Shallow horizontal convection with an insulated bottom ($\psi =-\mathrm{Ra}{ϵ}^{4}y(1-y{)}^2(2y+1)/48$)

For downward depth $z=ϵy$, take $(u,w)=({\psi }_z,-{\psi }_x)$, surface temperature $T=x$, a flat impermeable stress-free upper boundary and a no-slip insulated bottom. In the steady infinite-Prandtl number thin-layer interior, ${\psi }_{zzzz}=-\mathrm{Ra}$ with $\psi ={\psi }_{zz}=0$ at the surface and $\psi ={\psi }_z=0$ at the bottom. The displayed profile follows. Sidewall end layers close the circulation; the interior profile must not be imposed unchanged at those walls.

The interior first temperature correction obeys ${\Theta }_{zz}=u={\psi }_z$, $\Theta (0)=0$ and ${\Theta }_z(ϵ)=0$. Hence ${\Theta }_z=\psi$, which integrates to the displayed polynomial. Its contribution to depth-integrated horizontal advective heat flux is $\int u\Theta dz=-\int {\psi }^{2}dz=-19{\mathrm{Ra}}^2{ϵ}^9/1451520$. The leading conductive flux is $-ϵ$. These statements apply away from the sidewall end layers at fixed Rayleigh number as the layer becomes shallow.