 Solution

Set $S={\partial }_z\overline{u}>0$ and $N^2=-(g/{\rho }_0){\partial }_z\overline{\rho }>0$, using the constant reference mass density ${\rho }_0$ of the Boussinesq approximation. The mean-shear production is $\mathcal{P}=-\overline{u^{\prime }{w}^{\prime }}S>0$. The stable buoyancy destruction is $B=(g/{\rho }_0)\overline{{\rho }^{\prime }{w}^{\prime }}>0$. ThereforeThe Flux Richardson number compares stable buoyancy destruction with shear production; the gradient Richardson number compares restoring stratification with mean shear rate. They are generally different. The stationary energy budget gives $ϵ=\mathcal{P}(1-{\mathrm{Ri}}_f)$, so positive dissipation in this balance requires ${\mathrm{Ri}}_f<1$.