Axisymmetric pure plume

For a top-hat axisymmetric pure plume of radius $r$, speed $U$, and reduced gravity $g^{\prime }$, the conserved fluxes are $Q=\pi r^{2}U$, $M=\pi r^2{U}^2$, and $B=\pi r^{2}U{g}^{\prime }$. The Batchelor entrainment hypothesis gives $Q\propto B^{1/3}{z}^{5/3}$.