= Finite-volume inertial spreading in a triangular channel
{title2=$\ell\propto t^{4/5},\quad h\propto t^{-2/5}$}
With constant buoyancy and no settling or entrainment, the box volume is proportional to $h^2\ell$. Thus $h=h_0\sqrt{x_0/\ell}$ for an initial lock length $x_0$. A <gravity-current front condition> gives $\dot\ell=F\sqrt{g'h_0}(x_0/\ell)^{1/4}$, so $\ell^{5/4}$ increases linearly with time. This is a late-time integral approximation; it does not reproduce the initial rarefaction or the detailed nose.
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