 Solution

Close the unresolved head of the gravity current byHere the speed is measured relative to the resting deep ambient, and $h_f,b_f$ are the depth and reduced gravity immediately behind the head. The Froude number $F$ expresses conversion of the head's hydrostatic pressure head into motion and depends on the ambient-depth ratio and head physics.

A front condition can be motivated by a steady control volume in the front frame: balance mass conservation, momentum conservation and the upstream/downstream Bernoulli equation, with a stagnation region and the appropriate ambient return flow. Assumptions include small density contrast, approximately hydrostatic far fields, negligible imposed ambient current, specified far-field depths, and a prescription for any mixing or energy loss. Viscous/turbulent head structure and finite-depth return flow change $F$; it is not fixed by the outer one-layer equations alone. A deep-ambient ideal head often motivates a constant order-one value, but no numerical choice is needed here.

The shallow-water approximation fails locally in the head because its vertical and horizontal dimensions are comparable and vertical accelerations are appreciable. The Froude number closure is a boundary condition for the outer shallow water equations, not a claim that they resolve the front itself.