Science Heresy - April 2011

The figure shows wave height measured at two points in a wave tank. The upper graph, "Channel 1", shows a wave group 4.4m from the hydraulically controlled paddle which generated it. The lower graph, "Channel 2", shows the same group 30m further down the tank. The wave group appears to have split into two envelope solitons.

What happens when waves break?

1. Fluid mechanics cannot handle wave-breaking

For historical reasons fluid dynamics (and hence gravity wave dynamics) has always been the province of applied mathematicians rather than physicists. This is unfortunate because applied mathematicians share certain prejudices which make it difficult for them to deal with the real world. One such prejudice is the belief that fluid dynamic phenomena can always be described using deterministic equations. Another is their slavish adherence to the convention that relationships between variables must always be displayed in dimensionless form. This makes it almost impossible for them to test hypotheses using graphical methods, i.e. they cannot do experiments.

As a result fluid mechanics cannot deal with stochastic phenomena such as turbulence. Wave breaking involves turbulence. Thus, while fluid mechanics can predict when waves break, the effects of wave breaking are usually "swept under the carpet". In numerical wave models, wave breaking is parameterised using a crude rule of thumb and the unwarranted assumption of a fifth power roll-off with frequency.

2. An experimental physics approach to wave breaking

The 60m long towing tank at the Australian Maritime College was used to carry out an experiment. The towing tank had an hydraulically controlled paddle driven by a small computer which could be programmed to generate any wave shape. Wave groups comprising 12 or so sinusoidal waves with a raised cosine envelope were generated by the paddle and allowed to propagate down the tank. Only the amplitude was changed from group to group. As the group travelled down the tank individual waves were observed to move from the rear of the group to the front demonstrating that phase velocity is indeed greater than group velocity. The envelope of each group changed shape dramatically as can be seen in the diagram at the top of this page.

When the amplitude was set to larger values, wave breaking (i.e. white-capping) occurred. Individual waves broke as they passed through the centre of the envelope, i.e. when they were at their steepest.

How did wave breaking affect the downstream shape of the wave group?

The answer is - very little. The changing shape of the envelope appears to be due to the Benjamin-Feir sideband instability and was not caused by wave breaking. However wave breaking had an important effect on the frequency spectra of the downstream wave groups.

The images shows the "before" (light grey) and "after" (dark grey) frequency spectra for two groups with different amplitudes. The spectra of the time series of the diagram at the top of the page are shown on the right.

In both cases the spectra show that the original spectral peak has split into 2 sidebands in accordance with Benjamin-Feir theory. The downstream spectrum on the left has two approximately equal sidebands whereas the upper sideband of the downstream spectrum on the right is significantly smaller than the lower sideband. As a consequence, there was a net downward shift in frequency associated with wave breaking.

3. Hypothesis to account for frequency downshifting in the open sea

When wind starts to blow across still water small ripples are first formed by the Kelvin_Helmholtz instability. The ripple wavelength remains the same until the wind is strong enough to generate waves large enough for the Benjamin-Feir instability and white-capping to occur. When this happens energy is lost from the high frequency sideband and there is a net downward shift in frequency and lengthening of the waves. These downshifted waves accumulate more energy by this process until they too break, giving rise to further down-shifting.

This hypothesis can be confirmed by anyone willing to patiently observe waves forming on a dock or canal on a windy day. Downshifting only occurs when the ripples start to break.

4. Further Experiments

This simple experiment presents an exciting prospect. Much can still be learned experimentally about wave breaking and about the generation and non-linear interaction of solitons by performing similar experiments. There are many wave tanks around the world which could be used for this purpose in much the same way that physicists use particle accelerators to perform experiments in quantum mechanics. A whole new science of experimental non-linear surface gravity wave interactions could be opened up. Who knows where it might lead?

5. Rejection by the Ocean Wave Community

This paper describing the above experiment was presented to a symposium on wave-breaking held at the University of New South Wales in 1991. To a physicist this experiment provided a useful insight into how frequency downshifting is intimately associated with white-capping. However it was not well received by the symposium audience, largely comprised of applied mathematicians. Following the presentation, rather than ask a question from the floor, one prominent theoretician, Peter Janssen, was given permission by the Chair to take the podium. His passionate denunciation of the paper was based on the argument that much good theoretical work would become irrelevant if the presented argument were true. Therefore it could not be true.

The original paper transcribed from the conference proceedings can be found here (pdf 160 kB). It was never submitted to a peer reviewed journal. There was little point. Most of the potential reviewers were present when it was first presented. Given its reception at the symposium it is most unlikely to have been accepted for publication.

April 2011