Stability of axisymmetric Taylor-Couette flow in hydromagnetics

The linear marginal instability of an axisymmetric magnetohydrodynamics Taylor-Couette flow of infinite vertical extension is considered. We are only interested in those vertical wave numbers for which the characteristic Reynolds number is minimum. For hydrodynamically unstable flows minimum Reynolds numbers exist even without a magnetic field, but there are also solutions with smaller characteristic Reynolds numbers for certain weak magnetic fields. The magnetic field, therefore, destabilizes the rotating flow by the so-called magnetorotational instability (MRI). The MRI, however, can only exist for hydrodynamically unstable flow if the magnetic Prandtl number, Pr, is not too small. For too small magnetic Prandtl numbers (and too strong magnetic fields) only the well-known magnetic suppression of the Taylor-Couette instability can be found. The MRI is even more pronounced for hydrodynamically stable flows. In this case we can always find a magnetic field amplitude where the characteristic Reynolds number is minimum. These critical values are computed for different magnetic Prandtl numbers and for three types of geometry (small, medium, and wide gaps between the rotating cylinders). In all cases the minimum Reynolds numbers are running with 1/Pr for small enough Pr so that the critical Reynolds numbers may easily exceed values of 10(6) for the magnetic Prandtl number of sodium (10(-5)) or gallium (10(-6)). The container walls are considered either electrically conducting or insulating. For insulating walls with small and medium-size gaps between the cylinders (i) the critical Reynolds number is smaller, (ii) the critical Hartmann number is higher, and (iii) the Taylor vortices are longer in the direction of the rotation axis. For wider gaps the differences in the results between both sets of boundary conditions become smaller and smaller.