Scaling of high-Rayleigh-number convection based on internal convective boundary layer

We propose a phenomenological model for thermal convection at high Rayleigh numbers. It hypothesizes existence of a high-Reynolds-number turbulent boundary layer near each horizontal plate at such Rayleigh numbers, as done by Kraichnan. However, the boundary layer is shown for the first time to be convective. The convective logarithmic friction law of Tong and Ding (2020) is used to relate the large-scale velocity to the induced friction velocity. The predicted scaling relations of the Nusselt (Nu) and Reynolds numbers (Re) with the Rayleigh number (Ra) do not have power law forms. However, the predicted scaling is close to Ra$^{1/3}$ and Ra$^{4/9}$ respectively for Ra $\sim 10^{12}$ to $10^{16}$, in agreement with direct numerical simulation (DNS) and experimental results up to Ra $\sim 10^{15}$. The model predicts deviations from the apparent Ra$^{1/3}$ scaling beyond Ra$\sim 10^{17}$, slowly approaching the Ra$^{1/2}$ scaling. Therefore, contrary to previous models, the transition from the apparent Ra$^{1/3}$ scaling to the Ra$^{1/2}$ scaling occurs within the same regime of convective turbulent boundary layer, with no changes in the leading-order physics. The model provides conservative support to the Ra$^{1/2}$ scaling. It predicts that data at Ra $\sim 10^{17}$−$10^{18}$ is likely required to definitely validate the scaling.