Magnetorotational instability in protoplanetary disks. I. On the global stability of weakly ionized disks with ohmic dissipation

We investigate the stability of uniformly magnetized accretion disks, including the effect of ohmic dissipation. The growth of axisymmetric local and global modes is examined using linear perturbation theory. A simple local analysis shows that the dissipation process generally suppresses the growth of magnetorotational instability when the magnetic Reynolds number is less than unity. The characteristic length scale of unstable modes becomes longer than that of the ideal MHD case, and its unstable growth rate is inversely proportional to the magnetic diffusivity. We perform a global linear analysis in which the vertical structure of the disk is considered. The growth rate of the magnetorotational instability is obtained by solving eigen equations numerically. We find that the conditions for existing unstable global modes are beta(c) greater than or similar to 1 and beta(c) R-mc greater than or similar to 1, where beta(c) and R-mc are the plasma beta value and the magnetic Reynolds number defined at the midplane of the disk. The global stability criteria are approximately given by whether the minimum unstable wavelength expected by the local analysis would be shorter than the scale height of the disk or not. We also find unstable modes whose eigenfunctions of the perturbed velocities have amplitude localized near the surface layer of the disk. These unstable modes indicate layered accretion in the nonlinear regime. We apply the results of linear analysis to protoplanetary disks. For the case of the minimum-mass solar nebula, the magnetorotational instability occurs at the region farther out than 15 AU. This result suggests nonsteady accretion onto a central star in protoplanetary disks.