MHD Instability in cylindric Taylor-Couette flow

The linear marginal instability of an axisymmetric MHD Taylor-Couette flow of infinite vertical extension is considered. For flows with a resting outer cylinder there is a well-known characteristic Reynolds number even without magnetic field but for sufficiently weak magnetic fields there are solutions with smaller Reynolds numbers so that a characteristic minimum exists. The minimum only exists, however, for not too small magnetic Prandtl numbers. For small magnetic Prandtl numbers one only finds the typical magnetic suppression of the instability. We are here particularly interested in the case where the outer cylinder rotates so fast that the Rayleigh criterion for hydrodynamic stability is fulfilled. We find that for given magnetic Prandtl number now always a magnetic field amplitude exists where the characteristic Reynolds number is minimal. These critical values are computed for different magnetic Prandtl numbers. In all cases the Reynolds numbers are running with 1/Pm so that for the small magnetic Prandtl numbers of sodium (10^-5) or gallium (10^-6) the critical Reynolds numbers exceed values of 10^6 or 10^, resp.