 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 $\dot{V}=4\pi R^2\dot{R}$ 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.