= Fixed-correlation equilibrium model for stratified turbulence
{title2=$q^2/L_u-C_uSq+C_\rho^2L_\rho N^2=0$}
Let $S=U_z>0$, $N^2=-(g/\rho_0)\overline\rho_z>0$, $q^2=\overline{u_i'u_i'}$, and constant correlations $C_u=-\overline{u'w'}/q^2$ and $C_\rho^2=\overline{\rho'w'}^2/(q^2\overline{\rho'^2})$. Using $\epsilon=q^3/L_u$ and $\chi=\overline{\rho'^2}q/L_\rho$, the scalar-variance balance implies $\overline{\rho'w'}=-C_\rho^2L_\rho q\overline\rho_z$. Substituting in the energy budget gives the displayed quadratic for $q>0$.
Write $a=C_\rho^2L_\rho N^2/(C_uS)$ and $b=1/(L_uC_uS)$. The ratio of total sinks to production is $R=a/q+bq$. Positive equilibria exist only for $4ab\leq1$, equivalent to $\mathrm{Ri}\leq L_uC_u^2/(4L_\rho C_\rho^2)$. 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.
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