Destabilisation of hydrodynamically stable rotation laws by azimuthal magnetic fields

We consider the effect of toroidal magnetic fields on hydrodynamically stable Taylor-Couette differential rotation flows. For current-free magnetic fields a nonaxisymmetric m=1 magnetorotational instability arises when the magnetic Reynolds number exceeds O (100). We then consider how this `azimuthal magnetorotational instability' (AMRI) is modified if the magnetic field is not current-free, but also has an associated electric current throughout the fluid. This gives rise to current-driven Tayler instabilities (TI) that exist even without any differential rotation at all. The interaction of the AMRI and the TI is then considered when both electric currents and differential rotation are present simultaneously. The magnetic Prandtl number Pm turns out to be crucial in this case. Large Pm have a destabilizing influence, and lead to a smooth transition between the AMRI and the TI. In contrast, small Pm have a stabilizing influence, with a broad stable zone separating the AMRI and the TI. In this region the differential rotation is acting to stabilize the Tayler instabilities, with possible astrophysical applications (Ap stars). The growth rates of both the AMRI and the TI are largely independent of Pm, with the TI acting on the timescale of a single rotation period, and the AMRI slightly slower, but still on the basic rotational timescale. The azimuthal drift timescale is ~20 rotations, and may thus be a (flip-flop) timescale of stellar activity between the rotation period and the diffusion time.