An energy formula for nonlinear force-free magnetic fields

This paper describes the relation between energy and helicity integrals for nonlinear force-free fields, given arbitrary boundary conditions. First, the author reviews the relation of helicity integrals to structural properties of a field such as twist and shear. Helicity injection into an atmosphere due to boundary motions is also discussed. A gauge-invariant fieldline helicity is defined which measures the net twisting of magnetic flux about a given fieldline. The author then presents a formula for the energy of a force-free field in termns of these fieldine helicities. An expression for the energy of arbitrary fields, in terms of linking between magnetic fieldlines and the electrical current, is also given.