 Advection correction in shallow horizontal convection

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.