Plume flux-balance invariant ($M^{5/2}-5F{Q}^2/(8\alpha )=C$)

When $F$ and $\alpha$ are constant, eliminating height between $Q^{\prime }=2\alpha \sqrt{M}$ and $M^{\prime }=FQ/M$ gives $d(M^{5/2})/dQ=5FQ/(4\alpha )$. Thus $M^{5/2}-5F{Q}^2/(8\alpha )$ 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.

For positive source fluxes and positive constant buoyancy flux, the plume flux-balance invariant has a finite constant while $Q$ grows without bound. Dividing by $Q^2$ makes the source imbalance vanish relatively. The balances then give $Q\sim (6\alpha /5)(9\alpha F/10{)}^{1/3}{z}^{5/3}$ and $M\sim (9\alpha F/10{)}^{2/3}{z}^{4/3}$. This attraction concerns relative similarity and does not assert that different finite sources share the same virtual origin.