  The interaction of differential rotation and toroidal fields that are
current-free in the gap between two corotating axially unbounded cylinders is
considered. It is shown that nonaxisymmetric perturbations are unstable if the
rotation rate and Alfv\'en frequency of the field are of the same order, almost
independent of the magnetic Prandtl number Pm. For the very steep rotation law
\Omega\propto R^{-2} (the Rayleigh limit) and for small Pm the threshold values
of rotation and field for this Azimuthal MagnetoRotational Instability (AMRI)
scale with the ordinary Reynolds number and the Hartmann number, resp. A
laboratory experiment with liquid metals like sodium or gallium in a
Taylor-Couette container has been designed on the basis of this finding. For
fluids with more flat rotation laws the Reynolds number and the Hartmann number
are no longer typical quantities for the instability. For the weakly nonlinear
system the numerical values of the kinetic energy and the magnetic energy are
derived for magnetic Prandtl numbers \leq 1. We find that the magnetic energy
grows monotonically with the magnetic Reynolds number Rm, while the kinetic
energy grows with Rm/\sqrt{Pm}. The resulting turbulent Schmidt number, as the
ratio of the `eddy' viscosity and the diffusion coefficient of a passive scalar
(such as lithium) is of order 20 for Pm=1, but for small Pm it drops to order
unity. Hence, in a stellar core with fossil fields and steep rotation law the
transport of angular momentum by AMRI is always accompanied by an intense
mixing of the plasma, until the rotation becomes rigid.
