Defining and calculating self-helicity in coronal magnetic fields

We introduce two different 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 subvolume. Each definition is a natural application of the traditional relative helicity but relative to a different 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 minimization of magnetic energy to produce a piecewise constant-a 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. Piecewise constant-a fields generally have fewer unphysical properties than genuinely constant-a fields, whose twist a is uniform throughout the entire corona.