Modeling the thermal conduction in the solar atmosphere with the code MANCHA3D

Abstract

Context. Thermal conductivity is one of the important mechanisms of heat transfer in the solar corona. In the limit of strongly magnetized plasma, it is typically modeled by Spitzer's expression where the heat flux is aligned with the magnetic field. Aims. This paper describes the implementation of the heat conduction into the code MANCHA3D with an aim of extending singlefluid MHD simulations from the upper convection zone into the solar corona. Methods. Two different schemes to model heat conduction are implemented: (1) a standard scheme where a parabolic term is added to the energy equation, and (2) a scheme where the hyperbolic heat flux equation is solved. Results. The first scheme limits the time step due to the explicit integration of a parabolic term, which makes the simulations computationally expensive. The second scheme solves the limitations on the time step by artificially limiting the heat conduction speed to computationally manageable values. The validation of both schemes is carried out with standard tests in one, two, and three spatial dimensions. Furthermore, we implement the model for heat flux derived by Braginskii (1965) in its most general form, when the expression for the heat flux depends on the ratio of the collisional to cyclotron frequencies of the plasma, and, therefore on the magnetic field strength. Additionally, our implementation takes into account the heat conduction in parallel, perpendicular, and transverse directions, and provides the contributions from ions and electrons separately. The model recovers Spitzer's expression for parallel thermal conductivity in the strongly magnetized limit but also transitions smoothly between field-aligned conductivity and isotropic conductivity for regions with a low or null magnetic field. We describe the details of the implementation of Braginskii's thermal conductivity using a combination of the first scheme for the perpendicular and transverse directions and the second scheme for the parallel component. We estimate thermal conductivities in a quiet-Sun model. In this model, we find that the perpendicular and transverse components for electrons and ions and the parallel component for ions might have some significance below the transition region. Above the transition region only the parallel component for ions might be important. Finally, we present a two-dimensional test for heat conduction using realistic values of the solar atmosphere where we prove the robustness of the two schemes implemented and show that our adaptation of the hyperbolic treatment offers a great advantage over the computational cost of the simulations.