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.