  We model the Parker instability in vertically stratified isothermal gas using
non-ideal MHD three-dimensional simulations. Rotation, especially differential,
more strongly and diversely affects the nonlinear state than the linear stage
(where we confirm the most important conclusions of analytical models), and
stronger than any linear analyses predict. Steady state magnetic fields are
stronger and cosmic ray energy density higher than in comparable nonrotating
systems. Transient gas outflows induced by the nonlinear instability persist
longer, of order 2 Gyr, with rotation. Stratification combined with
(differential) rotation drives helical flows, leading to mean-field dynamo.
Consequently, the nonlinear state becomes oscillatory (while both the linear
instability and the dynamo are non-oscillatory). The horizontal magnetic field
near the midplane reverses its direction propagating to higher altitudes as the
reversed field spreads buoyantly. The spatial pattern of the large-scale
magnetic field may explain the alternating magnetic field directions in the
halo of the edge-on galaxy NGC 4631. Our model is unique in producing a
large-scale magnetic structure similar to such observation. Furthermore, our
simulations show that the mean kinetic helicity of the magnetically driven
flows has the sign opposite to that in the conventional non-magnetic flows.
This has profound consequences for the nature of the dynamo action and
large-scale magnetic field structure in the coronae of spiral galaxies which
remain to be systematically explored and understood. We show that the energy
density of cosmic rays and magnetic field strength are not correlated at scales
of order a kiloparsec.
