Accounting for surface temperature variations in Rayleigh-B\'{e}nard convection

Turbulent Rayleigh-B\'{e}nard convection is often modelled with a constant surface temperature. However, the surface temperature of many geophysical systems, such as lakes, is coupled to the atmospheric forcing. In this paper, we account for this dynamic surface temperature through an additional parameter $\beta$. Using an appropriately defined dynamical Rayleigh number $\RaD$, we recover many of the results from the standard Rayleigh-B\'{e}nard model. We hope that this work will simplify the application of Rayleigh-B\'{e}nard theory in geophysical contexts, such as lakes.