  Global axisymmetric stability of viscous, resistive, magnetized Couette flow
is re-examined, with the emphasis on flows that would be hydrodynamically
stable according to Rayleigh's criterion: opposing gradients of angular
velocity and specific angular momentum. In this regime, magnetorotational
instabilities [MRI] may occur. Previous work has focused on the
Rayleigh-unstable regime. To prepare for an experimental study of MRI, which
are of intense astrophysical interest, we solve for global linear modes in a
wide gap with realistic dissipation coefficients. Exchange of stability appears
to occur through marginal modes. Velocity eigenfunctions of marginal modes are
nearly singular at conducting boundaries, but magnetic eigenfunctions are
smooth and obey a fourth-order differential equation in the inviscid limit. The
viscous marginal system is of tenth order; an eighth-order approximation
previously used for Rayleigh-unstable modes does not permit MRI. Peak growth
rates are insensitive to boundary conditions. They are predicted with
surprising accuracy by WKB methods even for the largest-scale mode. We conclude
that MRI is achievable under plausible experimental conditions using
easy-to-handle liquid metals such as gallium.
