Convergence study of ambipolar diffusion in realistic simulations of magneto-convection

The aim of this paper is to improve our understanding of the heating mechanisms of the solar chromosphere via realistic three-dimensional (3D) modeling of solar magneto-convection, considering the fact that solar plasma contains a significant fraction of neutral gas. For that we performed simulations of the same physically volume of the Sun, namely 5.76x5.76x2.3 Mm^3 (with 1.4 Mm being above the optical surface), at three different resolutions: 20x20x14, 10x10x7 and 5x5x3.5 km^3. At all three resolutions we compare the time series of simulations with/without ambipolar diffusion, as the main non-ideal heating mechanism due to neutrals. We also compare simulations with three different magnetizations: (1) case of a small-scale dynamo; (2) an initially implanted vertical magnetic field of 50 G; (3) an initially implanted vertical field of 200 G, though not all of them are available at all resolutions. We obtain that the average magnetization of the simulations increases with improving resolution. So does the average magnetic Poynting flux, meaning that there is more magnetic energy in the simulation box at higher resolutions. Ambipolar diffusion operates at relatively large scales, which can be actually numerically resolved with the grid scale of the highest resolution simulations as the ones reported here. We consider two ways of evaluating where the ambipolar scales are numerically resolved: (i) a method to evaluate the numerical diffusion of the simulations and compare it to the physical ambipolar diffusion; (ii) an order of magnitude comparison of spatial scales given by the ambipolar diffusion to our grid resolution. At the resolved locations we compare the average temperature in the simulations with/without ambipolar diffusion, and conclude that the plasma is on average about 600 K hotter after 1200 sec of simulation time when the ambipolar diffusion is included.