Asymptotic Solutions of the 2D Oberbeck--Boussinesq Equations in the Large Rayleigh Number, Moderate Prandtl Number Limit

Boussinesq thermal convection in a horizontal layer between isothermal stress-free boundaries is the archetypal convection problem. In both natural and technological applications, the Rayleigh number $Ra$ generally exceeds the threshold for linear instability of the conduction state by orders of magnitude, so the high-$Ra$ limit of the governing equations is of particular interest. It is therefore remarkable that the structure of steady-state convection cells has not yet been established, except in the further limiting case of infinite Prandtl number ($Pr$). Here, we rectify this situation by presenting the first large-$Ra$ asymptotic analysis of the classical Rayleigh--B\'{e}nard convection problem with $Pr=\mathit{O}(1)$. We derive both details of the flow and a corresponding bulk heat transport coefficient, as a function of the cell aspect ratio. Predictions of our asymptotic theory are corroborated using full pseudospectral numerical simulations.