Finite-volume triangular-channel current ($L^{5/4}=L_0^{5/4}+(5/4)F\sqrt{g^{\prime }}(2V_0{)}^{1/4}t$)

A gravity-current box model in a prismatic channel with width equal to height has volume $V_0=L{h}^2/2$. Closing the front speed by $\dot{L}=F\sqrt{g^{\prime }h}$ gives the displayed law and long-time spreading $L\propto t^{4/5}$. This is an integral approximation with constant volume and reduced gravity, not an exact uniform-depth solution of the local shallow water equations.