= Solution
The layer equations describe long-wave adjustment behind the nose, but do not resolve the short region where the dense fluid pushes aside the ambient. A finite-height front cannot be determined by simply adjoining a vacuum solution of the one-layer momentum equation. The ambient <pressure> and turning flow provide additional nose physics.
In the nose frame, the material condition makes the adjacent current <velocity> equal to the front speed in the laboratory frame. The two interior <characteristic speeds> relative to that front are then $\pm c_f$: one characteristic brings information from the body and one brings a boundary condition from the nose. This is why a <gravity-current front condition> is needed. With a deep ambient, fixed channel shape and negligible viscosity, the local <buoyancy> depth supplies the <velocity> scale $\sqrt{g'h_f}$. A dimensionless closure therefore has the form
$$
\boxed{u_f=F\sqrt{g'h_f},}
$$
where a constant $F$ is an idealization for a fixed geometry and chosen nose model. Dimensional analysis identifies the form but does not determine the coefficient.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-83-front-control.png]
{title=Nose-fixed control volume and triangular cross-section for a gravity-current front condition}
To determine an appropriate coefficient theoretically, take a control volume moving with the nose and bounded by sections far enough ahead and behind for <hydrostatic pressure> and approximately uniform section velocities to apply. Use the actual triangular wetted areas and integrated pressures. Conservation of fluid volume fixes the ambient return-flow velocities; integrated horizontal momentum balances the incoming and outgoing momentum fluxes against the <pressure> forces. A Bernoulli relation on an ambient streamline, with an explicit nose-loss assumption if needed, supplies the <pressure> or energy closure. The density-anomaly <pressure> contribution of the lower layer contains the integral $g'kh_f^3/3$ derived above. Solving those balances gives $u_f^2/(g'h_f)$ and hence $F^2$.
Alternatively, measured front speeds and consistently defined adjacent heights in a geometrically similar experiment can calibrate the closure. A rectangular-channel numerical coefficient must not be transferred to a triangular channel without checking those balances. The height definition and dissipative nose structure also matter; https://paul-linden.com/wp-content/uploads/2017/05/104mtl05.pdf[a primary study of gravity-current front conditions] explains why different height conventions can lead to different inferred coefficients. No numerical value of $F$ is required for the following solution.
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