Relaxed states in a spheromak with inhomogeneous boundary fields

In this paper we consider force-free equilibrium solutions of the MHD equations in a spherical geometry for the case in which magnetic flux crosses the boundary of the containing vessel. The main motivation is to model more faithfully actual spheromak experiments in the laboratory, for which boundaries are unlikely to be magnetic surfaces. We show how a general inhomogeneous boundary field may be constructed from individual components. In particular, we consider the cases of a boundary field of dipolar form and one of quadrupolar form. We then go on to discuss solutions for fields embedded in point or ring electrodes using the ‘general solution’, some of which can be used to model experiments such as the PS-1- or CTX-type spheromaks.