Overshooting in simulations of compressible convection

Abstract

Context. Convective motions overshooting to regions that are formally convectively stable cause extended mixing. Aims. To determine the scaling of overshooting depth (dos) at the base of the convection zone as a function of imposed energy flux (Fn) 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 dos ∝ F 0.08 n for low Fn. A fixed step-profile of the heat conductivity at the bottom of the convection zone leads to a somewhat steeper dependency with dos ∝ F 0.12 n 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 Fn 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 substantial underestimation of overshooting depth. Conclusions. Extrapolating from the results obtained with smooth heat conductivity profiles, which are the most realistic of the setups considered, suggest that the overshooting depth for the solar energy flux is of the order of 20 per cent of the pressure scale height at the base of the convection zone which is two to four times higher than the estimates from helioseismology. The current simulations do not include rotation or magnetic fields which are known to reduce the convective overshooting.

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