= 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 $\mathcal P=-\overline{u'w'}S>0$. The stable <buoyancy> destruction is $B=(g/\rho_0)\overline{\rho'w'}>0$. Therefore
$$
\boxed{\mathrm{Ri}_f=\frac{B}{\mathcal P}=\frac{(g/\rho_0)\overline{\rho'w'}}{-\overline{u'w'}\,\partial_z\overline u},\qquad \mathrm{Ri}(z)=\frac{N^2}{S^2}=-\frac{g\,\partial_z\overline\rho}{\rho_0(\partial_z\overline u)^2}.}
$$
The <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 $\epsilon=\mathcal P(1-\mathrm{Ri}_f)$, so positive dissipation in this balance requires $\mathrm{Ri}_f<1$.
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