= Meltwater production over an advancing footprint
{title2=$\dot{\mathcal V}=s_m\mathcal L$}
For constant water-production rate per wetted area $s_m$, dry zero-flux noses and footprint length $\mathcal L(t)$, the moving-boundary <conservation of mass> gives $\dot{\mathcal V}=s_m\mathcal L$ per out-of-plane span. Consequently $\ddot{\mathcal V}=s_m\dot{\mathcal L}$. Equivalently the melt volume is $s_m\int_0^t\mathcal L(t\prime)dt\prime$; a point begins producing melt only after the front arrives.
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