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 phenomena 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 magnetic 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