Local turbulent kinetic energy balance ($ϵ=P+B_0,P=-\overline{u^{\prime }{w}^{\prime }}{U}_z$)

In stationary locally equilibrated turbulence, mean-shear production and buoyancy production balance viscous dissipation after transport terms are neglected. Stable buoyancy flux is negative and consumes part of the shear input. This approximation gives $ϵ=u_{\ast }^3/(\kappa z)$ in a neutral logarithmic layer and $ϵ\sim B_0$ in buoyancy-dominated free convection.

Let $S=U_z>0$, $N^2=-(g/{\rho }_0){\overline{\rho }}_z>0$, $q^2=\overline{u_i^{\prime }{u}_i^{\prime }}$, and constant correlations $C_u=-\overline{u^{\prime }{w}^{\prime }}/q^2$ and $C_{\rho }^2={\overline{{\rho }^{\prime }{w}^{\prime }}}^2/(q^2\overline{{\rho }^{\prime 2}})$. Using $ϵ=q^3/L_u$ and $\chi =\overline{{\rho }^{\prime 2}}q/L_{\rho }$, the scalar-variance balance implies $\overline{{\rho }^{\prime }{w}^{\prime }}=-C_{\rho }^2{L}_{\rho }q{\overline{\rho }}_z$. Substituting in the energy budget gives the displayed quadratic for $q>0$.

Write $a=C_{\rho }^2{L}_{\rho }{N}^2/(C_{u}S)$ and $b=1/(L_u{C}_{u}S)$. The ratio of total sinks to production is $R=a/q+bq$. Positive equilibria exist only for $4ab\le 1$, equivalent to $\mathrm{Ri}\le L_u{C}_u^2/(4L_{\rho }{C}_{\rho }^2)$. The larger-$q$ root has flux Richardson number $(1-\sqrt{1-4ab})/2$; the smaller root has the plus sign. Under the local kinetic-energy evolution the larger root is stable and the smaller root unstable. The critical value is closure-dependent and is not the universal inviscid linear-stability threshold.