(e.g., right panel in Figure \[fig2\]). As has been studied in detail since 1990’s (e.g., @herant94 [@burrows95; @jankamueller96]), buoyancy-driven convection supported by turbulence (e.g., @murphy13) transports heat radially outward, leading to a more extended entropy profile in the 2D (green line in the panel) and 3D (red line) model compared to the 1D model (blue line) (see also @hanke12).

Bottom panel of Figure 5 compares the turbulent energy spectra of the anisotropic velocity ([@takiwaki12]) as a function of wavenumber ($k$) between our 2D (green line) and 3D (red line) model, respectively. The green and red line crosses at around $k_{\rm cross}$ = 0.02/km (corresponding to $\sim$ 50 km in spacial scale), above which the amplitude for the 3D model (red line) dominates over that for the 2D model (green), below which the amplitude for the 2D model does for the 3D model. This is qualitatively consistent with the previous results using the light-bulb method (e.g., @hanke12 [@burrows12; @dolence12; @couch13; @fernandez13]. By comparing the 2D and 3D curve with spectral slopes labeled by the corresponding exponent (-5/3:dotted blue line, -3:dotted black line), the power-low dependence ($\propto k^{-5/3}$) approximately holds for $k_{\rm int} \lesssim 0.02$ /km in the 2D model (green line) 