= Solution
The <Batchelor entrainment hypothesis> makes the inward <entrainment velocity> proportional to the local plume speed:
$$
\boxed{u_e=\alpha |w|,}
$$
where $\alpha>0$ is the dimensionless <entrainment coefficient>. For an upward plume $w>0$, this is $u_e=\alpha w$. The hypothesis represents turbulent ingestion of ambient fluid; it is a closure rather than a consequence of <mass conservation>.
Let $A=b^2$, suppressing the common factor $\pi$ from cross-sectional integrals. In the <Boussinesq approximation>, use a constant reference <density> $\rho_r$ in the leading-order <mass conservation> equation. Dividing that equation by $\rho_r$ gives <volume conservation>:
$$
\boxed{A_t+(Aw)_z=2b u_e.}
$$
The factor two is the ratio of the perimeter $2\pi b$ to the suppressed area factor $\pi$. <Density> differences are retained in <buoyancy>, even though they are neglected in the leading inertial and volume terms.
To derive the density-deficit balance, first leave the small <density> differences explicit in the supplied mass equation. Because the ambient <density> $\rho_0(z)$ is time independent, the <product rule> gives
$$
\begin{aligned}
\partial_t[(\rho_0-\rho)gA]+\partial_z[(\rho_0-\rho)gAw]
&=g\rho_0[A_t+(Aw)_z]+g\rho_{0z}Aw\\
&\quad-g[(\rho A)_t+(\rho Aw)_z]\\
&=g\rho_0(2bu_e)+g\rho_{0z}Aw-g(2\rho_0bu_e)\\
&=g\rho_{0z}Aw.
\end{aligned}
$$
Thus the ambient mass entrained through the perimeter cancels exactly against its contribution to the volume budget, yielding the required <buoyancy> balance. It is understood to the accuracy of the underlying <Boussinesq approximation>; the volume equation is not an additional exact constraint on a fully variable-density fluid.
For later use define <reduced gravity> $g'=g(\rho_0-\rho)/\rho_r$ and the squared <buoyancy frequency> $N^2=-g\rho_{0z}/\rho_r$. With $B=Ag'$ and $Q=Aw$, the balance is
$$
\boxed{B_t+(Awg')_z=-N^2Q.}
$$
A stably stratified ambient has $N^2>0$, so upward motion consumes the plume's positive <density> deficit; an unstable ambient reverses this sign.
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