= Shallow horizontal convection with an insulated bottom
{title2=$\psi=-\operatorname{Ra}\epsilon^4y(1-y)^2(2y+1)/48$}
For downward depth $z=\epsilon y$, 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, $\psi_{zzzz}=-\operatorname{Ra}$ 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.
Back to article page
