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

TL;DR

This paper demonstrates that in rapidly rotating convection, fixed heat flux and fixed temperature boundary conditions lead to asymptotically equivalent interior convection dynamics, despite different boundary layer structures.

Contribution

It shows that the leading order interior convection is unaffected by boundary condition differences due to boundary layer effects, establishing asymptotic equivalence.

Findings

Boundary layers include Ekman and thermal wind layers.

Interior convection is asymptotically unaffected by boundary conditions.

Leading order equations are equivalent after rescaling thermal variables.

Abstract

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.