Computation of Relative Magnetic Helicity in Spherical Coordinates

TL;DR

This paper introduces a precise method for computing relative magnetic helicity in spherical coordinates, improving accuracy over previous approximate techniques, and is validated with models and solar active region data.

Contribution

The paper presents a new, accurate method for calculating magnetic helicity in spherical geometry, suitable for solar applications, outperforming previous approximate approaches.

Findings

Method shows excellent accuracy with semi-analytic models.

Application to solar active regions demonstrates improved precision.

Range of applicability of approximate methods is clarified.

Abstract

Magnetic helicity is a quantity of great importance in solar studies because it is conserved in ideal magneto-hydrodynamics. While many methods to compute magnetic helicity in Cartesian finite volumes exist, in spherical coordinates, the natural coordinate system for solar applications, helicity is only treated approximately. We present here a method to properly compute relative magnetic helicity in spherical geometry. The volumes considered are finite, of shell or wedge shape, and the three-dimensional magnetic field is considered fully known throughout the studied domain. Testing of the method with well-known, semi-analytic, force-free magnetic-field models reveals that it has excellent accuracy. Further application to a set of nonlinear force-free reconstructions of the magnetic field of solar active regions, and comparison with an approximate method used in the past, indicates that the proposed methodology can be significantly more accurate, thus making our method a promising tool in helicity studies that employ the spherical geometry. Additionally, the range of applicability of the approximate method is determined and discussed.