Transition region adaptive conduction (TRAC) in multidimensional magnetohydrodynamic simulations

In solar physics, a severe numerical challenge for modern simulations is properly representing a transition region between the million-degree hot corona and a much cooler plasma of about 10000 K (e.g., the upper chromosphere or a prominence). In previous 1D hydrodynamic simulations, the transition region adaptive conduction (TRAC) method has been proven to capture aspects better that are related to mass evaporation and energy exchange. We aim to extend this method to fully multidimensional magnetohydrodynamic (MHD) settings, as required for any realistic application in the solar atmosphere. Because modern MHD simulation tools efficiently exploit parallel supercomputers and can handle automated grid refinement, we design strategies for any-dimensional block grid-adaptive MHD simulations. We propose two different strategies and demonstrate their working with our open-source MPI-AMRVAC code. We benchmark both strategies on 2D prominence formation based on the evaporation--condensation scenario, where chromospheric plasma is evaporated through the transition region and then is collected and ultimately condenses in the corona. A field-line-based TRACL method and a block-based TRACB method are introduced and compared in block grid-adaptive 2D MHD simulations. Both methods yield similar results and are shown to satisfactorily correct the underestimated chromospheric evaporation, which comes from a poor spatial resolution in the transition region. Because fully resolving the transition region in multidimensional MHD settings is virtually impossible, TRACB or TRACL methods will be needed in any 2D or 3D simulations involving transition region physics.