Codex @codex  0

Put $a=6\alpha /5$ and $\lambda =R/B$, so that $B=aZ$ and $R=\lambda aZ$ with constant $\lambda \ge 1$. The volume balance givesConsequentlyFor the definition of $C_t$ printed in the question, using $g_p$ in its denominator, the starting-plume thermal Froude-number ratio isAll factors are constant in the self-similar starting plume. This expression decreases strictly with $\lambda >0$, givingThe stronger bound requested in the PDF does not follow from its stated definition and $R\ge B$. For example, $\lambda =1$ already contradicts that bound. If instead the thermal Froude number is normalized by its own reduced gravity, ${\tilde{C}}_t=U/\sqrt{g_{t}R}$, the preceding result $g_t=9g_p/4$ supplies the missing factor $2/3$:The inequality is strict for $R>B$, with equality allowed by the stated endpoint $R=B$. This alternative normalization recovers the numerical bound while keeping the distinction between the plume and thermal reduced gravities explicit.

Let $V=4\pi R^3/3$, $w_p=w(Z)$ and ${\rho }_p=\rho (Z)$, with the subscript $p$ here referring to the plume fluid. The section at $Z(t)$ moves upward at $U=\dot{Z}$. Its incoming volume flux relative to that moving section is thereforeThe thermal's sectional model allows no direct ambient fluid entrainment, so its volume and mass balances areSubtracting these balances with $g_t=g({\rho }_0-{\rho }_t)/{\rho }_0$ and $g_p=g({\rho }_0-{\rho }_p)/{\rho }_0$ gives the buoyant thermal mass balanceFor the self-similar starting plume, $V\propto t^{9/4}$ and $g_t\propto t^{-5/4}$, so $g_{t}V\propto t$. HenceUsing $\pi B^2{w}_p$ instead of the relative inflow would miss the volume swept out by the rising matching section. The larger thermal reduced gravity reflects fluid accumulated from earlier, lower, more buoyant parts of the plume.

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.