How shallow breaking waves lose energy

A few weeks ago my paper, Inertial energy dissipation in shallow-water breaking waves, was published in the Journal of Fluid Mechanics. It’s about how solitary waves (that is, individual waves not part of a larger wavetrain) approach a beach with a single, uniform slope, and break on it. Although the paper describes various stages in this process, including how the wave changes shape and steepens on approach to the beach, how it breaks, and how it then runs up the beach, it really focuses on how its energy is dissipated as it breaks.

This is a computational study, where the governing equations are solved down to small scales (a technique called direct numerical simulation), so that we can measure the velocity at all points in the fluid and thus determine its total kinetic energy at all points in time. Since we also know the shape of the wave, and hence the height of the water surface, we can also determine the gravitational potential energy. We can track the sum of these two energies as the wave breaks: it converts some potential energy into kinetic energy, and some of this is then dissipated by viscosity into heat. This study varies the height of the incoming wave and the slope of the beach, and measures the energy dissipation from the resulting breakers.

But this is all just a bunch of data unless we have some way to bind it together in an explanation. This is the core of the study: it presents a theoretical model based on related studies in deep water where one assumes the jet of the breaking wave generates a cloud of turbulence underneath the wave. Using information about the wave when it reaches the breaking point – its height, the local depth of the water, the slope of the beach – it turns out that its energy dissipation rate can be predicted quite well, explaining our data in a simple scaling model.

This is a nice result that, firstly, answers basic (“local”) questions about the breaking process in shallow water waves, and could help provide scaling model information for reduced-order models operating over large length and time scales, such as regional coastal wave models. It also opens up more possibilities in direct simulation of shallow-water waves, including extending to 3D to further characterize the nature of the dissipative structures underneath the breaking wave, or progressive wave systems which are more reflective of real coastal environments and hence allow for a more direct relation of dissipative properties to the wave field.

The paper can be read for free (though with some reduced resolution on the figures) here.

 

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