= Solution
In the <top-hat plume model>, vertical <velocity> $w(z)$ and <reduced gravity> $G(z)$ are uniform over $-b<x<b$. The ambient has zero vertical <velocity>, negligible ambient turbulence and a prescribed $T_0(z)$. Then
$$
q=2bw,\qquad m=2bw^2,\qquad\mathcal B=2bwG,
$$
and the equations become
$$
\boxed{(bw)'=\alpha w,\qquad (bw^2)'=bG,\qquad (bwG)'=-bwN^2.}
$$
Equivalently, the <ordinary differential equations> for the three unknowns are
$$
\boxed{w'=\frac Gw-\frac{\alpha w}{b},\qquad b'=2\alpha-\frac{bG}{w^2},\qquad G'=-N^2-\frac{\alpha G}{b}.}
$$
The last equation displays dilution by <fluid entrainment> as well as loss of relative <buoyancy> while rising through a stably stratified ambient. The model assumes a steady slender two-dimensional <turbulent plume>, large <Reynolds number>, the <Boussinesq approximation>, constant $\beta$ and $\alpha$, negligible heat loss, negligible imposed crossflow, and negligible <pressure> departure from ambient <hydrostatic pressure>. The factors of two count both entraining edges; $b$ is a half-width. A finite source also requires its volume, <momentum> and <buoyancy fluxes> as boundary data.
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