5.3 Maximum energy

There is in particular a large interest on force-free configurations for a given vertical magnetic field Bn on the lower boundary and in which range the energy content of these configurations can be. For such theoretical investigations, one usually assumes a so-called star-shaped volume, like the exterior of a spherical shell and the coronal magnetic field is unbounded but has a finite magnetic energy. (Numerical computations, on the other hand, are mainly carried out in finite computational volumes, like a 3D-box in Cartesian geometry.) It is not the aim of this review to follow the involved mathematical derivation, which the interested reader finds in Aly (1984Jump To The Next Citation Point). As we saw above, the minimum energy state is reached for a potential field. On the other hand, one is also interested in the maximum energy a force-free configuration can obtain for the same boundary conditions Bn. This problem has been addressed in the so-called Aly–Sturrock conjecture (Aly, 1984, 1991Jump To The Next Citation Point; Sturrock, 1991). The conjecture says that the maximum magnetic energy is obtained if all magnetic field lines are open (have one footpoint in the lower boundary and reach to infinity). This result implies that any non-open force-free equilibrium (which contains electric currents parallel to closed magnetic field lines, e.g., created by stressing closed potential field lines) contains an energy which is higher than the potential field, but lower than the open field. As pointed out by Aly (1991) these results imply that the maximum energy which can be released from an active region, say in a flare or coronal mass ejection (CME), is the difference between the energy of an open field and a potential field. While a flare requires free magnetic energy, the Aly–Sturrock conjecture does also have the consequence that it would be impossible that all field lines become open directly after a flare, because opening the field lines costs energy. This is in some way a contradiction to observations of CMEs, where a closed magnetic structure opens during the eruption. Choe and Cheng (2002Jump To The Next Citation Point) constructed force-free equilibria containing tangential discontinuities in multiple flux systems, which can be generated by footpoint motions from an initial potential field. These configurations contain energy exceeding the open field, a violation of the Aly–Sturrock conjecture, and would release energy by opening all field lines. Due to the tangential discontinuities, these configurations contain thin current sheets, which can develop micro-instabilities to convert magnetic energy into other energy forms (kinetic and thermal energy) by resistive processes like magnetic reconnection. It is not clear (Aly and Amari, 2007), however, which conditions are exactly necessary to derive force-free fields with energies above the open field: Is it necessary that the multiple flux-tubes are separated by non-magnetic regions like in Choe and Cheng (2002Jump To The Next Citation Point)? Or would it be sufficient that the field in this region is much weaker than in the flux tubes but remains finite? (See Sakurai, 2007, for a related discussion.)
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