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The pinch-type instability of helical magnetic 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 rotation. 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 Ω> Ω^A (Ω^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.