Defining and Calculating Self‐Helicity in Coronal Magnetic Fields

We introduce two dierent generalizations of relative helicity which may be applied to a portion of the coronal volume. Such a quantity is generally referred to as the self-helicity of the field occupying the sub-volume. Each definition is a natural application of the traditional relative helicity but relative to a dierent reference field. One of the generalizations, which we term additive self-helicity, can be considered a generalization of twist helicity to volumes which are neither closed nor thin. It shares with twist the property of being identically zero for any portion of a potential magnetic field. The other helicity, unconfined self-helicity, is independent of the shape of the volume occupied by the field portion and is therefore akin to the sum of twist and writhe helicity. We demonstrate how each kind of self-helicity may be evaluated in practice. The set of additive self-helicities may be used as a constraint in the minimiza- tion of magnetic energy to produce a piece-wise constant-fi equilibrium. This class of fields falls into a hierarchy, along with the flux constrained equilibria and potential fields, of fields with monotonically decreasing magnetic energies. Piece- wise constant-fi field generally have fewer unphysical properties than genuinely constant-fi fields, whose twist fi is uniform throughout the entire corona.