 Self-similar starting plume

For constant kinematic buoyancy flux $f_0$ from a point source, dimensional analysis gives head height and radius proportional to $f_0^{1/4}{t}^{3/4}$, head velocity proportional to $f_0^{1/4}{t}^{-1/4}$, and head reduced gravity proportional to $f_0^{1/4}{t}^{-5/4}$. Dimensionless coefficients require the chosen fluid entrainment and head closures.