Nonlinear force-free magnetic field extrapolations: comparison of the Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithm

TL;DR

This paper compares two algorithms, Grad-Rubin and Wheatland-Sturrock-Roumeliotis, for extrapolating nonlinear force-free magnetic fields, highlighting their performance, limitations, and applicability to different boundary conditions.

Contribution

The study implements both algorithms on a unified finite element framework and evaluates their effectiveness and limitations in reconstructing force-free magnetic fields.

Findings

Both algorithms perform well with known analytic fields.

Wheatland-Sturrock-Roumeliotis struggles with inconsistent boundary data.

Grad-Rubin loses convergence at high current densities.

Abstract

We compare the performance of two alternative algorithms which aim to construct a force-free magnetic field given suitable boundary conditions. For this comparison, we have implemented both algorithms on the same finite element grid which uses Whitney forms to describe the fields within the grid cells. The additional use of conjugate gradient and multigrid iterations result in quite effective codes. The Grad-Rubin and Wheatland-Sturrock-Roumeliotis algorithms both perform well for the reconstruction of a known analytic force-free field. For more arbitrary boundary conditions the Wheatland-Sturrock-Roumeliotis approach has some difficulties because it requires overdetermined boundary information which may include inconsistencies. The Grad-Rubin code on the other hand loses convergence for strong current densities. For the example we have investigated, however, the maximum possible current density seems to be not far from the limit beyond which a force free field cannot exist anymore for a given normal magnetic field intensity on the boundary.