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Under the Boussinesq approximation, total upper-layer kinetic energy is $K={\rho }_{l}Sh{u}_U^2/2$, where $u_U$ is its volume root-mean-square speed. For solid-body rotation of angular speed $\omega (t)$, direct radial averaging gives $u_U^2={\omega }^2{R}^2/2$, consistent with that energy expression. A constant $K$ implies $h{u}_U^2=h_0{u}_U(0{)}^2$. Define $C_K=u_U(0)/(\Omega R)$ to obtain the fixed-energy turbulent mixed layer relationWith $u_I=C_I{u}_U$, the interfacial power model for entrainment used in (b) givesMultiply by $h^{3/2}$ and integrate from $h_0$. This yieldsThe fixed-energy two-layer entrainment law is thereforeHere ${\mathrm{Ri}}_{B,0}=B/({\Omega }^2{R}^2)$ uses the same lid-speed normalization as part (b). All numerical geometry factors depend on the chosen characteristic stress/speed convention. The formula ceases to apply once $h=H$.

The exponent $2/5$ requires the local power to decrease as $u_I^3\propto h^{-3/2}$. It does not follow from constant upper-layer kinetic energy and a fixed fraction of unspecified lid power alone. For example, an imposed constant shaft power together with a constant fraction going to potential energy would give constant $\dot{h}$, not this exponent. The appropriate distinction between local interface power and global forcing appears explicitly in the primary study Entrainment and mixed layer dynamics of a surface-stress-driven stratified fluid, equations (3.6)–(3.7); the calculations above use that local interpretation.