Self-Similar Expansion Model of Cylindrical Flux Ropes Combined with an Alfvén Wave Current System

Magnetic flux ropes in space are generally connected to some regions electromagnetically. We consider the whole closed current system of the expanding flux ropes including the electric current associated with them. By combining the theories regarding the self-similar expansion of cylindrical flux ropes and the Alfvén wave current system, we examine conditions under which the electric current matches. These matching conditions are satisfied when the time dependence of the current flowing in the closed circuit agrees with that which maintains the expanding flux rope. As a consequence, we encountered three possible forms of expansion. The two-step eruption of solar filaments may be interpreted as a transition from one form of expansion to another. If this process works, increasing the diffusion outside of the flux rope is necessary to trigger the transition.