Lagrangian relaxation schemes for calculating force-free magnetic fields, and their limitations

TL;DR

This paper examines limitations of Lagrangian relaxation methods for computing force-free magnetic fields in astrophysics, highlighting potential inaccuracies and proposing an improved algorithm for better precision.

Contribution

It identifies a critical limitation in existing numerical solvers for force-free fields and introduces a new algorithm based on Stokes' theorem to enhance accuracy.

Findings

Limitations can invalidate numerical force-free field results.

Error estimates help assess solution quality.

Proposed algorithm improves current calculation accuracy.

Abstract

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. Here we report on a serious limitation on the computation of force-free fields that 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. In the present work we discuss this problem within the context of a Lagrangian relaxation scheme that conserves magnetic flux and div(B) identically. Error estimates are introduced to assess the quality of the calculated equilibrium. We go on to present an algorithm, based on re-writing the curl operation via Stokes' theorem, for calculating the current which holds great promise for improving dramatically the accuracy of the Lagrangian relaxation procedure.