= Advection correction in shallow horizontal convection
{title2=$T-x=-\operatorname{Ra}\epsilon^5(2y^5/5-3y^4/4+y^2/2)/48$}
The interior first temperature correction obeys $\Theta_{zz}=u=\psi_z$, $\Theta(0)=0$ and $\Theta_z(\epsilon)=0$. Hence $\Theta_z=\psi$, which integrates to the displayed polynomial. Its contribution to depth-integrated horizontal advective <heat flux> is $\int u\Theta\,dz=-\int\psi^2dz=-19\operatorname{Ra}^2\epsilon^9/1451520$. The leading conductive flux is $-\epsilon$. These statements apply away from the sidewall end layers at fixed Rayleigh number as the layer becomes shallow.
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