Axisymmetric magneto-rotational instability in viscous accretion disks

Axisymmetric MRI in viscous accretion disks is investigated. The linear growth of the viscous MRI is characterized by the Reynolds number R_MRI ≡ v²_A/νΩ, where v_A is the Alfvén velocity, ν is the kinematic viscosity, and Ω is the angular velocity of the disk. Although the linear growth of the MRI is suppressed as the Reynolds number decreases, its nonlinear behavior is found to be almost independent of R_MRI. At the nonlinear stage, the channel flow grows and the Maxwell stress increases even though R_MRI is much smaller than unity. Nonlinear behavior of the MRI in the viscous regime can be explained by the characteristics of the linear dispersion relation. Applying our results to the case with both viscosity and resistivity, it is anticipated that the critical value of the Lundquist number S_MRI ≡ v²_A/ηΩ for active turbulence would depend on the magnetic Prandtl number Pm ≡ ν/η, where η is the magnetic diffusivity.