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tokens: Rope System The evolution of a flux rope in a partially open bipolar background field as discussed in Sections 3.1 and 3.2 is summarized in Figure 3 , where h stands for the geometric parameter and λ the control parameter. There are two branches of equilibrium states for the flux rope: the lower branch ( D ∼ A in Figure 3 ) and the upper branch ( C ∼ B in Figure 3 ). The two branches are separated by the upward (from lower to upper branch) and downward (from upper to lower branch) catastrophes. The upward (downward) catastrophe is an irreversible but reproducible process, and it takes place when the solution jumps from the lower (upper) branch to the upper (lower) branch with increasing (decreasing) control parameter. Therefore, the two branches and the two catastrophes form a circulation, as shown by in Figure 3 (a). Figure 3. Schematic representations of the equilibrium solutions of the magnetic flux rope system: the profile of (a) a certain geometric parameter of the rope ( h ) and (b) the magnetic energy in the computational domain ( E ) vs. a certain control parameter of the system ( λ ). An upper catastrophe occurs at λ = λ u , whereas a downward catastrophe occurs at λ = λ d . The subscripts 1, 2, 3, and 4 label the state of the system immediately before and after the upward catastrophic point, and immediately before and after the downward catastrophic point, respectively. Although the jump of h is opposite in sign, the jump of E is always negative, implying a magnetic energy release for both of the two catastrophes. The evolution of magnetic energy (Figure 3 (b)) is also a circulation consisting of the two branches and two catastrophes, but it is quite different from that of the geometric parameter (Figure 3 (a)). At the lower branch (), magnetic energy is stored with increasing control parameter, whereas it is reduced during both upward and downward catastrophes, and during the shift of the solution along the upper branch from B to C as well. The constraint exerted by the background field on the flux rope is larger for solutions at the lower branch than that at the upper branch, so that the growth rate of the magnetic energy with λ , dE / dλ , is larger for lower-branch solutions than upper-branch solutions. Consequently, at the upward catastrophic point ( λ = λ u ), the magnetic energy of the upper-branch solution is smaller than that of the lower-branch solution ( E 2 E 1 ), whereas at the downward catastrophic point ( λ = λ d ), the opposite is true ( E 3 > E 4 ). This provides the essential energy condition under which the downward catastrophe could occur. 3.4. Mechanism of Magnetic Energy Release During the Catastrophe It is widely accepted that magnetic energy is released by magnetic reconnection in solar eruptive activities (Benz 2008 ; Shibata Magara 2011 ). However, magnetic reconnections are prohibited during our simulation (see Section 2 ). There should be other mechanisms of magnetic energy release in addition to magnetic