= Solution
Apply <volume conservation>, <mass conservation> and vertical <momentum flux> balance to a slice of the <triangular-profile line plume>. Ambient ingestion contributes $2e$ of <volume> and $2\rho_0e$ of <mass> per unit height and source length. The ambient is quiescent, so it supplies no leading vertical <momentum>. The integrated driving force is the stored <buoyancy> $\mathcal B$. Using the quantities from part (a), the <non-Boussinesq triangular-profile line-plume balances> are
$$
\boxed{\begin{aligned}
\partial_t(2b)+\partial_z(bW)&=2e,\\
\partial_t[b(\rho+\rho_0)]+\partial_z\left[\frac{bW}{3}(\rho_0+2\rho)\right]&=2\rho_0e,\\
\partial_t\left[\frac{bW}{3}(\rho_0+2\rho)\right]+\partial_z\left[\frac{bW^2}{6}(\rho_0+3\rho)\right]&=gb(\rho_0-\rho).
\end{aligned}}
$$
Together with <Batchelor entrainment>, these are three equations for $W,\rho,b$. They neglect viscous boundary stresses and streamwise pressure-force corrections within the integral-plume approximation.
Multiplying <volume> conservation by $g\rho_0$ and subtracting $g$ times <mass conservation> cancels the ambient sources exactly, giving
$$
\boxed{\partial_t[gb(\rho_0-\rho)]+\partial_z\left[\frac23gbW(\rho_0-\rho)\right]=0.}
$$
This is <buoyancy> conservation in the homogeneous ambient; it is a consequence of the first two balances, not an additional independent equation.
Now take the <Boussinesq approximation> in inertia and entrainment, retaining the small density deficit in <buoyancy>. Then $Q=\rho_0bW$, $M=(2/3)\rho_0bW^2$ and $F=(2/3)gbW(\rho_0-\rho)$. Consequently
$$
W=\frac32\frac MQ,\qquad 2\rho_0b=\frac43\frac{Q^2}{M},\qquad \mathcal B=\frac{FQ}{M}.
$$
The <mass> source is $2\alpha\rho_0W=3\alpha\rho_0M/Q$. Substitution into the <mass conservation>, <momentum> and <buoyancy> balances gives the <unsteady Boussinesq triangular-profile line-plume balances>
$$
\boxed{\begin{aligned}
\frac43\partial_t\left(\frac{Q^2}{M}\right)+\partial_zQ&=3\alpha\rho_0\frac MQ,\\
\partial_tQ+\partial_zM&=\frac{FQ}{M},\\
\partial_t\left(\frac{FQ}{M}\right)+\partial_zF&=0.
\end{aligned}}
$$
These use density-weighted <mass> and <momentum> fluxes. Removing $\rho_0$ from some flux definitions while retaining it in the source would mix incompatible conventions.
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