Appendix A
Recent Research on Extreme Wave Models

As awareness of extreme waves has grown, researchers have intensified efforts to understand the physics underlying their formation. If the occurrence of extreme waves can be linked to certain meteorological or sea state conditions, then it might be possible to forecast when they will occur. Such information would be invaluable to mariners and the marine insurance industry.

Much of what we know about extreme waves—indeed, the emphasis on learning more about them—has come from observations by shipmasters who have witnessed encounters between their vessel and an extreme wave or from survivors of vessels that foundered as a result of an extreme wave incident. Those observations are by their nature imprecise; generally they occurred too suddenly or under extremely stressful conditions where exact measurements were impossible.

In Chapter 8, I grouped extreme waves by probable cause, in the following order:

  • Strong currents

  • Storms

  • Continental shelves and shallow seas



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Extreme Waves Appendix A Recent Research on Extreme Wave Models As awareness of extreme waves has grown, researchers have intensified efforts to understand the physics underlying their formation. If the occurrence of extreme waves can be linked to certain meteorological or sea state conditions, then it might be possible to forecast when they will occur. Such information would be invaluable to mariners and the marine insurance industry. Much of what we know about extreme waves—indeed, the emphasis on learning more about them—has come from observations by shipmasters who have witnessed encounters between their vessel and an extreme wave or from survivors of vessels that foundered as a result of an extreme wave incident. Those observations are by their nature imprecise; generally they occurred too suddenly or under extremely stressful conditions where exact measurements were impossible. In Chapter 8, I grouped extreme waves by probable cause, in the following order: Strong currents Storms Continental shelves and shallow seas

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Extreme Waves Constructive interference (superposition) To these we may add several additional possible mechanisms for extreme wave formation that are the subject of current research: Nonlinear effects Spatial or temporal focusing effects Multidirectional effects Modulation and resonance The first category, strong currents, has indisputably been the source of extreme waves. When swell or storm waves encounter a fast-moving opposing current such as the Agulhas Current or the Gulf Stream, they tend to “pile up” as their velocity is reduced. Professor Chris Garrett pointed out that a wave with a phase speed of c meters per second can be stopped by an opposing current of only 1/4 c.1 When this happens, steep, high waves result, proceeded or followed by deep troughs. Thus, the evidence is clear that the probability of encountering an extreme wave is greater under these conditions and a prudent mariner should avoid this situation if possible. The increase in wave size as a function of wind velocity, fetch, and wind duration is a well-known phenomenon in storms. There are correlations that provide estimates of the significant wave heights under varying storm conditions, but none that predict the random occurrence of extreme waves. This suggests that some additional mechanism, yet to be fully understood, is at work. It would be useful to know if extreme wave formation is governed by a threshold effect; in other words, do seas have to build to a certain point before extreme waves are produced? Or is it purely a statistical effect? The evidence seems to indicate the latter possibility, because many mariners (myself included) recall sailing in relatively calm seas where the significant wave height was a few feet, but suddenly a wave two to three times as high struck the vessel. Likewise, shallow water, bottom effects, and refraction have the effect of slowing waves and causing wave heights to increase. Areas where there is a sharp transition in sea depth are potential danger zones

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Extreme Waves in rough seas. The question here is: Are there certain sea or wind conditions that combine to cause an extreme wave and how can they be anticipated? Constructive interference (sometimes called superposition) is the fourth category mentioned in Chapter 8. Here I include incidents (to the best of my knowledge) in which an extreme wave struck a vessel in the absence of the other conditions described above. Superposition is a well-known phenomenon, observable in many areas of physics, so its existence is not in doubt. The relevant research question is whether or not it is capable of causing waves that are 2.2 to 2.4 times as high as the significant wave. From the Rayleigh distribution, the answer would seem to be yes. (A factor of two seems obvious.) Turning to new areas of research, attempts to model nonlinear wave effects may shed light on how superposition can produce extreme waves. Nonlinearity makes mathematical modeling much more difficult, so scientists and engineers always try to develop linear models first and then turn to nonlinear models as a last resort. Some researchers go so far as to say that if rogue waves exist, they must inherently be nonlinear. Some of the features of extreme waves—their steepness and the shape of the wave crest—are modeled more accurately when second-order terms are included. Another interesting new approach is based on spatial and temporal focusing. Spatial focusing is another way of describing what happens when waves are refracted by the ocean bottom in coastal waters or by current gradients.2 Temporal focusing may result when waves disperse. Some wave groups may contract to a few wavelengths and then combine with others to produce short groups of very large waves.3 Some theorists believe that nonlinear focusing may allow a wave to “borrow” energy from its neighbors, becoming as much as 4.5 to 5 times greater than the average wave height.4 Multidirectional and multidimensional effects are also being studied to see if they can cause extreme waves. The idea is to investigate whether extreme waves result from wave trains interacting at an angle or from three-dimensional interactions—effects that would not be modeled by a one-dimensional analysis. When I spoke with Dr. Susanne Lehner, she indicated that waves from crossing seas arriving

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Extreme Waves from two different storms can add and build to an extreme size due to the continuous input of energy. Also, the Benjamin-Feir index, introduced in Chapter 11, measures a phenomenon called the Benjamin-Feir instability. It is defined as the ratio of the mean square slope of the frequency spectrum peak to its normalized width. Under the right conditions, instability causes the wave train to break up into periodic groups. Within each group a further focusing takes place, producing a very large and steep wave having a height roughly three times the initial height of the wave train.5 Finally, other research is directed at seeing whether frequency or amplitude modulation could be responsible for extreme waves. Or is it possible that certain wave periods and frequencies will resonate with a given sea state condition to create extreme waves? The analogy to this is easily demonstrated in a bathtub. A bathtub, or a harbor for that matter, has a series of resonant frequencies. If you take a piece of wood (or possibly just your hand) and get the water sloshing back and forth at just the right period, a large wave will occur.