in the disks threaded by vertical fields and can be seen if the effective magnetic Reynolds number is larger than about unity.