Nonlinear Parker Instability with the Effect of Cosmic-Ray Diffusion

We present the results of linear analysis and two-dimensional local magnetohydrodynamic (MHD) simulations of the Parker instability, including the effects of cosmic rays (CRs), in magnetized gas disks (galactic disks). As an unperturbed state for both the linear analysis and the MHD simulations, we adopted an equilibrium model of a magnetized, two-temperature, layered disk with constant gravitational acceleration parallel to the normal of the disk. The disk comprises a thermal gas, CRs, and a magnetic field perpendicular to the gravitational acceleration. CR diffusion along the magnetic field is considered; cross-field-line diffusion is supposed to be small and is ignored. We investigate two cases in our simulations. In the mechanical perturbation case, we add a velocity perturbation parallel to the magnetic field lines, while in the explosive perturbation case, we add CR energy into the sphere in which the CRs are injected. Linear analysis shows that the growth rate of the Parker instability becomes smaller if the coupling between the CRs and the gas is stronger (i.e., if the CR diffusion coefficient is smaller). Our MHD simulations of the mechanical perturbation confirm this result. We show that the falling matter is impeded by the CR pressure gradient; this causes a decrease in the growth rate. In the explosive perturbation case, the growth of the magnetic loop is faster when the coupling is stronger in the early stage. However, in the later stage the behavior of the growth rate becomes similar to the mechanical perturbation case.