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Large eddy simulations (LES) and implicit LES are wise and affordable alternatives to the unfeasible direct numerical simulations (DNS) of turbulent flows at high Reynolds numbers (Re). However, for systems with few observational constraints, it is a formidable challenge to determine whether or not the physics of the system is properly captured by these strategies. Here we address this problem through an analysis of numerical convergence of ILES of turbulent convection in 2D with 2 < Re < 10⁴. The thermodynamic atmosphere resembles the solar interior, including a fraction of the radiative zone and the convection zone. Our results indicate that the large-scale properties of convection do not change significantly with the increase of resolution and even a resolution of 128² grid points is sufficient to capture these dynamics. This happens because the relevant scales of the system are much larger than the Kolmogorov scales. Therefore, the small-scale structures dissipate on a time scale which depends on the turbulence and not on the effective dissipation corresponding to the resolution. Special attention is needed in regions with a sharp density contrast and the resolution cannot resolve small structures. This may lead to significant changes in the integral profiles of rms quantities. Most importantly, if the system includes a convective stable layer interfacing with the convection zone, the correct resolution of interacting internal gravity waves requires low viscosities, i.e., high resolution. Our simulations with resolution ≥ 1024² mesh points show the self-consistent emergence of solar QBO as a consequence of these waves.

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