Magnetorotational Instability in Protoplanetary Disks. I. On the Global Stability of Weakly Ionized Disks with Ohmic Dissipation
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Abstract
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 βc≳1 and βcRmc≳1, where βc and Rmc 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.