We present three-dimensional MHD simulations of the evolution of the magnetic field in the corona where the emergence of a twisted magnetic flux tube is driven at the lower boundary into a pre-existing coronal potential arcade field. Through a sequence of simulations in which we vary the amount of twisted flux transported into the corona before the emergence is stopped, we investigate the conditions that lead to a dynamic eruption of the resulting coronal flux rope. It is found that the critical condition for the onset of eruption is for the center of the flux rope to reach a critical height at which the corresponding potential field declines with height at a sufficiently steep rate, consistent with the onset of the torus instability of the flux rope. In some cases, immediately after the emergence is stopped, the coronal flux rope first settles into a quasi-static rise with an underlying sigmoid-shaped current layer developing. Preferential heating of field lines going through this current layer may give rise to the observed quiescent X-ray sigmoid loops before eruption. Reconnections in the current layer during the initial quasi-static stage is found to add detached flux to the coronal flux rope, allowing it to rise quasi-statically to the critical height and dynamic eruption of the flux rope then ensues. By identifying field lines whose tops are in the most intense part of the current layer during the eruption, we deduce the evolution and morphology of the post-flare X-ray loops and the flare ribbons at their footpoints.
