On measuring divergence for magnetic field modeling

TL;DR

This paper critiques the common divergence metric in magnetic field modeling, introduces a scale-independent alternative, and demonstrates that previous resolution-dependent results may be misleading, emphasizing the importance of multiple metrics.

Contribution

The authors propose a modified divergence metric that does not depend on mesh size and analyze its implications on existing NLFFF model data.

Findings

The average fractional flux metric scales with mesh size, affecting interpretation.

Different divergence metrics can yield contrasting results on the same data.

Multiple divergence metrics provide a more comprehensive assessment of magnetic field models.

Abstract

A physical magnetic field has a divergence of zero. Numerical error in constructing a model field and computing the divergence, however, introduces a finite divergence into these calculations. A popular metric for measuring divergence is the average fractional flux $\langle |f_{i}| \rangle$. We show that $\langle |f_{i}| \rangle$ scales with the size of the computational mesh, and may be a poor measure of divergence because it becomes arbitrarily small for increasing mesh resolution, without the divergence actually decreasing. We define a modified version of this metric that does not scale with mesh size. We apply the new metric to the results of DeRosa et al. (2015), who measured $\langle |f_{i}| \rangle$ for a series of Nonlinear Force-Free Field (NLFFF) models of the coronal magnetic field based on solar boundary data binned at different spatial resolutions. We compute a number of divergence metrics for the DeRosa et al. (2015) data and analyze the effect of spatial resolution on these metrics using a non-parametric method. We find that some of the trends reported by DeRosa et al. (2015) are due to the intrinsic scaling of $\langle |f_{i}| \rangle$. We also find that different metrics give different results for the same data set and therefore there is value in measuring divergence via several metrics.