= Solution
Let the closed box have ceiling height $H$ above the source. The printed width $2R$ alone does not specify this height, so it must appear as an additional geometrical parameter. Take $t=0$ here when the plume has first reached the ceiling and its thin outflow has spread across the box. The leading <filling box model> assumes a negligible source volume, insulated boundaries, fast horizontal spreading, slow evolution compared with the plume transit time, negligible diffusion, and $b\ll R$. Below the interface the ambient still has its original <temperature>, so the local plume flux remains $q(z)=2\alpha w_*z$ from part (c).
The ceiling outflow creates a warm upper region. <Fluid entrainment> draws the original lower fluid into the plume, making that upper region deepen. The <filling-box first front> is the lower material boundary of fluid already modified by plume discharge, separating it from the initially unmodified ambient; it need not bound a uniformly mixed upper layer. With negligible occupied plume area, <volume conservation> at its height $h(t)$ gives
$$
2R\dot h=-q(h)=-2\alpha w_*h.
$$
Therefore
$$
\boxed{h(t)=H e^{-t/t_f},\qquad t_f=\frac{R}{\alpha w_*}=\frac{2R}{(2\alpha)^{2/3}\mathcal B_0^{1/3}}.}
$$
If source switch-on is the time origin instead, replace $t$ by elapsed time after the initial transit and spreading stage; the quasi-steady model does not resolve that stage.
The <buoyancy> jump requires distinguishing ambient fluid at the front from the plume crossing it. Just after the first ceiling outflow, the ambient <buoyancy> immediately above the nascent front is
$$
G_H=\frac{\mathcal B_0}{q(H)}=\frac{w_*^2}{H}.
$$
Outside the narrow plume, ambient <temperature> is transported without diffusion. If $G_a$ is its <buoyancy> relative to the original fluid, its equation is
$$
\partial_tG_a-\frac{q(z)}{2R}\partial_zG_a=0.
$$
The first front follows precisely these material trajectories, so the <temperature> of the fluid immediately above it is conserved. The initially unmodified fluid below has $G_a=0$. Thus the ambient jump is
$$
\boxed{[G_a]_{\text{above}-\text{below}}=\frac{w_*^2}{H},\qquad [T_a]=\frac{w_*^2}{g\beta H}.}
$$
By contrast, the contemporaneous plume immediately below the front has $G_p(h,t)=w_*^2/h(t)$, which grows as the front descends. That plume value is not the ambient jump across the material first front. Equating the two would inadvertently impose a uniformly mixed upper layer and discard the <stable density stratification> of the standard <filling box model>. Diffusion or extensive interfacial mixing can smooth and alter the ideal jump.
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