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For the upward incident internal gravity wave, $P={\rho }_0\omega m_{0}W/k^2$. Using real physical fields, its time-averaged vertical energy flux per unit area isThe assumed mixing power is $\mathcal{I}/4$. A deepening law also requires the gravitational potential energy of the mixed layer. The wave problem specifies a well-mixed half-space above the interface, but supplies neither a finite upper mixed-layer depth nor a replacement buoyancy/energy boundary condition. There is therefore no unique finite-layer $d(t)$ from the printed information alone.

The missing parameter can be displayed explicitly. Let the initially mixed upper region have finite depth $a$, with a fixed upper boundary at $z=a$, and conserve its integrated buoyancy while homogenizing down to $z=-d$. Its new buoyancy isIntegrate $-{\rho }_{0}zb$ over the initial and final profiles in $-d<z<a$. The potential energy of mixed-layer deepening with an initial buoyancy jump isWith $d(0)=0$ and a maintained constant incident flux, the resulting law isThe right-hand side is positive and determines a unique depth once $a$ is supplied. Initially $\dot{d}=\omega m_0|w_0{|}^2/(4k^2\Delta ba)$ for $a>0$, already showing why the upper depth matters. If the added assumptions instead specify a zero initial mixed depth, the cubic term gives $d=[3\omega m_0|w_0{|}^{2}t/(2k^2{N}_0^2){]}^{1/3}$; the initial jump then has no finite volume above it to mix with. Neither assumption is contained in the original half-space wave statement. Literal homogenization of the entire upper half-space is the $a\to \infty$ limit and costs infinite energy for any fixed positive $d$. This limitation should not be concealed by a guessed entrainment depth or by keeping the upper buoyancy fixed while claiming conservation.