= Solution
At the <gravity current> nose, the depth and <velocity> change over a distance comparable with the depth, with appreciable vertical acceleration and mixing. The <hydrostatic approximation> and the depth-uniform interior model do not resolve that structure. A suitable <gravity-current front condition> is
$$
\boxed{\dot X=\operatorname{Fr}\sqrt{G\phi h},}
$$
with a positive, order-one <Froude number> determined by the nose closure and geometry. It is not fixed by the interior equations alone. This definition uses $\sqrt{g'h}$; if the front condition is expressed using $c=\sqrt{g'h/2}$, its numerical coefficient changes by $\sqrt2$.
For a late-time <gravity-current box model>, take the depth and <particle volume fraction> to be uniform over $0<x<X(t)$, and neglect entrainment and resuspension. The initial <volume> is $\mathcal V=\beta h_0^2L/2$, so
$$
\frac{\beta h^2X}{2}=\mathcal V,\qquad h=h_0\sqrt{\frac LX}.
$$
The total particle <volume> is $\mathcal V\phi$. Deposition over the projected boundary width gives $d(\mathcal V\phi)/dt=-\beta hXw\phi$. With the specified <hindered settling> factor, the <prismatic triangular-channel gravity-current box model> is
$$
\boxed{\dot X=\operatorname{Fr}\sqrt{G\phi h},\qquad
\dot\phi=-\frac{2w_0}{h}\phi(1-\phi),\qquad h=h_0\sqrt{L/X}.}
$$
Initializing this approximate model with $X(0)=L$ and $\phi(0)=\phi_0$ matches the original <volume> and <concentration>; it does not assert that the immediate dam-break flow is a uniform box.
Eliminate time to obtain
$$
\frac{d\phi}{\sqrt\phi(1-\phi)}=-\frac{2w_0X^{3/4}}{\operatorname{Fr}\sqrt G\,h_0^{3/2}L^{3/4}}\,dX.
$$
Since the left primitive is $2\operatorname{artanh}\sqrt\phi$, the <hindered-settling runout invariant in a prismatic triangular channel> is
$$
\boxed{\operatorname{artanh}\sqrt\phi
=\operatorname{artanh}\sqrt{\phi_0}-\frac{4w_0}{7\operatorname{Fr}\sqrt G\,h_0^{3/2}L^{3/4}}(X^{7/4}-L^{7/4}).}
$$
This relation, the <volume> constraint and $\dot X$ determine the late-time evolution. If desired, the time is given by the quadrature $t=\int_L^{X(t)}[\operatorname{Fr}\sqrt{G\phi(s)h_0\sqrt{L/s}}]^{-1}\,ds$. The physical branch has nonnegative right-hand side in the boxed relation; it ends when that expression reaches zero.
Thus the limiting <runout length of a gravity current>, measured from the closed end, is
$$
\boxed{X_\infty=\left[L^{7/4}+\frac{7\operatorname{Fr}\sqrt G\,h_0^{3/2}L^{3/4}}{4w_0}\operatorname{artanh}\sqrt{\phi_0}\right]^{4/7}.}
$$
The advance beyond the dam is $X_\infty-L$. In the dilute limit, $\operatorname{artanh}\sqrt{\phi_0}\simeq\sqrt{\phi_0}$. The ideal model approaches its finite <runout length of a gravity current> only as $t\to\infty$: near the limit $\phi$ decays exponentially with rate $2w_0/h_\infty$ and the front speed tends to zero. The exponent $7/4$ uses the prismatic triangular geometry; a channel widening with $x$ has a different <volume> constraint and runout law.
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