Overshooting in simulations of compressible convection

Context. Convective motions that overshoot into regions that are formally convectively stable cause extended mixing. Aims. We aim to determine the scaling of the overshooting depth (d(os)) at the base of the convection zone as a function of imposed energy flux (F-n) and to estimate the extent of overshooting at the base of the solar convection zone. Methods. Three-dimensional Cartesian simulations of hydrodynamic compressible non-rotating convection with unstable and stable layers were used. The simulations used either a fixed heat conduction profile or a temperature-and density-dependent formulation based on Kramers opacity law. The simulations covered a range of almost four orders of magnitude in the imposed flux, and the sub-grid scale diffusivities were varied so as to maintain approximately constant supercriticality at each flux. Results. A smooth heat conduction profile (either fixed or through Kramers opacity law) leads to a relatively shallow power law with d(os) proportional to F-n(0.08) for low F-n. A fixed step-profile of the heat conductivity at the bottom of the convection zone leads to a somewhat steeper dependency on d(os) proportional to F-n(0.12) in the same regime. Experiments with and without subgrid-scale entropy diffusion revealed a strong dependence on the effective Prandtl number, which is likely to explain the steep power laws as a function of F-n reported in the literature. Furthermore, changing the heat conductivity artificially in the radiative and overshoot layers to speed up thermal saturation is shown to lead to a substantial underestimation of the overshooting depth. Conclusions. Extrapolating from the results obtained with smooth heat conductivity profiles, which are the most realistic set-up we considered, suggest that the overshooting depth for the solar energy flux is about 20% of the pressure scale height at the base of the convection zone. This is two to four times higher than the estimates from helioseismology. However, the current simulations do not include rotation or magnetic fields, which are known to reduce convective overshooting.