= 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 gives
$$
\boxed{Z=c_Zf_0^{1/4}t^{3/4},\qquad
R=c_Rf_0^{1/4}t^{3/4},\qquad
U=\dot Z=\frac34c_Zf_0^{1/4}t^{-1/4},\qquad
g_t=c_gf_0^{1/4}t^{-5/4}.}
$$
The 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.
Back to article page
