A finite-volume <gravity-current box model> in a widening triangular channel has volume proportional to $h^2L^2$, so $hL=K$ is constant. A <gravity-current front condition> and absorbing-bed <particle deposition flux> give
$$
\dot L=\operatorname{Fr}\sqrt{dK\phi/L},\qquad \dot\phi=-2W_sL\phi/K.
$$
Eliminating time makes $\sqrt\phi$ affine in $L^{5/2}$, producing a finite limiting <runout length of a gravity current> despite an infinite idealized stopping time.


 Triangular-channel gravity-current box model

ID: triangular-channel-gravity-current-box-model