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.

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.