Damping of the Shear Instability in Magnetized Disks by Ohmic Diffusion

We study damping of the axisymmetric magnetic shear instability by ohmic diffusion in accretion disks. The dispersion relation for axisymmetric perturbations in a magnetized disk is derived, including the effects of ohmic diffusion. The instability is damped when the diffusion rate is greater than the growth rate of the instability. We calculate the critical value of the diffusion coefficient at which the instability is suppressed. Comparing the critical value with the ohmic diffusion coefficients in different types of disks, we find that ohmic diffusion can damp the instability in the solar nebula (late stage of a protostellar disk) in which the conductivity is low, particularly in its middle region where neither thermal nor cosmic-ray ionizations are efficient. The damping is not significant in accretion disks around neutron stars, stellar black holes, massive black holes, or white dwarfs.