 Source-volume length of a turbulent plume

Source volume flux is negligible only when entrained volume greatly exceeds it. In the jet limit of the top-hat plume model, this requires $z\gg Q_0/(c\sqrt{M_0})$. In a pure plume with $Q=C_Q{z}^{5/3}$, it requires $z\gg (Q_0/C_Q{)}^{3/5}$, where $C_Q=(3c/5)(9c{F}_0/20{)}^{1/3}$ for full-cross-section fluxes and $c=2{\alpha }_e\sqrt{\pi }$. These scales differ from the jet length which measures imposed momentum relative to buoyancy. Neglecting source volume does not automatically permit neglecting source momentum.