A bound for relative magnetic helicity in terms of free magnetic energy

The topological complexity of magnetic fields in astrophysics can be described with the aid of magnetic helicity. Amongst its many properties, magnetic helicity provides a bound on the magnetic energy. Whilst this is a very elegant result, classical magnetic helicity, which applies to closed magnetic fields, is not suitable in many astrophysical problems, e.g. modelling active regions in the solar atmosphere. Instead, the classical definition must be replaced by relative helicity, which is able to include open magnetic fields. The purpose of this note is to extend the classical helicity-energy bound to show that an analogous relationship holds between relative magnetic helicity and the free magnetic energy, i.e. the magnetic energy available above the minimum-energy field. The bound is constructed by considering a selfmutual decomposition of relative helicity, which provides information on how the structure of the magnetic field is constrained for a given value of free energy