Magnetic helicity and the relaxation of fossil fields

In the absence of an active dynamo, purely poloidal magnetic field configurations are unstable to large-scale dynamical perturbations, and decay via reconnection on an Alfvenic time-scale. Nevertheless, a number of classes of dynamo-free stars do exhibit significant, long-lived, surface magnetic fields. Numerical simulations suggest that the large-scale poloidal field in these systems is stabilized by a toroidal component of the field in the stellar interior. Using the principle of conservation of total helicity, we develop a variational principle for computing the structure of the magnetic field inside a conducting sphere surrounded by an insulating vacuum. We show that, for a fixed total helicity, the minimum energy state corresponds to a force-free configuration. We find a simple class of axisymmetric solutions, parametrized by angular and radial quantum numbers. However, these solutions have a discontinuity in the toroidal magnetic field at the stellar surface which will exert a toroidal stress on the surface of the star. We then describe two other classes of solutions, the standard spheromak solutions and ones with fixed surface magnetic fields, the latter being relevant for neutron stars with rigid crusts. We discuss the implications of our results for the structure of neutron star magnetic fields, the decay of fields, and the origin of variability and outbursts in magnetars.