= Solution
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 $\phi_m\sim c_m\zeta$ and $\phi_h\sim c_h\zeta$, with positive closure constants. Using $\ell_O=u_*^3/(\kappa|B_0|)$ gives
$$
\boxed{U_z\sim c_m\frac{|B_0|}{u_*^2},\qquad U(z)-U(z_r)\sim c_m\frac{|B_0|}{u_*^2}(z-z_r),\qquad N^2\sim c_h\frac{|B_0|^2}{u_*^4}.}
$$
Thus the mean velocity is approximately linear in height and the real <buoyancy frequency> is approximately constant, $N\sim\sqrt{c_h}|B_0|/u_*^2$. Under the <Reynolds analogy>, $c_h=c_m$.
The <local turbulent kinetic energy balance> gives
$$
\boxed{\epsilon=P-|B_0|\sim(c_m-1)|B_0|,\qquad \ell_\epsilon\sim\frac{u_*^3}{\epsilon}=\frac{u_*^3}{(c_m-1)|B_0|}.}
$$
Positive dissipation requires $c_m>1$. The turbulent energy-turnover length is of order $u_*^3/|B_0|$, hence of order the <Obukhov length>, independent of observation height. A related buoyancy-limited length is $u_*/N$. The <Ozmidov length> follows by equating an eddy turnover time $\ell^{2/3}\epsilon^{-1/3}$ with $N^{-1}$:
$$
\ell_{\rm Oz}=(\epsilon/N^3)^{1/2}\sim\frac{\sqrt{c_m-1}}{c_h^{3/4}}\frac{u_*^3}{|B_0|}.
$$
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> give
$$
\boxed{\eta_K=(\nu^3/\epsilon)^{1/4}\sim\left[\frac{\nu^3}{(c_m-1)|B_0|}\right]^{1/4}.}
$$
Without $\nu$, 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,
$$
\boxed{\mathrm{Ri}_f\longrightarrow\frac1{c_m},\qquad \mathrm{Ri}_g\longrightarrow\frac{c_h}{c_m^2}.}
$$
They become height-independent constants. The commonly used continuation $c_m=c_h=5$ gives $\mathrm{Ri}_f=\mathrm{Ri}_g\simeq0.2$ and $\epsilon\simeq4|B_0|$. 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 https://arxiv.org/abs/1404.1397[Grachev and colleagues' similarity study].
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