= Solution
Use $G=g\beta(T-T_0(z))$ for the plume <reduced gravity>, and take $z$ upwards. For a symmetric <line plume> define fluxes per unit source length,
$$
q=\int w\,dx,\qquad m=\int w^2\,dx,\qquad \mathcal B=\int wG\,dx.
$$
The integrals span the plume; its edges can move horizontally with $z$. The <fluid entrainment> rate at one moving edge is the inward normal transport relative to that edge. For an edge $x=b(z)$ it is $E=wb'-u$; symmetry supplies equal transport through the other edge. The <Batchelor entrainment hypothesis> closes its mean value as $E=\alpha w_c$, where $w_c$ is a representative vertical <velocity> and $\alpha$ is the <entrainment coefficient>. The integrated <volume conservation> equation is therefore $q'=2\alpha w_c$.
Using <volume conservation>, the vertical acceleration equation has conservative form $\partial_x(uw)+\partial_z(w^2)=G$. Incoming ambient has zero vertical <momentum>; the integrated <momentum conservation> equation is consequently $m'=\int G\,dx$. For the <temperature> anomaly, $T_0$ depends on height and
$$
\partial_x(uG)+\partial_z(wG)=-N^2w,\qquad N^2=g\beta\frac{dT_0}{dz}.
$$
Ambient fluid enters with $G=0$, giving the <buoyancy flux> balance. The integral model is
$$
\boxed{q'=2\alpha w_c,\qquad m'=\int G\,dx,\qquad \mathcal B'=-N^2q.}
$$
Here $N$ is the <buoyancy frequency> when $N^2>0$; the same algebra permits unstable <temperature> gradients with $N^2<0$.
These are mean integral balances. The advective equations printed in the question omit explicit turbulent stresses and heat fluxes; a <turbulent plume> interpretation requires the usual entrainment closure for that unresolved transverse transport. Assuming vanishing edge stresses and heat fluxes, and negligible additional axial turbulent-flux contributions, gives the displayed bulk model. It would be inconsistent to treat a discontinuous top-hat profile as a smooth exact pointwise solution of the inviscid temperature-advection equation.
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