Radius growth of an axisymmetric pure plume ($b(z)=b_s+6\alpha z/5$)

For pi-normalized top-hat fluxes, $b=Q/\sqrt{M}$. The pure similarity solution therefore has $b=(6\alpha /5)(z+z_0)$, so the radius grows linearly with slope $6\alpha /5$. The corresponding plume virtual origin is $-z_0$ and a finite source has $z_0=5b_s/(6\alpha )$. This law translates geometric plume-contact criteria into heights.