  2.5D time-dependent ideal magnetohydrodynamic (MHD) models in Cartesian
coordinates were used in previous studies to seek MHD equilibria involving a
magnetic flux rope embedded in a bipolar, partially open background field. As
demonstrated by these studies, the equilibrium solutions of the system are
separated into two branches: the flux rope sticks to the photosphere for
solutions at the lower branch but is suspended in the corona for those at the
upper branch. Moreover, a solution originally at the lower branch jumps to the
upper, as the related control parameter increases and reaches a critical value,
and the associated jump is here referred to as upward catastrophe. The present
paper advances these studies in three aspects. First, the magnetic field is
changed to be force-free. The system still experiences an upward catastrophe
with an increase in each control parameter. Secondly, under the force-free
approximation, there also exists a downward catastrophe, characterized by a
jump of a solution from the upper branch to the lower. Both catastrophes are
irreversible processes connecting the two branches of equilibrium solutions so
as to form a cycle. Finally, the magnetic energy in the numerical domain is
calculated. It is found that there exists a magnetic energy release for both
catastrophes. The Amp\`{e}re's force, which vanishes everywhere for force-free
fields, appears only during the catastrophes and does a positive work, which
serves as a major mechanism for the energy release. The implications of the
downward catastrophe and its relevance to solar activities are briefly
discussed.
