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 modeling of solar magneto-convection, considering the fact that solar plasma contains a significant fraction of neutral gas. To that end, we performed simulations of the same physical volume of the Sun, namely 5.76×5.76×2.3 Mm3 (with 1.4 Mm being above the optical surface), at three different resolutions: 20×20×14, 10×10×7 and 5×5×3.5 km3. At all three resolutions, we compared the time series of simulations with and without ambipolar diffusion, the main non-ideal heating mechanism due to neutrals. We also compared simulations with three different magnetizations: (1) a case of a small-scale dynamo; (2) an initially implanted vertical magnetic field of 50 G; and (3) an initially implanted vertical field of 200 G, though not all are available at each resolution. We find that the average magnetization of the simulations increases with improving resolution, both in the dynamo and in the unipolar cases, and 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 actually be numerically resolved with the grid scale of the highest resolution simulations as the ones reported here. We considered two ways of evaluating where the ambipolar scales are numerically resolved. On the one hand, we provide a method to evaluate the numerical diffusion of the simulations and compare it to the physical ambipolar diffusion. On the other hand, we compare an order of magnitude evaluation of spatial scales given by the ambipolar diffusion to our grid resolution. At those resolved locations, we compared the average temperature in the simulations with and without ambipolar diffusion, and we conclude that the plasma is on average about 600 K hotter after 1200 s of simulation time when the ambipolar diffusion is included. The amount of temperature enhancement increases with the resolution and with time, and there are no signs of saturation at our best horizontal resolution of 5 km.