  Using an isothermal MHD code, we have performed three-dimensional,
high-resolution simulations of the Parker instability. The initial equilibrium
system is composed of exponentially-decreasing isothermal gas and magnetic
field (along the azimuthal direction) under a uniform gravity. The evolution of
the instability can be divided into three phases: linear, nonlinear, and
relaxed. During the linear phase, the perturbations grow exponentially with a
preferred scale along the azimuthal direction but with smallest possible scale
along the radial direction, as predicted from linear analyses. During the
nonlinear phase, the growth of the instability is saturated and flow motion
becomes chaotic. Magnetic reconnection occurs, which allows gas to cross field
lines. This, in turn, results in the redistribution of gas and magnetic field.
The system approaches a new equilibrium in the relaxed phase, which is
different from the one seen in two-dimensional works. The structures formed
during the evolution are sheet-like or filamentary, whose shortest dimension is
radial. Their maximum density enhancement factor relative to the initial value
is less than 2. Since the radial dimension is too small and the density
enhancement is too low, it is difficult to regard the Parker instability alone
as a viable mechanism for the formation of giant molecular clouds.
