Axisymmetry vs. nonaxisymmetry of a Taylor-Couette flow with azimuthal magnetic fields

The instability of a supercritical Taylor-Couette flow of a conducting fluid with resting outer cylinder under the influence of a uniform axial electric current is investigated for magnetic Prandtl number Pm=1. In the linear theory the critical Reynolds number for axisymmetric perturbations is not influenced by the current-induced axisymmetric magnetic field but all axisymmetric magnetic perturbations decay. The nonaxisymmetric perturbations with m=1 are excited even without rotation for large enough Hartmann numbers ("Tayler instability"). For slow rotation their growth rates scale with the Alfv\'en frequency of the magnetic field but for fast rotation they scale with the rotation rate of the inner cylinder. In the nonlinear regime the ratio of the energy of the magnetic m=1 modes and the toroidal background field is very low for the non-rotating Tayler instability but it strongly grows if differential rotation is present. For super-Alfv\'enic rotation the energies in the m=1 modes of flow and field do not depend on the molecular viscosity, they are almost in equipartition and contain only 1.5% of the centrifugal energy of the inner cylinder. The geometry of the excited magnetic field pattern is strictly nonaxisymmetric for slow rotation but it is of the mixed-mode type for fast rotation -- contrary to the situation which has been observed at the surface of Ap stars.