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The instability of a supercritical Taylor-Couette flow of a conducting fluidwith resting outer cylinder under the influence of a uniform axial electriccurrent is investigated for magnetic Prandtl number Pm=1. In the linear theorythe critical Reynolds number for axisymmetric perturbations is not influencedby the current-induced axisymmetric magnetic field but all axisymmetricmagnetic perturbations decay. The nonaxisymmetric perturbations with m=1 areexcited even without rotation for large enough Hartmann numbers ("Taylerinstability"). For slow rotation their growth rates scale with the Alfv\'enfrequency of the magnetic field but for fast rotation they scale with therotation rate of the inner cylinder. In the nonlinear regime the ratio of theenergy of the magnetic m=1 modes and the toroidal background field is very lowfor the non-rotating Tayler instability but it strongly grows if differentialrotation is present. For super-Alfv\'enic rotation the energies in the m=1modes of flow and field do not depend on the molecular viscosity, they arealmost in equipartition and contain only 1.5% of the centrifugal energy of theinner cylinder. The geometry of the excited magnetic field pattern is strictlynonaxisymmetric for slow rotation but it is of the mixed-mode type for fastrotation -- contrary to the situation which has been observed at the surface ofAp stars.

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