= Parabolic-channel sediment-current box model
{title2=$h^{3/2}L=\mathrm{constant},\quad\dot\phi=-3W_s\phi/(2h)$}
A well-mixed <particle-laden gravity current> in a prismatic channel with $b(z)=C\sqrt z$ has volume $V=2Ch^{3/2}L/3$. Negligible entrainment gives fixed $V$. Vertical <particle deposition flux> is $W_s\phi b(h)L$, so $\dot\phi=-3W_s\phi/(2h)$. Close the front by $\dot L=F\sqrt{G\phi h}$, where $G=g(\rho_p-\rho_0)/\rho_0$. Eliminating time gives
$$
\sqrt\phi+\frac{3W_sL^2}{8F\sqrt G\,h_0^{3/2}L_0}
=\sqrt{\phi_0}+\frac{3W_sL_0^2}{8F\sqrt G\,h_0^{3/2}L_0}.
$$
The concentration tends to zero at a finite limiting <runout length of a gravity current>, although reaching that limit takes infinite time in this model.
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