Plume flux-balance invariant
= Plume flux-balance invariant
{title2=$M^{5/2}-5FQ^2/(8\alpha)=C$}
When $F$ and $\alpha$ are constant, eliminating height between $Q'=2\alpha\sqrt M$ and $M'=FQ/M$ gives $d(M^{5/2})/dQ=5FQ/(4\alpha)$. Thus $M^{5/2}-5FQ^2/(8\alpha)$ is constant. Its sign classifies forced and lazy departures from <pure plume balance>, and its vanishing makes a finite-source plume exactly self-similar after a virtual-origin shift.