= Unsteady top-hat line-plume balances
Use half-width $b$, upward speed $w$, constant inertial reference density $R$, and positive buoyant <reduced gravity> $g'$. Half-section fluxes are $Q=Rbw$, $M=Rbw^2$ and $F=Rg'bw$. With two-sided <entrainment> $u_e=\alpha w$, the <Boussinesq> balances are
$$
(Q^2/M)_t+Q_z=R\alpha M/Q,\qquad
Q_t+M_z=FQ/M,\qquad
(FQ/M)_t+F_z=-N^2Q.
$$
Full-width fluxes are twice these definitions. The shape coefficients differ from triangular transverse profiles, so their unsteady balances cannot be used unchanged here.
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