Three-dimensional solar active region magnetohydrostatic models and their stability using Euler potentials

¹ Departament de Física, Universitat de les Illes Balears (UIB), 07122 Palma, Spain
² Institute of Applied Computing & Community Code (IAC 3), UIB, 07122 Palma, Spain
e-mail: jaume.terradas@uib.es
³ School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK

Received: 5 August 2022
Accepted: 3 December 2022

Abstract

Active regions (ARs) are magnetic structures typically found in the solar atmosphere. We calculated several magnetohydrostatic (MHS) equilibrium models that include the effect of a finite plasma-β and gravity and that are representative of AR structures in three dimensions. The construction of the models is based on the use of two Euler potentials, α and β, that represent the magnetic field as B = ∇α × ∇β. The ideal MHS nonlinear partial differential equations were solved numerically using finite elements in a fixed 3D rectangular domain. The boundary conditions were initially chosen to correspond to a potential magnetic field (current-free) with known analytical expressions for the corresponding Euler potentials. The distinctive feature of our model is that we incorporated the effect of shear by progressively deforming the initial potential magnetic field. This procedure is quite generic and allowed us to generate a vast variety of MHS models. The thermal structure of the ARs was incorporated through the dependence of gas pressure and temperature on the Euler potentials. Using this method, we achieved the characteristic hot and overdense plasma found in ARs, but we demonstrate that the method can also be applied to study configurations with open magnetic field lines. Furthermore, we investigated basic topologies that include neutral lines. Our focus is on the force balance of the structures, and we do not consider the energy balance in the constructed models. In addition, we addressed the difficult question of the stability of the calculated 3D models. We find that if the plasma is convectively stable, then the system is not prone, in general, to develop magnetic Rayleigh-Taylor instabilities. However, when the plasma-β is increased or the density at the core of the AR is high, then the magnetic configuration becomes unstable due to magnetic buoyancy.