Maximum energy of semi-infinite magnetic field configurations

This article addresses the conjecture by Aly (1984) that the maximum-energy state of magnetic field configurations that have a given flux distribution on a given surface is the open-field configurations. It is first shown that the existence of a maximum-energy configuration depends upon the topology of the source surface: for a multiply connected surface there is no maximum-energy state. However, the magnetic energy is known to be bounded above for a wide class of simply connected surfaces, and it is shown that in this case there must be a field configuration that has maximum energy. Furthermore, for this configuration the energy is stationary under arbitrary small footprint displacements that leave the surface flux distribution unchanged. It is then shown that this condition leads to the result that either Bn, the normal component of magnetic field, of jn, the normal component of current, must vanish at each point on the source surface. This condition is met by the open configuration but not by any other configuration. The implications of this result for the understanding of flares and of coronal mass ejections are discussed.