= Solution
Let the fixed current volume per unit channel width be
$$
V_0=Lh=L_0h_0.
$$
A suitable high-<Reynolds number> deep-ambient <gravity-current front condition> is
$$
\dot L=\operatorname{Fr}\sqrt{g'h},
$$
where the order-one <Froude number> $\operatorname{Fr}$ records the selected front closure. In the dilute limit the mixture's <reduced gravity> is
$$
g'=\frac g{\rho_a}
\sum_{i=1}^2(\rho_i-\rho_a)\phi_i,
$$
with ambient and carrier-fluid density $\rho_a$.
The well-mixed particle volume of species $i$ is $V_0\phi_i$. Its deposition rate through the base of length $L$ is $L\widehat W_i\phi_i$, where $\widehat W_i<0$. The resulting <gravity-current box model> is therefore
$$
h=\frac{V_0}{L},
\qquad
\dot L=\operatorname{Fr}\sqrt{\frac{V_0g'}L},
\qquad
\dot\phi_i=\frac{L\widehat W_i}{V_0}\phi_i.
$$
It conserves fluid volume while suspended particle volume, and therefore the driving buoyancy, decreases by deposition.
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