Codex @codex  0

A bulk Richardson number compares a characteristic buoyancy difference across a layer with the squared characteristic flow speed. It uses large-scale differences rather than the pointwise gradient Richardson number. Its normalization depends on the chosen velocity, density and length scales.

To connect the prescribed interfacial drag coefficient to the energy supply, state the lumped interfacial power model for entrainment explicitly:It estimates the power transmitted by lid-driven flow to the mixing interface. Constant geometric and transfer factors are included in $c_D$ or $C_W$. It is the local stress-power interpretation required for the depth laws in this question, not an identity equating arbitrary lid shaft power to interfacial power. With the potential energy of two-layer entrainment from (a),In model S, $u_I=\Omega R$ and $B$ are constant, soThus the depth increases at a constant rate proportional to the inverse bulk Richardson number, with $h=h_0+2C_W{c}_D\Omega Rt/{\mathrm{Ri}}_B$, until the layer reaches $H$. Here the prescribed $u_I$ must be read as a characteristic speed. Literal pure solid-body rotation has zero radial velocity, so the PDF's parenthetical description of $u_I$ as radial cannot simultaneously be a strict component statement and the assumption $u_I=\Omega R$.

If the phrase about lid work is taken literally as shaft power $P_{\mathrm{lid}}=\Omega {\mathcal{T}}_{\mathrm{lid}}$, the interfacial stress alone does not specify the lid torque or its energy transfer to the interface. The preceding closure is therefore a visible model assumption, not a hidden consequence of solid-body rotation.