Abstrakt:
In this paper we continue investigations of the Parker-shearing instability performing numerical simulations of the magnetic flux tube dynamics in the thin flux-tube approximation. We show that evolution of flux tubes resulting from numerical simulations is very similar to that of linear solutions if the vertical displacements are smaller than the vertical scale height $H$ of the galactic disc. If the vertical displacements are comparable to $H$, the vertical growth of perturbations is faster in the nonlinear range than in the linear one and we observe a rapid inflation of the flux tube at its top, which leads to a singularity in numerical simulations, if only the cosmic rays are taken into account. Then we perform simulations for the case of nonuniform external medium, which show that the dominating wavelength of the Parker instability is the same as the wavelength of modulations of external medium. As a consequence of this fact, in the case of dominating cosmic ray pressure, the dynamo $\alpha$ effect related to these short wavelength modulations is much more efficient than that related to the linearly most unstable long wavelengths modes of the Parker instability. Under the influence of differential forces resulting from differential rotation and the density waves, the $\alpha$-effect is essentially magnified in the spiral arms and diminished in the interarm regions, what confirms our previous results obtained in the linear approximation.
