 Solution

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