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Write $S=U_z$ and $N^2=d\bar{b}/dz$. The gradient Richardson number and Flux Richardson number areThe first compares the restoring buoyancy gradient with mean shear; the second is the fraction of shear production consumed by stable buoyancy work. With this convention, both are positive for stable mixing and negative for unstable stratification. In local steady equilibrium, $ϵ=P+B_0=P(1-{\mathrm{Ri}}_f)$, so a positive dissipation requires ${\mathrm{Ri}}_f<1$. The inviscid Miles–Howard theorem concerning ${\mathrm{Ri}}_g\ge 1/4$ is a linear stability criterion, not a universal equality for turbulent fluxes.

Use the gradient-diffusion closures $-\overline{u^{\prime }{w}^{\prime }}=K_{m}S$ and $B_0=-K_b{N}^2$. The turbulent Prandtl number is ${\mathrm{Pr}}_t=K_m/K_b$, givingUnder the ideal Reynolds analogy $K_m=K_b$, the two Richardson numbers coincide. It is an additional closure, not an automatic consequence of the definitions. If the buoyancy-gradient similarity function is defined by $N^2=-B_0{\varphi }_h/(\kappa z{u}_{\ast })$, then

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.

For very stable continuously turbulent conditions, $z\gg {\ell }_O>0$, local eddies lose leading dependence on their distance from the wall. In very stable surface-layer similarity, the dimensionless functions must therefore scale as ${\varphi }_m\sim c_m\zeta$ and ${\varphi }_h\sim c_h\zeta$, with positive closure constants. Using ${\ell }_O=u_{\ast }^3/(\kappa |B_0|)$ givesThus the mean velocity is approximately linear in height and the real buoyancy frequency is approximately constant, $N\sim \sqrt{c_h}|B_0|/u_{\ast }^2$. Under the Reynolds analogy, $c_h=c_m$.

The local turbulent kinetic energy balance givesPositive dissipation requires $c_m>1$. The turbulent energy-turnover length is of order $u_{\ast }^3/|B_0|$, hence of order the Obukhov length, independent of observation height. A related buoyancy-limited length is $u_{\ast }/N$. The Ozmidov length follows by equating an eddy turnover time ${\ell }^{2/3}{ϵ}^{-1/3}$ with $N^{-1}$:These are turbulent turnover or buoyancy-transition scales. If “the scale at which dissipation occurs” means the actual viscous cutoff, molecular viscosity is also required: the Kolmogorov microscales giveWithout $\nu$, the viscous length cannot be fixed by the supplied stress and flux alone. Separating it from the turbulent turnover length avoids confusing an energy-budget estimate with the molecular cutoff.

Finally,They become height-independent constants. The commonly used continuation $c_m=c_h=5$ gives ${\mathrm{Ri}}_f={\mathrm{Ri}}_g\simeq 0.2$ and $ϵ\simeq 4|B_0|$. This numerical value is a closure choice, not a deduction from units; setting $c_m=1$ would incorrectly leave zero dissipation. Extremely stable intermittent or collapsed turbulence need not satisfy this local-equilibrium model. The relation between the Obukhov and buoyancy-dissipation lengths in the local regime is also analyzed in Grachev and colleagues' similarity study.