= Filling-box first front
{title2=$h(t)$}
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.
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