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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.