  Magneto-rotational instability (MRI) is an important instability mechanism
for rotating flows with magnetic fields. In particular, when the strength of
the magnetic field tends to zero, the stability criterion for rotating flows is
generally different from the classical Rayleigh criterion for rotating flows
without a magnetic field. MRI has wide applications in astrophysics,
particularly to the turbulence and enhanced angular momentum transport in
accretion disks. For the case of vertical magnetic fields, we give rigorous
proof of linear MRI and a complete description of the spectra and semigroup
growth of the linearized operator. Moreover, we prove nonlinear stability and
instability from the sharp linear stability/instability criteria.
