In this paper we study a class of three-dimensional magnetohydrodynamic model problems that may be useful to understand the role of twisted flux ropes in coronal mass ejections. We construct in a half-space a series of force-free bipolar configurations with different helicity contents and bring them into an evolution by imposing to their footpoints on the boundary slow motions converging toward the inversion line. For all the cases that have been computed, this process leads, after a phase of quasi-static evolution, to the formation of a twisted flux rope by a reconnection process and to the global disruption of the configuration. In contrast with the results of some previous studies, however, the rope is never in equilibrium. It thus appears that the presence of a rope in the preeruptive phase is not a necessary condition for the disruption but may be the product of the disruption itself. Moreover, the helicity keeps an almost constant value during the evolution, and the problem of the origin of the helicity content of an eruptive configuration appears to be that of the initial force-free state. In addition to these numerical simulations, we report some new relations for the time variations of the energy and the magnetic helicity and develop a simple analytical model in which the magnetic field evolution exhibits essential features quite similar to those observed during the quasi-static phase in the numerics.
