  The influence of fixed temperature and fixed heat flux thermal boundary
conditions on rapidly rotating convection in the plane layer geometry is
investigated for the case of stress-free mechanical boundary conditions. It is
shown that whereas the leading order system satisfies fixed temperature
boundary conditions implicitly, a double boundary layer structure is necessary
to satisfy the fixed heat flux thermal boundary conditions. The boundary layers
consist of a classical Ekman layer adjacent to the solid boundaries that adjust
viscous stresses to zero, and a layer in thermal wind balance just outside the
Ekman layers adjusts the temperature such that the fixed heat flux thermal
boundary conditions are satisfied. The influence of these boundary layers on
the interior geostrophically balanced convection is shown to be asymptotically
weak, however. Upon defining a simple rescaling of the thermal variables, the
leading order reduced system of governing equations are therefore equivalent
for both boundary conditions. These results imply that any horizontal thermal
variation along the boundaries that varies on the scale of the convection has
no leading order influence on the interior convection.
