 Unsteady top-hat line-plume balances

Use half-width $b$, upward speed $w$, constant inertial reference density $R$, and positive buoyant reduced gravity $g^{\prime }$. Half-section fluxes are $Q=Rbw$, $M=Rb{w}^2$ and $F=R{g}^{\prime }bw$. With two-sided entrainment $u_e=\alpha w$, the Boussinesq balances areFull-width fluxes are twice these definitions. The shape coefficients differ from triangular transverse profiles, so their unsteady balances cannot be used unchanged here.