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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.

Write $g^{\prime }=g({\rho }_f-{\rho }_a)/{\rho }_f>0$. The Saint-Venant dry-front condition describes a liquid advancing over a dry bed, or into an ambient whose inertia is negligible, as when ${\rho }_a/{\rho }_f\ll 1$. The edge has $h_f=0$; its velocity is obtained by continuing the appropriate shallow water Riemann invariant to zero depth. There is no independent universal finite-depth head Froude number. In the triangular geometry a release of a resting layer gives $u_f=2\sqrt{2g^{\prime }{h}_0}$, whereas a rectangular release gives $2\sqrt{g^{\prime }{h}_0}$.

The Benjamin deep-ambient front condition includes the inertia of an ambient that must turn around a finite-depth gravity current head. In the ideal deep-ambient Boussinesq approximation, ${\rho }_a/{\rho }_f$ is close to one andHere $h_f$ is the current depth just behind its head, $u_f$ its speed relative to the far ambient, and $F$ the ideal head Froude number. The ambient is much deeper than $h_f$; a finite upper-layer depth changes the closure. The lower density must exceed the ambient density for a bottom current. A non-Boussinesq head requires an appropriate density-ratio correction rather than using $F=\sqrt{2}$ at every ratio. The Saint-Venant and Benjamin models therefore describe different front physics, even though both use the same long-wave equations in the interior.

Put $c_0=\sqrt{g^{\prime }{h}_0/2}$. The undisturbed reservoir has $u=0$ and Riemann invariants $u\pm 4c=\pm 4c_0$. The right-going characteristic curves supply the invariant $u+4c=4c_0$ throughout the release. The spreading disturbance is the other family, whose speed is $u-c$. In its rarefaction wave let $\xi =x/t=u-c$. ThenAt the reservoir edge $u=0,c=c_0$, so the back of the rarefaction wave travels at $-c_0$. At the dry edge $c=0$ the Saint-Venant dry-front condition givesThus the fan occupies $-c_{0}t\le x\le 4c_{0}t$, while $x<-c_{0}t$ remains undisturbed. Its full length grows at $5c_0$; there is no finite-depth uniform shelf between the fan and the dry front. The optional interior depth is $h=h_0[(4c_0-x/t)/(5c_0){]}^2$.

The fan's minus-family characteristic curves are rays $x=\xi t$. For the plus family in the fan, $dx/dt=u+c=(8c_0+3x/t)/5$; integration gives $x=4c_{0}t+C{t}^{3/5}$, with constants selected at entry from the reservoir. These curved paths transport the constant plus Riemann invariant.