How high is the highest wave? Which properties does it have? How often and under what circumstances do extreme waves occur? We have let these questions be the basis for a continuous research effort since 1995. On these pages we try to give a summary of the topic and of our activities.
It is well known that extreme waves often occur in areas were waves propagate into a strong opposing current. A well known example where many large ships have encountered difficulties is the Agulhas current outside South Africa. The strong current going south meets strong swell from storms in the Antarctic Ocean.
The picture below was taken on the oil freighter Esso Languedoc outside the coast of Durban (1980). The man who took it, Philippe Lijour, estimated the mean wave height when this occurred to be about 5-10 m. The mast on the starboard side is 25 m above the mean sea level. The wave approached from behind and broke over deck, but caused only minor damage.
In areas where waves from storms in the open ocean approach shallower waters (e.g. several locations along the Norwegian coast), the waves will be refracted and diffracted.
There may be focusing of wave energy in certain areas such that the probability of encountering large waves is much greater than in other areas. Such refraction and diffraction of waves, either due to currents or bathymetry, can be computed. In a certain sense these waves may therefore be predicted. At the end of the 1970's the research project "Skip i sjøgang" (S. P. Kjeldsen) localized 24 dangerous areas along the Norwegian coast; see also "Den norske los".
It is far more difficult to avoid, as well as to explain, extreme waves occurring in the open ocean far from variable bathymetry or ocean currents. On January 1st 1995 an extreme wave was measured under the Draupner platform (16/11-E) in the North Sea providing indisputable evidence that such waves do indeed exist. This wave has been known in the international scientific community as the "new year wave". The maximal amplitude of 18.5 m is more than three times the significant amplitude for the wave train! The maximal wave height of 25.6 m is much more than twice the significant wave height of about 10.8 m. The time series is reproduced below with the surface elevation in meters as a function of the time in seconds.
Analysis of the ocean state around this waves shows that the wave train as a whole is weakly nonlinear and has relatively small bandwidth. This justifies the use of nonlinear Schrödinger equations as simplified mathematical models for wave description.
If we suppose that the wave above is long crested, we can simulate it numerically forward and backward in space. Below we show how the time series develops upstream (left figure) and downstream (right figure) in intervals of 50 meters. One characteristic wave length is about 260 meters.
The following important observations can be made from the numerical simulation above:
FOCUSING OF WAVE ENERGY
As can be seen in the figure of the "new year wave" there is a significant concentration of wave energy in comparison with the mean energy (a factor of about 18 in this case). Are there physical effects that can cause such a concentration or focusing of wave energy in the open ocean? Three known effects have been proposed as possible candidates: (1) Time-space focusing. (2) Current focusing. (3) Nonlinear focusing. The first two are described by so-called linear theory and have been known since the beginning of the past century.
Focusing in time and space. This effect is utilized in large wave tanks for testing of ship models. With a wave maker at the end of the tank one creates a signal in the form of a wave train where the wave length varies, with the shortest waves in front. Long waves propagate faster and will catch up on the shorter waves. Thereby a few large waves are created over a short time and within a limited area. The animation below shows the focusing effect at the top. If this is superposed on an irregular sea state as shown in the middle, one could end up with the behavior shown at the bottom.
The problem of this explanation of freak waves is the following: How can an orderly signal as the one on top be created spontaneously in the ocean? Nobody has so far answered this convincingly.
Current focusing. Even though the current velocities in the open ocean (far from coastal areas) are small, typically about 10 cm/s, they can give small deflections of the waves when they act over long distances. The result can be local focusing or defocusing of wave energy, in the same way as one can see at the bottom of a swimming pool when the sun shines. White & Fornberg (1998) have proposed this as an explanation of freak waves. The figure below (from their work) shows wave trajectories through an area of variable current. The current field is faintly marked in the background. It can be seen that all wave trajectories are parallel initially. The deflection due to the current produces both areas of both increased and decreased wave intensity.
The problem with this explanation of freak waves is the following: To have a significant effect from this process it is required that the waves enter the zone of variable currents with a single direction. If they have natural directional distribution one would end up with the same situation as observed at the bottom of the swimming pool when the sun goes behind a cloud such that the light becomes more diffuse; the effect disappears.
Nonlinear focusing. As opposed to the effects above, this one cannot be explained by linear theory. It was shown in the middle of the 1960s that if you generate uniform periodic waves in one end of a long wave tank, the waves will spontaneously split into groups, which get more prominent as they propagate along the tank. According to linear theory these waves should remain uniform and periodic. One developed a wave equation (the so-called nonlinear Schrödinger equation) capable of explaining this strange behavior qualitatively. This equation has later been modified and improved to also give good quantitative agreement with experiments.
The effect of weakly nonlinear effects on the evolution of a wave train on deep water can be seen on the figures below which are all simulated.
The figures show time series "measured" at four different "stations" along a "numerical wave tank". The lower (blue) curve shows linear evolution, or rather the lack of evolution. The middle (green) curve shows evolution according to the cubic nonlinear Schrödinger equation. The upper (red) curve shows the evolution according to the higher order modified nonlinear Schrödinger equation. Experiments reveal that the waves in reality behave very similarly to the upper curves.
The simplest nonlinear Schrödinger equation has many exact solutions. One of them has been particularly popular as candidate to explain freak waves. It is called a "breather" and starts out as a periodic wave train where the amplitude is weakly modulated. After some time it develops a particularly strong focusing of wave energy by which a small part of the wave train "breathes" itself up at the expense of the neighborhood.
A "breather" is shown above a three different stages. Notice how the central waves grow by extracting energy from its neighborhood.
The problem with this explanation of freak waves is the following: To get this pure effect one must start with a periodic wave. If one starts out with waves of various periods (or lengths) and with various directions, the picture becomes much more complicated and unpredictable.
Our work has been funded by the Research Council of Norway (NFR 139177/431, NFR 109328/410), the European Union (EVG1-CT-2001-00051, MAS3-CT96-5016), Norsk Hydro and Statoil. The time series for the freak wave at the Draupner platform has been made available from Statoil by J. I. Dalane and O. T. Gudmestad.
RCN Strategic University Program: Modelling of currents and waves for sea structures (UiO, UiB, NTNU)