Overshooting in simulations of compressible convection

Abstract: (abridged) Context: Convective motions overshooting to regions that are formally convectively stable cause extended mixing. Aims: To determine the scaling of overshooting depth ($d_{\rm os}$) at the base of the convection zone as a function of imposed energy flux ($\mathscr{F}{\rm n}$) and to estimate the extent of overshooting at the base of the solar convection zone. Methods: Three-dimensional Cartesian simulations of compressible non-rotating convection with unstable and stable layers are used. The simulations use either a fixed heat conduction profile or a temperature and density dependent formulation based on Kramers opacity law. The simulations cover a range of almost four orders of magnitude in the imposed flux. Results: A smooth heat conduction profile (either fixed or through Kramers opacity law) leads to a relatively shallow power law with $d{\rm os}\propto \mathscr{F}{\rm n}^{0.08}$ for low $\mathscr{F}{\rm n}$. A fixed step-profile of the heat conductivity at the bottom of the convection zone leads to a somewhat steeper dependency with $d_{\rm os}\propto \mathscr{F}{\rm n}^{0.12}$. 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 $\mathscr{F}{\rm n}$ reported in the literature. Furthermore, changing the heat conductivity artificially below the convection zone is shown to lead to substantial underestimation of overshooting depth. Conclusions: Extrapolating from the results obtained with smooth heat conductivity profiles suggest that the overshooting depth for the solar flux is of the order of $0.2$ pressure scale heights at the base of the convection zone which is two to four times higher than estimates from helioseismology.