= Solution
The <entrainment coefficient> $\alpha$ is the ratio of the inward ambient-fluid <velocity> normal to a <buoyant plume> edge to the characteristic <buoyant plume> <velocity>. For an upward <buoyant plume>, take $u_e=\alpha w>0$. There are two edges at half-width $b$, so the full-section volume source is $2u_e$. Dividing every integral balance by two gives consistent half-section equations.
For <top-hat plume> profiles, <volume conservation> and <mass conservation> are
$$
\boxed{b_t+(bw)_z=\alpha w,\qquad
(\rho b)_t+(\rho bw)_z=\rho_0(z)\alpha w.}
$$
The mass entering the <buoyant plume> is ambient mass with local <mass density> $\rho_0(z)$. The <Boussinesq approximation> uses a common reference <mass density> in inertia while retaining the <mass density> deficit in <buoyancy>. The mass equation may be kept to the first order in that deficit to derive its transport.
Multiply the volume equation by the time-independent ambient <mass density>. The product rule gives
$$
(\rho_0b)_t+(\rho_0bw)_z=\rho_0\alpha w+\rho_0'(z)bw.
$$
Subtract the mass equation and multiply by $g$. The <entrainment> sources cancel exactly, leaving
$$
\boxed{\partial_t[(\rho_0-\rho)gb]+\partial_z[(\rho_0-\rho)gbw]
=g\rho_0'(z)bw=-\rho_0N^2bw,}
$$
where
$$
\boxed{N^2=-\frac{g}{\rho_0}\frac{d\rho_0}{dz}.}
$$
For stable <stratification>, $\rho_0'<0$ and the <buoyancy frequency> $N$ is real. Rising fluid moves into lighter ambient fluid and loses its <mass density> deficit relative to that local ambient. Thus environmental <stratification> reduces the <buoyant plume>'s <buoyancy flux>, even though the ambient fluid entering by <entrainment> contributes no <mass density> deficit at its own height.
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