Magnetic helicity estimations in models and observations of the solar magnetic field. Part I: Finite volume methods

TL;DR

This paper systematically compares six finite volume methods for estimating magnetic helicity in models and observations, demonstrating their accuracy, reliability, and sensitivity to errors through various tests, including realistic solar simulations.

Contribution

It provides the first comprehensive benchmarking of multiple finite volume methods for magnetic helicity estimation, highlighting their mutual consistency and limitations.

Findings

Most methods agree within a few percent across tests.

Reliability of methods in realistic solar simulations is high with only 3% spread.

Sensitivity to numerical resolution and solenoidal errors varies among methods.

Abstract

Magnetic helicity is a conserved quantity of ideal magneto-hydrodynamics characterized by an inverse turbulent cascade. Accordingly, it is often invoked as one of the basic physical quantities driving the generation and structuring of magnetic fields in a variety of astrophysical and laboratory plasmas. We provide here the first systematic comparison of six existing methods for the estimation of the helicity of magnetic fields known in a finite volume. All such methods are reviewed, benchmarked, and compared with each other, and specifically tested for accuracy and sensitivity to errors. To that purpose, we consider four groups of numerical tests, ranging from solutions of the three-dimensional, force-free equilibrium, to magneto-hydrodynamical numerical simulations. Almost all methods are found to produce the same value of magnetic helicity within few percent in all tests. In the more solar-relevant and realistic of the tests employed here, the simulation of an eruptive flux rope, the spread in the computed values obtained by all but one method is only 3%, indicating the reliability and mutual consistency of such methods in appropriate parameter ranges. However, methods show differences in the sensitivity to numerical resolution and to errors in the solenoidal property of the input fields. In addition to finite volume methods, we also briefly discuss a method that estimates helicity from the field lines' twist, and one that exploits the field's value at one boundary and a coronal minimal connectivity instead of a pre-defined three-dimensional magnetic-field solution.