LAGRANGIAN RELAXATION SCHEMES FOR CALCULATING FORCE-FREE MAGNETIC FIELDS, AND THEIR LIMITATIONS

Force-free magnetic fields are important in many astrophysical settings. Determining the properties of such force-free fields—especially smoothness and stability properties—is crucial to understanding many key phe-nomena in astrophysical plasmas, for example, energy release processes that heat the plasma and lead to dynamic or explosive events. In the present work we discuss a serious limitation on the computation of force-free fields, within the context of a Lagrangian relaxation scheme that conserves magnetic flux and ∇ · B identically. This issue has the potential to invalidate the results produced by numerical force-free field solvers even for cases in which they appear to converge (at fixed grid resolution) to an equilibrium mag-netic field. Error estimates are introduced to assess the quality of the calculated equilibrium. We go on to present an algorithm, based on rewriting the curl operation via Stokes ’ theorem, for calculating the current which holds great promise for improving dramatically the accuracy of the Lagrangian relaxation procedure