mean opacity. $$\label{fsgs} \vec{F}_{\rm sgs}=-\frac{\mu}{\sigma_{\rm sgs}}C_{\rm
p}(\nabla T-\nabla_{\rm a}\frac{T}{p}\nabla p)$$ stands for the diffusive energy transport by the SGS turbulence. $\sigma_{\rm sgs}$ is the effective Prandtl number for the SGS model (hereafter, $\sigma_{\rm sgs}$ is called SGS Prandtl number), $C_{\rm p}$ is the specific heat of the gas at constant pressure, and $\nabla_{\rm a}$ is the adiabatic gradient.

The SGS Prandtl number $\sigma_{\rm sgs}$ is taken to be $1/3$; the Deardorff number $c_{\rm \mu}$ in Eq. (\[mu\]) is fixed at $0.2$. Near the solid boundaries, the SGS viscosity is enhanced to absorb irrelevant acoustic waves. In the stable region, the turbulent diffusive flux (\[fsgs\]) is taken to be zero. The equations (\[ns1\])$\sim$(\[ns3\]) are solved in the Cartesian coordinates $(x,y,z)$, where $z=(r-r_{\rm b})/d$ for $r \rm{(radius)}\in [r_{\rm
b},r_{\rm b}+d]$. $r_{\rm b}$ and $d$ are the radial location of the bottom and the height of the computational domain, respectively (see Section 2.3). The computed domain is a rectangular box with periodic boundaries on the sides and solid boundaries at the top ($z=1$) and bottom ($z=0$). The aspect ratio of the domain (width/height) is $1.5$. The total energy flux $F_{\rm b}$ is fixed at the bottom; at the top the entropy is fixed. 