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The nonaxisymmetric Tayler instability (TI) of toroidal magnetic fields due to axial electric currents is studied for conducting incompressible fluids between two infinitely long corotating cylinders. 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 instability does not depend on Pm hence the TI also exists in the limit Pm → 0. By rigid rotation the instability is suppressed where for Pm = 1 the rotational quenching takes its maximum. Rotation laws with negative shear (i.e. d Ω  /d R  < 0) strongly destabilize the toroidal field if the rotation is not too fast. In galaxies with their quadrupolar magnetic field geometry this effect could have drastic implications. For sufficiently high Reynolds numbers of rotation, however, the TI completely disappears. For the considered magnetic constellation the superrotation laws support the rotational stabilization. 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.  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 kA (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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