Rotating Rayleigh–Bénard convection with fixed-flux thermal boundary conditions

We derive the asymptotic solution for the onset of steady, linear, Boussinesq convection in a rapidly rotating system with stress-free, fixed-flux boundary conditions. While the fixed-temperature (FT) case is attainable analytically with relative ease, the fixed-flux (FF) configuration presents greater complexity. However, in the rapidly rotating limit, the leading-order interior solution remains unaffected by the choice of thermal boundary conditions. We exploit this property by employing an asymptotic approach to characterise the differences between the FT and FF systems. Specifically, this involves constructing a composite boundary layer structure comprising an Ekman layer of thickness $ {\textit{Ta}}^{-1/4}$ , where $ \textit{Ta}$ is the Taylor number ( $ \textit{Ta} \gg 1$ for rapid rotation), and a thermal boundary layer of thickness $ {\textit{Ta}}^{-1/6}$ , to accommodate the FF boundary condition. To capture both scales systematically, we introduce the small parameter ${\varepsilon } = {\textit{Ta}}^{-1/12}$ , representing the ratio between the two boundary layer thicknesses, and use it to guide the asymptotic expansion. The asymptotic corrections capturing the differences between the two systems are combined with the FT system to construct the corresponding solution for the FF system. We find an asymptotic correction of ${\mathcal{O}} ( {\textit{Ta}}^{-1/2} )$ to the critical Rayleigh number, corresponding wavenumber, vertical velocity and temperature, along with a correction of ${\mathcal{O}} ( {\textit{Ta}}^{-1/6} )$ to the vertical vorticity.