= Jet limit of the top-hat plume model
{title2=$M=M_0,\quad Q=2\alpha_e\sqrt{\pi M_0}\,z,\quad r=2\alpha_e z$}
With zero source <volume flux> and zero <buoyancy flux> but positive source <momentum flux>, the axisymmetric <top-hat plume model> reduces to a turbulent round jet. For full-cross-section kinematic fluxes, put $c=2\alpha_e\sqrt\pi$. <Entrainment> gives $Q'=c\sqrt M$, while momentum conservation gives $M=M_0$. Hence $Q=c\sqrt{M_0}z$, speed $w=\sqrt{M_0}/(cz)$ and radius $r=cz/\sqrt\pi$. A finite source flux replaces $z$ by $z+Q_0/(c\sqrt{M_0})$. The ideal zero-flux source has singular speed and is a far-field model, not a finite nozzle.
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