= Starting-plume thermal Froude-number ratio
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 $\dot V=4\pi R^2\dot R$ imply
$$
\frac{\dot Z}{w_p}=\frac5{5+24\alpha\lambda^3}.
$$
The 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.
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