 Solution

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.