Limitations of force-free magnetic field extrapolations: revisiting basic assumptions

TL;DR

Force-free magnetic field extrapolations in the solar corona often neglect plasma-beta, leading to significant errors in estimating currents and magnetic energy, which limits their reliability for understanding coronal heating and eruptions.

Contribution

This paper critically examines the assumptions behind force-free extrapolations, demonstrating that neglecting plasma-beta results in unreliable estimates of currents and magnetic energy.

Findings

Pressure gradients can balance Lorentz force when plasma-beta is significant.

Errors in magnetic energy estimates are comparable to the free magnetic energy itself.

Force-free extrapolations accurately depict magnetic structure but not current or energy quantities.

Abstract

Force-free extrapolations are widely used to study the magnetic field in the solar corona based on surface measurements. The extrapolations assume that the ratio of internal energy of the plasma to magnetic energy, the plasma-beta is negligible. Despite the widespread use of this assumption observations, models, and theoretical considerations show that beta is of the order of a few percent to more than 10%, and thus not small. We investigate what consequences this has for the reliability of extrapolation results. We use basic concepts starting with the force and the energy balance to infer relations between plasma-beta and free magnetic energy, to study the direction of currents in the corona with respect to the magnetic field, and to estimate the errors in the free magnetic energy by neglecting effects of the plasma (beta<<1). A comparison with a 3D MHD model supports our basic considerations. If plasma-beta is of the order of the relative free energy (the ratio of the free magnetic energy to the total magnetic energy) then the pressure gradient can balance the Lorentz force. This is the case in the solar corona, and therefore the currents are not properly described. In particular the error in terms of magnetic energy by neglecting the plasma is of the order of the free magnetic energy, so that the latter can not be reliably determined by an extrapolation. While a force-free extrapolation might capture the magnetic structure and connectivity of the coronal magnetic field, the derived currents and free magnetic energy are not reliable. Thus quantitative results of extrapolations on the location and amount of heating in the corona (through current dissipation) and on the energy storage of the magnetic field (e.g. for eruptive events) are limited.