Fig. \[flx\], the first vertical dashed line from right is $1$ PSHs away from the top boundary. $H_{\rm p}=-d r/d\ln{p}$ sometimes does not increase monotonously with depth as the gravity $g$ is variable and the EOS has a complex form. In some plots (e.g., Fig. \[flx\]), the overshooting zone $\Delta_c$, defined in Section \[sec:ovsh\], is indicated by a double-headed arrow.

Stationary Solutions
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![Velocity field projected onto the $x$–$z$ plane at $y=0.75$. Upper panel: Case I2; lower panel: Case M1. The regions confined by solid lines indicate the overshooting zones, i.e., $\Delta_c$ defined in Section \[sec:ovsh\].[]{data-label="vfs"}](./figures/vfyz-im.eps "fig:"){width="8.2cm"} ![Velocity field projected onto the $x$–$z$ plane at $y=0.75$. Upper panel: Case I2; lower panel: Case M1. The regions confined by solid lines indicate the overshooting zones, i.e., $\Delta_c$ defined in Section \[sec:ovsh\].[]{data-label="vfs"}](./figures/vfyz-ms.eps "fig:"){width="8.2cm"}

Figure \[flx\] shows the distributions of averaged vertical fluxes for Case I2 and Case M1, from which we can see that the systems are almost completely relaxed. The different energy fluxes are defined as follows: $$F_{\rm e}=\overline{v_{z}(e+p)}$$ is the enthalpy flux, $$F_{\rm k}=\frac{1}{2}\overline{v_{z}\rho v^2}$$ is the flux of kinetic energy, $$F_{\rm v}=\overline{v_{i}\Sigma_{iz}}$$ is the small-scale eddy viscosity flux, where the Einstein summation convention is used. The diffusive flux $F_{\rm 