Entrainment exponent from a local interfacial Richardson closure ($E\propto {\mathrm{Ri}}_i^{-3/2}$)

Combine a fixed-energy mixed-layer relation $u^{2}h=A{\Omega }^2{R}^3$, jump $g^{\prime }=N^{2}h/2$, and stress power proportional to $u^3$ with an energy-transfer factor $(\Omega /N{)}^{\alpha }(\delta /R{)}^{\beta }$. Eliminating depth in favor of the interfacial Richardson number gives $E\propto (\Omega /N{)}^{\alpha -1}(\delta /R{)}^{\beta +3/2}{\mathrm{Ri}}_i^{-3/2}$. A local-only entrainment closure requires $\alpha =1$, $\beta =-3/2$ and exponent $3/2$ on the Richardson number.