= Solution
For the upward incident <internal gravity wave>, $P=\rho_0\omega m_0W/k^2$. Using real physical fields, its time-averaged vertical <energy flux> per unit area is
$$
\mathcal I=\frac12\operatorname{Re}(PW^*)=\frac{\rho_0\omega m_0|w_0|^2}{2k^2}.
$$
The 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 is
$$
b_m=b_1-\frac{\Delta b\,d+N_0^2d^2/2}{a+d}.
$$
Integrate $-\rho_0zb$ over the initial and final profiles in $-d<z<a$. The <potential energy of mixed-layer deepening with an initial buoyancy jump> is
$$
\frac{\Delta\mathcal P}{\rho_0}=\frac{\Delta b\,a d}{2}+\frac{N_0^2a d^2}{4}+\frac{N_0^2d^3}{12}.
$$
With $d(0)=0$ and a maintained constant incident flux, the resulting law is
$$
\boxed{\frac{\Delta b\,a d}{2}+\frac{N_0^2a d^2}{4}+\frac{N_0^2d^3}{12}
=\frac{\omega m_0|w_0|^2}{8k^2}t,\qquad
\dot d=\frac{\omega m_0|w_0|^2/(8k^2)}{\Delta b\,a/2+N_0^2a d/2+N_0^2d^2/4}.}
$$
The right-hand side is positive and determines a unique depth once $a$ is supplied. Initially $\dot d=\omega m_0|w_0|^2/(4k^2\Delta b\,a)$ 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 $d=[3\omega m_0|w_0|^2t/(2k^2N_0^2)]^{1/3}$; 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.
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