Theory of current instability experiments in magnetic Taylor-Couette   flows
  We consider the linear stability of dissipative MHD Taylor-Couette flow with imposed toroidal magnetic fields. The inner and outer cylinders can be either insulating or conducting; the inner one rotates, the outer one is stationary. The magnetic Prandtl number can be as small as 10-5, approaching realistic liquid-metal values. The magnetic field destabilizes the flow, except for radial profiles of B$_\phi$(R) close to the current-free solution. The profile with B$_{in}$=B$_{out}$ (the most uniform field) is considered in detail. For weak fields the TC-flow is stabilized, until for moderately strong fields the m=1 azimuthal mode dramatically destabilizes the flow again. There is thus a maximum value for the critical Reynolds number. For sufficiently strong fields (as measured by the Hartmann number) the toroidal field is always unstable, even for Re=0.   The electric currents needed to generate the required toroidal fields in laboratory experiments are a few kA if liquid sodium is used, somewhat more if gallium is used. Weaker currents are needed for wider gaps, so a wide-gap apparatus could succeed even with gallium. The critical Reynolds numbers are only somewhat larger than the nonmagnetic values, so such an experiment would require only modest rotation rates. 
G. Ruediger, M. Schultz, D. Shalybkov, R. Hollerbach