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.