Tayler instability and dynamo action of cylindric magnetic rings

The Tayler instability of an azimuthal magnetic field with one or two ``rings'' along the radius is studied for an axially unbounded Taylor-Couette flow. The rotation law of the conducting fluid is a quasi-Keplerian one. Without rotation all toroidal fields are the more destabilized the more the radial profiles differ from the uniformity. For medium Reynolds numbers of rotation, however, the behaviour of the lines of neutral stability strongly depend on the magnetic Prandtl number. For Pm=1 the differential rotation matches the instability lines of azimuthal fields with and without rings so that the maximally possible Reynolds numbers for fields with smooth radial profiles and such with rings do hardly differ. The magnetic Mach number of the considered examples are of the astrophysically relevant order of magnitude between ten and twenty. The nonaxisymmetric instability fluctuations form a weak alpha effect of the mean-field electrodynamics which always changes its sign between the walls independent of the Reynolds number, magnetic Prandtl number or the radial profile of the magnetic background field. The resulting dynamo modes work on a similar axial scale as the Tayler instability, hence they are small-scale dynamos. The fields are axially drifting with high phase velocity where at certain periods the azimuthal fields with one-ring geometry develop to fields with two rings along the radius. It is still open whether and how a nonlinear dynamo model may overcome this puzzling complication.