Codex @codex  0

Use the kinematic fluxes $q=Q/(\pi {\rho }_0)$, $m=M/(\pi {\rho }_0)$ and $f_0=F_0/(\pi {\rho }_0)$. In the steady Boussinesq approximation, the non-Boussinesq top-hat plume equations reduce toEliminating $z$ gives $m^{5/2}=5f_0{q}^2/(8\alpha )$, where the integration constant vanishes for a pure plume from a point source. Write $b=az$ and $w=K{z}^{-1/3}$. The two balances give $a=6\alpha /5$ and $K^3=3f_0/(4a^2)=25f_0/(48{\alpha }^2)$. The Boussinesq point-source plume is thereforeHere $q=a^{2}K{z}^{5/3}$ and $m=a^2{K}^2{z}^{4/3}$ tend to zero at the source, while $f_0$ remains positive. The plume Froude number is independent of height:The ideal point source is a far-field similarity idealization. Since $g^{\prime }\propto z^{-5/3}$, the Boussinesq approximation fails near $z=0$ and is valid only when $g^{\prime }/g\ll 1$. In terms of $K$, this requires $z\gg [4K^2/(3g){]}^{3/5}$; the density formula must not be extrapolated into its unphysical negative-density region.

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.

For a spherical buoyant thermal fed only through the moving top of a Boussinesq point-source plume, put $\lambda =R/B$ and $B=(6\alpha /5)Z$. Relative inflow $\pi B^2(w_p-\dot{Z})$ and $\dot{V}=4\pi R^2\dot{R}$ implyThe ratio of Froude numbers is $5/[\sqrt{\lambda }(5+24\alpha {\lambda }^3)]$ when both use plume reduced gravity $g_p$. Normalizing the thermal with its own $g_t=9g_p/4$ multiplies this by $2/3$. These two definitions must not be interchanged when stating bounds.