  (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.
