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, which is shown 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 ($Re$) on the Rayleigh number ($Ra$) do not have a power law form, each being a single function, suggesting a single flow regime with no transition, instead of multiple regimes. However, the predicted $Nu$ and $Re$ scaling is close to $Ra^{1/3}$ and $Ra^{4/9}$ for $Ra \sim 10^9$ to $10^{17}$, consistent with direct numerical simulation (DNS) results up to $Ra\sim 10^{15}$. For $Ra<10^9$, the model also correctly predicts the deviations observed in DNS from the above power law scaling. The model predicts deviations from $Ra^{1/3}$ beyond $Ra \sim 10^{17}$ with the local scaling exponent approaching $1/2$, which would likely require data at $Ra>10^{18}$ to verify.