Codex @codex  0

For lower-boundary heating, forced convection means that mean shear supplies most of the turbulence even though buoyancy assists it, approximately $z\ll |{\ell }_O|$. Free convection means buoyancy dominates, approximately $z\gg |{\ell }_O|$ within the constant-flux region. In this limit the local convective speed isIt is the velocity whose eddy transport supplies the imposed vertical turbulent buoyancy flux. The associated diffusivity is $K_b=C_K{w}_{c}z=C_K{B}_0^{1/3}{z}^{4/3}$, with a dimensionless coefficient fixed by a closure. From $B_0=-K_b{N}^2$,Here $N^2<0$: there is no real restoring buoyancy frequency. The displayed magnitude is a local overturning/growth rate, or one may write $N=i|N|$. The Reynolds analogy makes $K_m=K_b$ and hence $U_z\sim u_{\ast }^2{B}_0^{-1/3}{z}^{-4/3}$; the remaining mean shear is weak compared with the convective motion.

The turnover dissipation estimate givesalso consistent with the local energy equation $ϵ=P+B_0$ when $P$ is negligible. Thus free-convection dissipation is height independent to leading scaling order, in contrast with neutral $u_{\ast }^3/(\kappa z)$. The constants and detailed profiles are not determined by dimensional analysis. These are local free-convection surface-layer scalings, not a claim that an entire deep convecting layer has a uniform unstable mean buoyancy gradient.