  According to Rayleigh's criterion, rotating flows are linearly stable when
their specific angular momentum increases radially outward. The celebrated
magnetorotational instability opens a way to destabilize those flows, as long
as the angular velocity is decreasing outward. Using a short-wavelength
approximation we demonstrate that even flows with very steep positive shear can
be destabilized by azimuthal magnetic fields which are current-free within the
fluid. We illustrate the transition of this instability to a rotationally
enhanced kink-type instability in case of a homogeneous current in the fluid,
and discuss the prospects for observing it in a magnetized Taylor-Couette flow.
