 Solution

Let $f_0=F_0/(\pi {\rho }_0)$ as above. A self-similar starting plume has no source length or time scale; dimensional analysis therefore givesThe dimensionless constants depend on the entrainment coefficient and thermal closure. Thus $R/B$ is constant because the underlying Boussinesq point-source plume has $B=b(Z)=(6\alpha /5)Z$. The steadily increasing thermal volume scales as $t^{9/4}$, even though its reduced gravity decreases.