Parasitic instabilities in magnetized, differentially rotating disks

Velikhov, Chandrasekhar, and Balbus & Hawley have discovered an MHD instability of differentially rotating fluids that may explain the effective viscosity of accretion disks. If the unperturbed magnetic energy density is much less than the gas pressure, this "magnetorotational" instability (MRI) arises on small scales and is approximately incompressible. If the unperturbed field is sufficiently, strong, the MRI is suppressed. We therefore ask whether the MRI mechanism can amplify even very weak initial fields until they reach equipartition with the gas pressure. We show that in the local incompressible limit, the MRI modes are exact solutions of the nonlinear fluid equations, even if the perturbed magnetic field is much larger than the unperturbed field. Also, we present a new exact solution in the opposite limit that the magnetic pressure is much larger than that of the gas. On the other hand, we show that the incompressible MRI modes are themselves subject to parasitic instabilities with instantaneous growth rates proportional to the MRI amplitude. Some of the parasitic instabilities are of the Kelvin-Helmholtz type, while others are less familiar. The eigenfunctions of the latter group suggest that they may promote rapid reconnection of the MRI field. Thus, parasitic modes may stop MRI growth at subequipartition amplitudes if it develops from a sufficiently weak initial field.