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2.5D time-dependent ideal magnetohydrodynamic (MHD) models in Cartesiancoordinates were used in previous studies to seek MHD equilibria involving amagnetic flux rope embedded in a bipolar, partially open background field. Asdemonstrated by these studies, the equilibrium solutions of the system areseparated into two branches: the flux rope sticks to the photosphere forsolutions at the lower branch but is suspended in the corona for those at theupper branch. Moreover, a solution originally at the lower branch jumps to theupper, as the related control parameter increases and reaches a critical value,and the associated jump is here referred to as upward catastrophe. The presentpaper advances these studies in three aspects. First, the magnetic field ischanged to be force-free. The system still experiences an upward catastrophewith an increase in each control parameter. Secondly, under the force-freeapproximation, there also exists a downward catastrophe, characterized by ajump of a solution from the upper branch to the lower. Both catastrophes areirreversible processes connecting the two branches of equilibrium solutions soas to form a cycle. Finally, the magnetic energy in the numerical domain iscalculated. It is found that there exists a magnetic energy release for bothcatastrophes. The Amp\`{e}re's force, which vanishes everywhere for force-freefields, appears only during the catastrophes and does a positive work, whichserves as a major mechanism for the energy release. The implications of thedownward catastrophe and its relevance to solar activities are brieflydiscussed.

Downward catastrophe of solar magnetic flux ropes