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In the prismatic triangular-channel gravity-current box model, take $h=h_0\sqrt{L/X}$ and $w(\varphi )=w_0(1-\varphi )$. Eliminating time gives $d\varphi /[\sqrt{\varphi }(1-\varphi )]=-2w_0{X}^{3/4}dX/(\mathrm{Fr}\sqrt{G}{h}_0^{3/2}{L}^{3/4})$. Integration yields the displayed invariant with $K=4w_0/(7\mathrm{Fr}\sqrt{G}{h}_0^{3/2}{L}^{3/4})$. The positive-concentration branch approaches a finite runout length of a gravity current as time tends to infinity.

Initially the same expansion fan and constant-speed front apply. Its left edge reaches the closed rear wall at $t_w=x_0/c_0$. After that, the infinite-reservoir solution would give a nonzero velocity at the wall and violates the no-through-flow condition. A reflected disturbance adjusts the interior while the finite initial volume is progressively spread over a longer, shallower current.

The nose does not respond instantaneously when the fan reaches the wall. Its first return signal follows a $+$ characteristic from $(x,t)=(-x_0,t_w)$ through the undisturbed fan. There $u+c=(8c_0+3x/t)/5$, so integrating $dx/dt=u+c$ givesIt reaches the uniform nose-adjacent region at $t_n=t_w(c_0/c_f{)}^{5/2}$, where $x=(u_f-c_f)t_n$. Crossing that region at speed $u_f+c_f$, it catches the front at $2t_n$. Thus the initial nose remains uninformed about the rear wall until this return time. Subsequently its speed must adjust rather than remain constant indefinitely.

For late times use a prismatic triangular-channel gravity-current box model with uniform representative depth $\bar{h}$ over length $\ell =x_f+x_0$. The initial volume is $V=k{h}_0^2{x}_0$. With no entrainment or loss,Apply the same front closure using that representative depth:Starting from a matching late-time state $(t_b,{\ell }_b)$, integration gives finite-volume inertial spreading in a triangular channel:Hence $\ell \propto t^{4/5}$, $\bar{h}\propto t^{-2/5}$ and the nose speed decays like $t^{-1/5}$. A simple initialized box approximation can instead set ${\ell }_b=x_0$ and $t_b=0$, but it then deliberately replaces the exact early rarefaction rather than reproducing it.

The box's volume equation is compatible with a velocity increasing linearly from zero at the rear wall to the nose speed: $u(x,t)=\dot{\ell }(x+x_0)/\ell$. A flat-depth box with this velocity is not an exact solution of the full momentum equation during deceleration. It is an integral approximation that leaves the interior shape and nose dynamics unresolved while predicting late-time bulk scaling.

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 iswith 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 $\sqrt{g^{\prime }h}$; if the front condition is expressed using $c=\sqrt{g^{\prime }h/2}$, its numerical coefficient changes by $\sqrt{2}$.

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 $\mathcal{V}=\beta h_0^{2}L/2$, soThe total particle volume is $\mathcal{V}\varphi$. Deposition over the projected boundary width gives $d(\mathcal{V}\varphi )/dt=-\beta hXw\varphi$. With the specified hindered settling factor, the prismatic triangular-channel gravity-current box model isInitializing this approximate model with $X(0)=L$ and $\varphi (0)={\varphi }_0$ matches the original volume and concentration; it does not assert that the immediate dam-break flow is a uniform box.

Eliminate time to obtainSince the left primitive is $2\mathrm{artanh}\sqrt{\varphi }$, the hindered-settling runout invariant in a prismatic triangular channel isThis relation, the volume constraint and $\dot{X}$ determine the late-time evolution. If desired, the time is given by the quadrature $t={\int }_L^{X(t)}[\mathrm{Fr}\sqrt{G\varphi (s)h_0\sqrt{L/s}}{]}^{-1}ds$. 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, isThe advance beyond the dam is $X_{\infty }-L$. In the dilute limit, $\mathrm{artanh}\sqrt{{\varphi }_0}\simeq \sqrt{{\varphi }_0}$. The ideal model approaches its finite runout length of a gravity current only as $t\to \infty$: near the limit $\varphi$ 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.