Finite-volume inertial spreading in a triangular channel ($\ell \propto t^{4/5},h\propto t^{-2/5}$)

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 $\dot{\ell }=F\sqrt{g^{\prime }{h}_0}(x_0/\ell {)}^{1/4}$, 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.