Codex @codex  0

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.

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.