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For a positive approximately constant vertical turbulent buoyancy flux, dimensional turnover scaling gives $K_b\sim B_0^{1/3}{z}^{4/3}$ and $N^2\sim -B_0^{2/3}{z}^{-4/3}$. The negative squared frequency describes unstable overturning, not a real restoring oscillation. The leading dissipation is height-independent, unlike neutral wall turbulence.

The signed wall-turbulence length formed from stress velocity and the vertical turbulent buoyancy flux. With upward buoyancy transport positive, stable cooling has $B_0<0$ and ${\ell }_O>0$; heating from below gives ${\ell }_O<0$. The magnitude marks the transition between shear-dominated and buoyancy-dominated scaling.

For stable shear-driven turbulence, let $P$ be the rate of production of turbulent kinetic energy by mean shear and $B_m>0$ the rate consumed by buoyancy work. The Flux Richardson number isWith an upward coordinate and vertical turbulent buoyancy flux $\overline{w^{\prime }{b}^{\prime }}$, $B_m=-{\rho }_l\int \overline{w^{\prime }{b}^{\prime }}d\mathcal{V}$, while shear production is $P=-{\rho }_l\int \overline{u^{\prime }{w}^{\prime }}{\partial }_{z}Ud\mathcal{V}$. In a locally steady balance without net turbulent-energy transport or storage, $P=B_m+\mathcal{D}$, so $0\le {\mathrm{Ri}}_f\le 1$ and the dissipation-normalized mixing coefficient is $B_m/\mathcal{D}={\mathrm{Ri}}_f/(1-{\mathrm{Ri}}_f)$. A global ratio $C_W={\dot{E}}_P/P_I$ need not equal this local production-normalized Flux Richardson number when energy transport, storage, or bulk acceleration is important.

The interfacial Richardson number in this tank isIn model S,Thus the relative importance of interface stratification changes continually. If mixing efficiency depends on interfacial Richardson number, constant $C_W$ is not generally self-consistent over substantial deepening; it can at best describe a regime where that dependence is weak. In model K,A local efficiency depending only on this ratio can therefore remain constant, provided the fixed interface thickness and the other empirical coefficients remain appropriate.

The energetic reason to expect intermediate behavior is that entrainment must both lift and homogenize dense fluid and mobilize it. The whole upper-layer kinetic energy budget has the form $P_{\mathrm{lid}}=\dot{K}+{\dot{E}}_P+\mathcal{D}$, apart from additional boundary losses. Keeping the characteristic speed constant makes $K\propto h$ and requires a continuing power supply to accelerate new fluid, together with dissipation in an ever larger layer. Keeping $K$ constant instead forces the speed to decline as $h^{-1/2}$. A finite lid forcing plausibly produces an initial regime near the former and a deeper regime near the latter, rather than indefinite constant-speed entrainment.

One can make this expected bracket quantitative after matching the models' initial interface speed $u_0$. Suppose the actual speed lies between $u_0\sqrt{h_0/h}$ and $u_0$, and the same local coefficients are used. Set $D_0=2C_W{c}_D{u}_0^3/B$. ThenIntegrating these inequalities givesEquivalently, if $K\propto h^q$ with $0\le q\le 1$, the local power closure yields a late-time depth exponent $2/(5-3q)$ between $2/5$ and $1$. This is the intended energetic interpolation between K and S, not a universal bound following from the printed assumptions alone. Different efficiencies or unmatched initial coefficients can invalidate pointwise ordering; quantitative prediction requires the lid-to-interface power transfer and the actual energy budget.

Let $B_0=\overline{w^{\prime }{b}^{\prime }}$ be the vertical turbulent buoyancy flux, positive for heating from below. Its dimensions are length squared per time cubed. The stress velocity and height therefore admit a new dimensionless group $B_{0}z/u_{\ast }^3$. Define the signed Obukhov lengthStable cooling has $B_0<0,{\ell }_O>0$; unstable heating has $B_0>0,{\ell }_O<0$. Monin-Obukhov similarity theory expresses the mean shear asThe similarity hypothesis assumes approximately steady, horizontally homogeneous, constant-stress and constant-flux conditions, with the observation level outside the inner sublayer but below the outer-layer scales. Dimensional analysis fixes the arguments and prefactor, not the entire function ${\varphi }_m$. A measured or modeled turbulence closure supplies its form.

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.