 Solution

For stationary horizontally homogeneous means, incompressibility gives ${\bar{w}}_z=0$ and the impermeable boundary makes $\bar{w}=0$ throughout. With no imposed horizontal mean pressure gradient, the balances reduce toThe last equation includes the vertical Reynolds stress; omitting it requires an additional small-normal-stress approximation. The total vertical buoyancy flux is $B_0$ and the horizontal momentum flux is a signed constant $J_M$ with $|J_M|=U_{\ast }^2$. These are fluxes per unit reference density.

The local down-gradient closure defines eddy viscosity and eddy diffusivity byA parcel moving upwards typically retains the smaller momentum or scalar value from its previous height in an increasing mean profile; its fluctuation then correlates negatively with upward motion. This motivates positive coefficients, but is a closure assumption, not a universal theorem about turbulent or convective fluxes. The mean gradients obeyFor positive applied traction on a fluid above a bottom boundary, $J_M=+U_{\ast }^2$ and the mean velocity decreases away from the forcing boundary. The usual atmospheric convention specifies the stress absorbed by the boundary and has the opposite sign of $J_M$; scalar and Richardson-number results are unchanged. At large Reynolds number and Péclet number, outside the molecular wall sublayers, $K_M\gg \nu$ and $K_B\gg \kappa$, so molecular transport can be neglected in these flux-gradient relations. Large bulk numbers do not justify doing this exactly at a smooth solid wall.