 Pure plume balance

For the constant-entrainment axisymmetric plume equations $Q^{\prime }=2\alpha \sqrt{M}$, $M^{\prime }=FQ/M$ with $F>0$, pure balance means $M^{5/2}=5F{Q}^2/(8\alpha )$ at every height. This is a similarity relation between fluxes, not a requirement that the physical source have zero radius. A finite source on this curve is represented by a plume virtual origin. For fluxes normalized by pi and top-hat fields, the same relation is $g^{\prime }b/w^2=8\alpha /5$.