The Saturation Limit of the Magnetorotational Instability

Simulations of the magnetorotational instability (MRI) in a homogeneous shearing box have shown that the asymptotic strength of the magnetic field declines steeply with increasing resolution. Here I model the MRI driven dynamo as a large scale dynamo driven by the vertical magnetic helicity flux. This growth is balanced by large scale mixing driven by a secondary instability. The saturated magnetic energy density depends almost linearly on the vertical height of the typical eddies. The MRI can drive eddies with arbitrarily large vertical wavenumber, so the eddy thickness is either set by diffusive effects, by the magnetic tension of a large scale vertical field component, or by magnetic buoyancy effects. In homogeneous, zero magnetic flux, simulations only the first effect applies and the saturated limit of the dynamo is determined by explicit or numerical diffusion. The exact result depends on the numerical details, but is consistent with previous work, including the claim that the saturated field energy scales as the gas pressure to the one quarter power (which we interpret as an artifact of numerical dissipation). The magnetic energy density in a homogeneous shearing box will tend to zero as the resolution of the simulation increases, but this has no consequences for the dynamo or for angular momentum transport in real accretion disks. The claim that the saturated state depends on the magnetic Prandtl number may also be an artifact of simulations in which microphysical transport coefficients set the MRI eddy thickness. Finally, the efficiency of the MRI dynamo is a function of the ratio of the Alfv\'en velocity to the product of the pressure scale height and the local shear. As this approaches unity from below the dynamo reaches maximum efficiency.