= Finite-source forced-plume quadrature
{title2=$z=c^{-1}\int_{Q_0}^{Q}[M_0^{5/2}+5F_0(q^2-Q_0^2)/(4c)]^{-1/5}dq$}
For an axisymmetric <top-hat plume model> in uniform ambient density, use full kinematic fluxes and $c=2\alpha_e\sqrt\pi$. The equations $Q'=c\sqrt M$, $M'=F_0Q/M$ conserve $M^{5/2}-5F_0Q^2/(4c)$. Solve this invariant for $M(Q)$, then integrate $dz=dQ/(c\sqrt M)$ to obtain the displayed exact quadrature. It determines the transition from the near-source <jet limit of the top-hat plume model> to the distant <pure plume>. The two regimes cannot be added by linear superposition because the flux equations are nonlinear.
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