Turbulent kinetic energy ($e_t=\frac{1}{2}\overline{u_i^{\prime }{u}_i^{\prime }}$)

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 work of the Reynolds stress against the mean velocity gradient transfers kinetic energy from the mean flow into fluctuations. In a plane shear flow $U(y)$ this is $P=-\overline{u^{\prime }{v}^{\prime }}{U}_y$. The eddy viscosity closure gives $P={\nu }_T{U}_y^2\ge 0$. Buoyancy production and transport are separate terms in the turbulent kinetic energy budget.

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.