Tayler instability of toroidal magnetic fields in MHD Taylor-Couette flows

The nonaxisymmetric ‘kink-type’ Tayler instability (TI) of toroidal magnetic fields is studied for conducting incompressible fluids of uniform density between two infinitely long cylinders rotating around the same axis. The electric current flows within the gap between the cylinders is axial direction. For given Reynolds number of rotation the magnetic Prandtl number Pm of the liquid conductor and the ratio of the cylinder’s rotation rates are the free parameters. It is shown that for resting cylinders the critical Hartmann number for the unstable modes does not depend on Pm. By rigid rotation the instability is suppressed where the critical ratio of the rotation velocity and the Alfvén velocity of the field (only) slightly depends on Pm. For Pm = 1 the rotational quenching of TI takes its maximum. One also finds that rotation laws with negative shear (i.e. dΩ/dR < 0) strongly destabilize the toroidal field if the rotation is not too fast. In radiative zones of young stars, galaxies and in the fluid crust of neutron stars this effect could have drastic implications. For sufficiently high Reynolds numbers of rotation the suppression of the nonaxisymmetric magnetic instability always dominates. Superrotation laws support the rotational stabilization but only for not too high Pm. The angular momentum transport of the instability is anticorrelated with the shear so that an eddy viscosity can be defined which proves to be positive. For negative shear the Maxwell stress of the perturbations remarkably contributes to the angular momentum transport. We have also shown the possibility of laboratory TI experiments with a wide-gap container filled with fluid metals like sodium or gallium. Even the effect of the rotational stabilization can be reproduced in the laboratory with electric currents of only a few kAmp.