Can Shallow-Water Theory Describe Breaking?

Summary

The conventional answer is no, except in the sense of allowing approximate discussion of shock-like post-breaking events. But solutions by characteristics of the nonlinear shallow-water equations do give a rather impressive picture of an apparent plunger during the actual breaking itself, and one wonders if there is anything in it, nevertheless. Shallow-water theory should demand only that the fluid velocity vector be nearly horizontal, nothing else. In particular, it should not make any assumption that the free-surface slope is small. The fluid particles in a near-vertical wall of water could be moving nearly horizontally. However, even if this were so, unfortunately there is still a kinematic barrier to acceptance of shallow-water theory beyond the point where the free surface first beomes vertical, since most derivations assume that the free-surface height is a single-valued function of the horizontal co-ordinate. There are alternative derivations, e.g. Lagrangian, that do not demand such an assumption. An apparent singularity occurs when the free surface becomes vertical, but this singularity is not present for pure simple waves. The opportunity is also taken to discuss similar breaking-like problems for lubrication equations, which are the viscous-fluid equivalent of the shallow-water equations, with applications such as to dripping of freshly-painted vertical surfaces.