Helicity and alpha effect driven by the nonaxisymmetric Tayler instability of toroidal magnetic fields in stellar radiation zones are computed. In the linear approximation a purely toroidal field always excites pairs of modes with identical growth ra
tes but with opposite helicity so that the net helicity vanishes. If the magnetic background field has a helical structure by an extra (weak) poloidal component then one of the modes dominates producing a net kinetic helicity anticorrelated to the current helicity of the background field. The mean electromotive force is computed with the result that the alpha effect by the most rapidly growing mode has the same sign as the current helicity of the background field. The alpha effect is found as too small to drive an alpha^{2} dynamo but the excitation conditions for an alphaOmega dynamo can be fulfilled for weak poloidal fields. Moreover, if the dynamo produces its own alpha effect by the magnetic instability then problems with its sign do not arise. For all cases, however, the alpha effect shows an extremely strong concentration to the poles so that a possible alphaOmega dynamo might only work at the polar regions. Hence, the results of our linear theory lead to a new topological problem for the existence of large-scale dynamos in stellar radiation zones on the basis of the current-driven instability of toroidal fields.
To find out whether toroidal field can stably exist in galaxies the current-driven instability of toroidal magnetic fields is considered under the influence of an axial magnetic field component and under the influence of both rigid and differential r
otation. The MHD equations are solved in a simplified model with cylindric geometry. We assume the axial field as uniform and the fluid as incompressible. The stability of a toroidal magnetic field is strongly influenced by uniform axial magnetic fields. If both field components are of the same order of magnitude then the instability is slightly supported and modes with m>1 dominate. If the axial field even dominates the most unstable modes have again m>1 but the field is strongly stabilized. All modes are suppressed by a fast rigid rotation where the m=1 mode maximally resists. Just this mode becomes best re-animated for Omega > Omega^A (Omega^A the Alfven frequency) if the rotation has a negative shear. -- Strong indication has been found for a stabilization of the nonaxisymmetric modes for fluids with small magnetic Prandtl number if they are unstable for Pm=1. For rotating fluids the higher modes with m>1 do not play an important role in the linear theory. In the light of our results galactic fields should be marginally unstable against perturbations with m<= 1. The corresponding growth rates are of the order of the rotation period of the inner part of the galaxy.
The stability problem of MHD Taylor-Couette flows with toroidal magnetic fields is considered in dependence on the magnetic Prandtl number. Only the most uniform (but not current-free) field with B_in = B_out has been considered. For high enough Hart
mann numbers the toroidal field is always unstable. Rigid rotation, however, stabilizes the magnetic (kink-)instability. The axial current which drives the instability is reduced by the electromotive force induced by the instability itself. Numerical simulations are presented to probe this effect as a possibility to measure the turbulent conductivity in a laboratory. It is shown numerically that in a sodium experiment (without rotation) an eddy diffusivity 4 times the molecular diffusivity appears resulting in a potential difference of ~34 mV/m. If the cylinders are rotating then also the eddy viscosity can be measured. Nonlinear simulations of the instability lead to a turbulent magnetic Prandtl number of 2.1 for a molecular magnetic Prandtl number of 0.01. The trend goes to higher values for smaller Pm.
