Filling-box first front ($h(t)$)

In a filling box model, the first front separates fluid already modified by plume discharge from the initially unmodified fluid below. If the plume occupies a negligible fraction of horizontal area $A$, its volume flux across the front is $P(h)$, and the ceiling exhaust is $F$, volume conservation gives $A\dot{h}=F-P(h)$. For a closed box with a negligible-volume buoyancy source, $F=0$. For a finite injected source flux $Q$ with ceiling overflow $Q$, $F=Q$. The first front need not delimit a uniformly mixed upper layer throughout the unventilated transient.

An ideal two-sided line plume has half-width $b=\alpha z$, constant velocity $w_{\ast }=(\mathcal{B}/(2\alpha ){)}^{1/3}$ and volume flux $q=2\alpha w_{\ast }z$. In a closed box of width $2R$, neglecting the plume's occupied area gives $2R\dot{h}=-q(h)$ for the filling-box first front, initially at ceiling height $H$. With negligible diffusion the front is a material interface; its ambient buoyancy jump retains the initial ceiling-discharge value $\mathcal{B}/q(H)=w_{\ast }^2/H$. The contemporaneous plume reduced gravity $w_{\ast }^2/h(t)$ is a different quantity. The upper ambient generally becomes stratified rather than remaining one uniformly mixed layer.