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The influence of fixed temperature and fixed heat flux thermal boundaryconditions on rapidly rotating convection in the plane layer geometry isinvestigated for the case of stress-free mechanical boundary conditions. It isshown that whereas the leading order system satisfies fixed temperatureboundary conditions implicitly, a double boundary layer structure is necessaryto satisfy the fixed heat flux thermal boundary conditions. The boundary layersconsist of a classical Ekman layer adjacent to the solid boundaries that adjustviscous stresses to zero, and a layer in thermal wind balance just outside theEkman layers adjusts the temperature such that the fixed heat flux thermalboundary conditions are satisfied. The influence of these boundary layers onthe interior geostrophically balanced convection is shown to be asymptoticallyweak, however. Upon defining a simple rescaling of the thermal variables, theleading order reduced system of governing equations are therefore equivalentfor both boundary conditions. These results imply that any horizontal thermalvariation along the boundaries that varies on the scale of the convection hasno leading order influence on the interior convection.

The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection