Investigating numerical stability by scaling heat conduction in a 1D hydrodynamic model of the solar atmosphere

¹ Institute of Physics, University of Graz, Universitätspl. 5, 8010 Graz, Austria
² Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, 8042 Graz, Austria

^⋆ Corresponding author; vartika.pandey@uni-graz.at

Received: 28 March 2024
Accepted: 8 December 2024

Abstract

Context. Numerical models of the solar atmosphere are widely used in solar research and provide insights into unsolved problems such as the heating of coronal loops. A prerequisite for such simulations is an initial condition for the plasma temperature and density. Many explicit numerical schemes employ high-order derivatives that require some diffusion, for example isotropic diffusion, for each independent variable to maintain numerical stability. Otherwise, significant numerical inaccuracies and subsequent wiggles will occur and grow at steep temperature gradients in the solar transition region.

Aims. We tested how to adapt the isotropic heat conduction to the grid resolution so that the model is capable of resolving varying temperature gradients. Our ultimate goal is to construct an atmospheric stratification that can serve as an initial condition for multi-dimensional models.

Methods. Our temperature stratification spans from the solar interior to the outer corona. From that, we computed the hydrostatic density stratification. Since numerical and analytical derivatives are not identical, the model needs to settle to a numerical equilibrium to fit all model parameters, such as mass diffusion and radiative losses. To compensate for energy losses in the corona, we implemented an artificial heating function that mimics the expected heat input from the 3D field-line braiding mechanism.

Results. Our heating function maintains and stabilises the obtained coronal temperature stratification. However, the diffusivity parameters need to be adapted to the grid spacing. Unexpectedly, we find that higher grid resolutions may need larger diffusivities – contrary to the common understanding that high-resolution models are automatically more realistic and would need less diffusivity.

Conclusions. Smaller grid spacing causes larger temperature gradients in the solar transition region and hence a greater potential for numerical problems. We conclude that isotropic heat conduction is an efficient remedy when using explicit schemes with high-order numerical derivatives.