Gravity-current front condition

A common high-Reynolds number closure for a deep-ambient gravity current is $\dot{L}=\mathrm{Fr}\sqrt{g^{\prime }h}$, where $L$ is front position and the order-one Froude number depends on the chosen front model.

In an ideal Boussinesq gravity current advancing beneath a much deeper ambient, the local head Froude number is $u_f/\sqrt{g^{\prime }{h}_f}=\sqrt{2}$. The ambient must be displaced around the finite-depth head. This differs from the Saint-Venant dry-front condition, which neglects ambient inertia. The coefficient belongs to the ideal deep-ambient head model; confinement, mixing and drag can change it.

The shallow water equations at a dry bed, with negligible ambient mass density, end in a vanishing-depth edge, rather than a finite-depth gravity current head. In a prismatic triangular channel the Riemann invariant $u+4\sqrt{g^{\prime }h/2}$ gives the speed $2\sqrt{2g^{\prime }{h}_0}$ after release of a resting semi-infinite layer of depth $h_0$. In a rectangular channel the corresponding speed is $2\sqrt{g^{\prime }{h}_0}$; the channel geometry matters.