Magnetic Energy and Helicity Budgets in the Active-Region Solar Corona. II. Nonlinear Force-Free Approximation

TL;DR

This paper introduces a practical method to estimate the free magnetic energy and helicity in solar active regions using only boundary magnetic data, improving efficiency and interpretability over traditional 3D extrapolation techniques.

Contribution

It develops a nonlinear force-free approximation method that derives magnetic energy and helicity budgets from boundary data without detailed 3D magnetic field knowledge.

Findings

Method accurately reproduces energies and helicities compared to traditional extrapolation.

Corrects for overestimation in linear force-free models for larger active regions.

Provides physical insights into eruptive phenomena in solar active regions.

Abstract

Expanding on an earlier work that relied on linear force-free magnetic fields, we self-consistently derive the instantaneous free magnetic energy and relative magnetic helicity budgets of an unknown three-dimensional nonlinear force-free magnetic structure extending above a single, known lower-boundary magnetic field vector. The proposed method does not rely on the detailed knowledge of the three-dimensional field configuration but is general enough to employ only a magnetic connectivity matrix on the lower boundary. The calculation yields a minimum free magnetic energy and a relative magnetic helicity consistent with this free magnetic energy. The method is directly applicable to photospheric or chromospheric vector magnetograms of solar active regions. Upon validation, it basically reproduces magnetic energies and helicities obtained by well-known, but computationally more intensive and non-unique, methods relying on the extrapolated three-dimensional magnetic field vector. We apply the method to three active regions, calculating the photospheric connectivity matrices by means of simulated annealing, rather than a model-dependent nonlinear force-free extrapolation. For two of these regions we correct for the inherent linear force-free overestimation in free energy and relative helicity that is larger for larger, more eruptive, active regions. In the third studied region, our calculation can lead to a physical interpretation of observed eruptive manifestations. We conclude that the proposed method, including the proposed inference of the magnetic connectivity matrix, is practical enough to contribute to a physical interpretation of the dynamical evolution of solar active regions.