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Magnetic Flux Emergence into the Solar Corona. III. The Role of Magnetic Helicity Conservation

This paper treats the reconfiguration of a twisted magnetic field, from an initial two-flux system containing a current sheet to a minimum-energy state, under the conservation of total relative magnetic helicity. In the specific model presented, we assume that a fresh magnetic field of the opposite polarity has emerged into a corona containing a preexisting magnetic field, both represented by constant-α force-free fields with the same constant α. The magnetic reconnection that takes place between the two twisted magnetic flux systems during a relaxation is assumed to take the field to a minimum-energy state that keeps the total relative magnetic helicity conserved. Our calculations suggest that this kind of relaxation may result in the formation of magnetic flux ropes and may change the twist directions in flux ropes in situations where flux ropes exist in the emerging or preexisting fields. These effects are all due to the interplays between the internal magnetic helicities of the two flux systems and their mutual magnetic helicity, with redistribution of these helicities through magnetic reconnection. In the absence of an interior current sheet, the lowest α force-free field always has the minimum magnetic energy for a given magnetic helicity, as Berger has shown. When an interior current sheet is present, this result breaks down. The lowest α force-free magnetic field with an interior equilibrium current sheet does not always have the minimum magnetic energy for a given total magnetic helicity. Implications of our results for flux emergence in the solar corona are also addressed.