The small magnetic Prandtl number approximation suppresses magnetorotational instability

Axisymmetric stability of viscous resistive magnetized Couette flow is re-examined, with emphasis on flows that would be hydrodynamically stable according Rayleigh's criterion: opposing gradients of angular velocity and specific angular momentum. In this regime, magnetorotational instability (MRI) may occur. The governing system in cylindrical coordinates is of tenth order. It is proved, by methods based on those of Synge and Chandrasekhar, that by dropping one term from the system, MRI is suppressed, in fact no instability at all occurs, with insulating boundary conditions. This term is often neglected because it has the magnetic Prandtl number, which is very small, as a factor; nevertheless it is crucially important.