= Triangular-channel gravity-current box model
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.
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