On the Relation Between Reconnected Flux and Parallel Electric Fields in the Solar Corona

The complicated magnetic topology in the solar corona often does not lend itself to the ready differentiation of topologically different regions. Instead, magnetic reconnection often happens in coronal fields that do not vanish, and for which no separator or separatrix is readily identifiable. For this, rather generic, situation, the question of how to relate observed magnetic flux changes to parallel electric fields remains unanswered. In order to shed light on this question, we apply to this problem the theory of General Magnetic Reconnection. We prove that a unique relation exists between the time evolution of the reconnected magnetic flux on one hand, and the integral along magnetic field lines of the parallel electric field on the other. Furthermore, we will use two explicit examples to illustrate how parallel electric fields are related to the formation of closed loops by the reconnection process, and how flux rope-like topologies result from reconnection if photospheric electric fields are negligible on flare time scales.