Prismatic triangular-channel gravity-current box model

A uniform-depth particle-laden gravity current in fixed triangular cross section has conserved volume proportional to $h^{2}X$. A gravity-current front condition and absorbing-boundary particle deposition flux give $\dot{X}=\mathrm{Fr}\sqrt{G\varphi h}$ and $\dot{\varphi }=-2w(\varphi )\varphi /h$. The box describes late-time bulk evolution; it does not resolve the nose or immediate dam-break flow.

With constant buoyancy and no settling or entrainment, the box volume is proportional to $h^2\ell$. Thus $h=h_0\sqrt{x_0/\ell }$ for an initial lock length $x_0$. A gravity-current front condition gives $\dot{\ell }=F\sqrt{g^{\prime }{h}_0}(x_0/\ell {)}^{1/4}$, so ${\ell }^{5/4}$ increases linearly with time. This is a late-time integral approximation; it does not reproduce the initial rarefaction or the detailed nose.

For a dilute well-mixed particle current of conserved bulk volume $W$, triangular section $\lambda h^2$, constant particle settling speed $w_s$ and initial reduced gravity $g_0^{\prime }$, put $s=\sqrt{\varphi /{\varphi }_0}$. The balances give $\dot{X}=Bs{X}^{-1/4}$ and $\dot{\varphi }=-2w_s\varphi /h$, where $B=\mathrm{Fr}\sqrt{g_0^{\prime }}(W/\lambda {)}^{1/4}$. Eliminating time gives $s=1-(X/R{)}^{7/4}$ and the displayed limiting runout. It is approached at infinite time. Early spreading has the inertial $t^{4/5}$ exponent, while settling limits the eventual reach.

Under the same box-model assumptions, $M_0={\rho }_p{\varphi }_{0}W$ is initial particle mass and $m_d$ is final deposited mass per downstream length. Integrate the vertical particle deposition flux across the instantaneous triangular-valley width $2\lambda h$ from arrival of the front at $x$ until settling is complete. This gives the displayed profile on $0\le x\le R$ and zero beyond the runout. Its integral is $M_0$, verifying particle mass conservation. Areal deposit density also depends on the lateral coordinate because the wetted valley width changes as the current thins.

// Target: geophysics.bigb

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.