Horizontal convection

Horizontal convection is buoyancy-driven circulation forced by lateral variations of temperature along one horizontal boundary. Hot and cold surface regions drive a circulation even though all thermal forcing is applied at the same height. An insulated bottom and sides require zero net heat exchange through the remaining surface in steady state, but permit heat to enter its warmer region and leave its colder region.

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.