A model for magnetic energy storage and Taylor’s relaxation in the solar corona. I: Helicity-constrained minimum energy state in a half-cylinder

Abstract

The problem of the existence of a minimum energy state is studied in the set ℋ of all the magnetic fields B: (i) occupying the half-cylinder D={r<R,z≳0}; (ii) having a normal component vanishing on the vertical part {r=R} of the boundary of D and taking given values Q(r) on its horizontal part {z=0}; (iii) having a relative helicity equal to a prescribed value H. It is first shown that the only field that may possibly be an energy minimizer in ℋ is the unique (and therefore axisymmetric) constant-α force-free field Bα contained in that set. Thus it is proved that Bα minimizes the energy indeed if and only if 0≤‖H‖≤Hc≤∞, where Hc is an estimated critical value. For Hc≤‖H‖≤∞, on the contrary, it is possible to construct in ℋ nonaxisymmetric fields with an energy smaller than that of Bα and no minimum energy state does exist. However, Bα still minimizes the energy for Hc ≤‖ H‖ ≤ Haxc (with possibly Haxc = ∞) if attention is restricted to the axisymmetric fields of ℋ. These results are used to put a limit on the validity of a popular model of the heating of the solar corona, in which the field of a coronal structure is supposed to release sporadically, by Taylor’s relaxation, a part of the energy it continuously extracts from the kinetic energy of the photospheric motions. It is argued that, as a consequence of the results above, one of the basic assumptions of the model breaks down when the field becomes highly sheared. It is speculated that, in such a situation, a completely new regime should set up, in which helicity and energy are continuously ejected at large distances by the system.