Magnetic Viscosity by Localized Shear Flow Instability in Magnetized Accretion Disks

Differentially rotating disks are subject to the axisymmetric instability for perfectly conducting plasma in the presence of poloidal magnetic fields (Balbus & Hawley 1991). For nonaxisymmetric perturbations, we find localized unstable eigenmodes whose eigenfunction is confined between two Alfven singularities at omega_D = +/- omega_A, where omega_D is the Doppler-shifted wave frequency and omega_A = k_parallel nu_A is the Alfven frequency. The radial width of the unstable eigenfunction is Delta x is approximately omega_A/(AK_y), where A is Oort's constant and k_y is the azimuthal wavenumber. The growth rate of the fundamental mode is larger for smaller values of k_y/k_z. The maximum growth rate when k_y/k_z is approximately 0.1 is approximately 0.2 Omega for the Keplerian disk with local angular velocity Omega. It is found that the purely growing mode disappears when k_y/k_z is greater than 0.12. In a perfectly conducting disk, the instability grows even when the seed magnetic field is infitesimal. Inclusion of the resistivity, however, leads to the appearance of an instability threshold. When the resistivity eta depends on the instability-induced turbulent magnetic fields delta B as eta (mean value of delta B squared), the marginal stability condition self consistently determines the alpha-parameter of the angular momentum transport due to magnetic stress. For fully ionized disks, the magnetic viscosity parameter alpha_B is between 0.001 and 1. Our three-dimensional MHD simulation confirms these unstable eigenmodes. It also shows that the alpha-parameter observed in simulation is between 0.01 and 1, in agreement with theory. The observationally required smaller alpha in the quiescent phase of accretion disks in dwarf novae may be explained by the decreased inoization due to the temperature drop.