### 5.3 Maximum energy

There is in particular a large interest on force-free configurations for a given vertical magnetic field
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 (1984). 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
. This problem has been addressed in the so-called Aly–Sturrock conjecture (Aly, 1984,
1991; 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 (2002) 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 (2002)? 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.)