Simulations of magnetorotational instability in a magnetized couette flow

In preparation for an experimental study of magnetorotational instability (MRI) in liquid metal, we present non-ideal two-dimensional magnetohydrodynamic simulations of the nonlinear evolution of MRI in the experimental geometry. The simulations adopt initially uniform vertical magnetic fields, conducting radial boundaries, and periodic vertical boundary conditions. No-slip conditions are imposed at the cylinders. Our linear growth rates compare well with existing local and global linear analyses. The MRI saturates nonlinearly with horizontal magnetic fields comparable to the initial axial field. The rate of angular momentum transport increases modestly but significantly over the initial state. For modest fluid and magnetic Reynolds numbers Re, Re_m ∼ 10 ² -10³, the final state is laminar reduced mean shear except near the radial boundaries, and with poloidal circulation scaling as the square root of resistivity, in partial agreement with the analysis of Knobloch and Julien. A sequence of simulations at Re_m = 20 and 10² ≲ Re ≲10^4.4 enables extrapolation to the experimental regime (Re_m ≈ 20, Re ∼ 10⁷), albeit with unrealistic boundary conditions. MRI should increase the experimentally measured torque substantially over its initial purely hydrodynamic value.