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Define the friction velocity by $u_{\ast }=(|{\tau }_w|/\rho {)}^{1/2}$, where ${\tau }_w$ is the total tangential wall stress. It is a stress-derived velocity scale, not the local mean velocity. In the overlap region, $y$ is small relative to the channel half-width, so the total stress is approximately its wall value. Molecular stress is small there, giving $-\overline{u^{\prime }{v}^{\prime }}\simeq u_{\ast }^2$.

Combining this Reynolds stress with the logarithmic shear gives $\mathcal{P}\simeq u_{\ast }^3/(\kappa y)$. For locally equilibrated wall turbulence, energy transfer to small scales occurs over the local eddy turnover time and is balanced by molecular dissipation; energy storage and the net energy-transport divergence are small in the leading overlap-layer balance. ThusThis is the local turbulent kinetic energy balance. It is a modeling approximation rather than a general identity for all wall positions. The constant-energy solution derived next is consistent with it: its modeled $k$ diffusion term is exactly zero, while the $ϵ$ diffusion term remains essential.

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.

Kinetic energy per unit mass in velocity fluctuations. Mean shear, buoyancy and transport supply or redistribute it, while molecular viscosity dissipates it. Its local budget motivates the local turbulent kinetic energy balance when storage and transport are negligible.

The covariance of vertical velocity and buoyancy fluctuations has dimensions length squared per time cubed. It supplies buoyancy production in the local turbulent kinetic energy balance. This local flux density differs dimensionally from the section-integrated buoyancy flux of a plume, which includes an area or width integration.