  Parker instability arises from the presence of magnetic fields in a plasma in
a gravitational field such as the interstellar medium (ISM), wherein the
magnetic buoyant pressure expels the gas and causes the gas to move along the
field lines. The process of mixing of this instability in the ISM near the
Galactic plane is investigated. The initial ISM is assumed to consist of two
fluids: plasma gas and cosmic-ray particles, in hydrostatic equilibrium,
coupled with a uniform, azimuthally-aligned magnetic field. The evolution of
the instability is explored in two models: an isothermal exponential-declining
density model and a two-layered, hyperbolic tangent temperature model. After a
small perturbation, the unstable gas aggregates at the bottom of the magnetic
loops and forms dense blobs. The growth rate of the instability decreases as
the coupling between the cosmic rays and the plasma becomes stronger (meaning a
smaller CR diffusion coefficient). The mixing is enhanced by the cosmic-ray
diffusion, while the shape of the condensed gas depends sensitively on the
initial equilibrium conditions. The hyperbolic tangent temperature model
produces a more concentrated and round shape of clumps at the foot points of
rising magnetic arches, like the observed giant molecular cloud, whereas the
exponential density model gives rise to a filamentary morphology of the clumpy
structure. When considering a minimum perpendicular or cross field diffusion of
cosmic rays, which is often substantially smaller than the parallel coefficient
$\kappa_{\|}$, around $2%-4%$ of $\kappa_{\|}$, the flow speed is significantly
increased such that the magnetic loops extend to a greater altitude. We
speculate that the galactic wind flow perpendicular to the galactic disk may be
facilitated by Parker instabilities through the cross field diffusion of cosmic
rays.
