Modelling the solar transition region using an adaptive conduction method

Abstract

Modelling the solar Transition Region with the use of an Adaptive Conduction (TRAC) method permits fast and accurate numerical solutions of the field-aligned hydrodynamic equations, capturing the enthalpy exchange between the corona and transition region, when the corona undergoes impulsive heating. The TRAC method eliminates the need for highly resolved numerical grids in the transition region and the commensurate very short time steps that are required for numerical stability. When employed with coarse spatial resolutions, typically achieved in multi-dimensional magnetohydrodynamic codes, the errors at peak density are less than 5% and the computation time is three orders of magnitude faster than fully resolved field-aligned models. This paper presents further examples that demonstrate the versatility and robustness of the method over a range of heating events, including impulsive and quasi-steady footpoint heating. A detailed analytical assessment of the TRAC method is also presented, showing that the approach works through all phases of an impulsive heating event because (i) the total radiative losses and (ii) the total heating when integrated over the transition region are both preserved at all temperatures under the broadening modifications of the method. The results from the numerical simulations complement this conclusion.

Transition region adaptive conduction (TRAC) in multidimensional magnetohydrodynamic simulations

“…To restrict this cutoff temperature to a reasonable range, an upper bound, 0.2 max (T (s)), and a lower bound, 2×10 4 K, were set, as recommended by Johnston & Bradshaw (2019). Other restrictions to prevent some sudden jumps can be found in the Appendix A of Johnston et al (2020). For the region in which the temperature is lower than T c , the TRAC method applies the following modifications:…”

“…This TRAC method is expected to be both efficient and accurate, eliminating the shortcomings of the previous two methods. Johnston et al (2020) gave a detailed explanation and demonstration of the physical idea behind this method. Based on this, an additional modification to the heating term was proposed.…”

Linking computational models to follow the evolution of heated coronal plasma

“…The details of the coronal models and the methods used are presented by Reid et al (2020) for 3D MHD, and by Johnston et al (2020) for 1D hydrodynamic, simulations; only a brief summary is given here. The 3D coronal model is based on the three-strand avalanche model of Reid et al (2018) and Reid (2020), sketched in Fig.…”

“…addressing thermal conduction, such as super-time-stepping (implemented by Johnston et al 2017a), ∆t MHD and ∆t c contribute roughly equally to the time step restriction required for numerical stability. As a rule of thumb, a grid having |∆x| = ∆s = 100 km can model accurately coronal temperatures of the order of 1 MK (Bradshaw & Cargill 2013;Johnston et al 2017aJohnston et al , 2020 and a number of contemporary MHD models that incorporate energy transport (such as those of Dahlburg et al 2016Dahlburg et al , 2018 satisfy this criterion. However, this is a 'best case' scenario.…”

“…This heating can then be used as input for a one-dimensional (hereafter, 1D), field-aligned hydrodynamic model, which does include thermal conduction and optically thin radiation. We also use the new 'Transition Region using Adaptive Conduction' method (TRAC; Johnston et al 2020) in order to ensure rapid computation, although we note here that the TRAC method is not a universal panacea to the problems of 3D coronal MHD. We return to this aspect in the discussion.…”