The solar tachocline is shown as hydrodynamically stable against nonaxisymmetric disturbances if it is true that no cos4θ term exists in its rotation law. We also show that the toroidal field of 200 Gauss amplitude which produces the tachocline in the magnetic theory of Rüdiger & Kitchatinov (1997) is stable against nonaxisymmetric MHD disturbances – but it becomes unstable for rotation periods slightly slower than 25 days. The instability of such weak fields lives from the high thermal diffusivity of stellar radiation zones compared with the magnetic diffusivity. The growth times, however, result as very long (of order of 105 rotation times). With estimations of the chemical mixing we find the maximal possible field amplitude to be ~500 Gauss in order to explain the observed lithium abundance of the Sun. Dynamos with such low field amplitudes should not be relevant for the solar activity cycle.

With nonlinear simulations of MHD Taylor-Couette flows it is shown that for the rotation-dominated magnetic instability the resulting eddy viscosity is only of the order of the molecular viscosity. The Schmidt number as the ratio of viscosity and chemical diffusion grows to values of ~20. For the majority of the stellar physics applications, the magnetic-dominated Tayler instability will be quenched by the stellar rotation.