On Measuring Divergence for Magnetic Field Modeling

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 $\left\langle | {f}_{i}| \right\rangle $ . We show that $\left\langle | {f}_{i}| \right\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., who measured $\left\langle | {f}_{i}| \right\rangle $ for a series of nonlinear force-free field 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. data and analyze the effect of spatial resolution on these metrics using a nonparametric method. We find that some of the trends reported by DeRosa et al. are due to the intrinsic scaling of $\left\langle | {f}_{i}| \right\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.