Comparison of methods for modelling coronal magnetic fields

¹ School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife, KY16 9SS, UK
e-mail: eeg2@st-andrews.ac.uk; awh@st-andrews.ac.uk
² Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK
³ Space and Atmospheric Physics, The Blackett Laboratory, Imperial College, London SW7 2BW, UK

Received: 28 April 2017
Accepted: 3 November 2017

Abstract

Aims. Four different approximate approaches used to model the stressing of coronal magnetic fields due to an imposed photospheric motion are compared with each other and the results from a full time-dependent magnetohydrodynamic (MHD) code. The assumptions used for each of the approximate methods are tested by considering large photospheric footpoint displacements.

Methods. We consider a simple model problem, comparing the full non-linear MHD, determined with the Lare2D numerical code, with four approximate approaches. Two of these, magneto-frictional relaxation and a quasi-1D Grad-Shafranov approach, assume sequences of equilibria, whilst the other two methods, a second-order linearisation of the MHD equations and Reduced MHD, are time dependent.

Results. The relaxation method is very accurate compared to full MHD for force-free equilibria for all footpoint displacements, but has significant errors when the plasma β₀ is of order unity. The 1D approach gives an extremely accurate description of the equilibria away from the photospheric boundary layers, and agrees well with Lare2D for all parameter values tested. The linearised MHD equations correctly predict the existence of photospheric boundary layers that are present in the full MHD results. As soon as the footpoint displacement becomes a significant fraction of the loop length, the RMHD method fails to model the sequences of equilibria correctly. The full numerical solution is interesting in its own right, and care must be taken for low β₀ plasmas if the viscosity is too high.