One of the most fruitful and intriguing avenues for developing novel scientific research is through cross-pollination with other fields of study.  This is one of the reasons I’m proud of my excessively liberal arts-focused education, as well as one of the reasons I like reading blogs on diverse subjects outside of my field: interesting ideas can often come from unexpected sources.

An example of this I found a few months ago in Physical Review Letters, in an article entitled, “Freak waves in the linear regime: a microwave study,” by Höhmann, Kuhl, Stöckman, Kaplan and Heller.  Freak waves, also known as rogue waves*, are anomalously large — and deadly — isolated oceanic waves that can shatter and overturn ships, and they have only been acknowledged relatively recently as a genuine and unusual phenomenon, albeit one that is still not completely understood.

Hokusai’s 1832 The Great Wave off Kanagawa, via Wikipedia.   Not necessarily a freak wave, but probably close to what most people would envision one to be.

It was probably inevitable that researchers in optics would become interested in freak waves: broadly speaking, a wave is a wave, and an effect that appears in water waves is likely reproducible in electromagnetic waves.  Experts on oceanic freak waves have even been invited to speak at optics meetings; a session at the 2009 Optical Society of America’s Frontiers in Optics meeting was opened with the invited talk, “Freak Ocean Waves in One and Two Dimensions,” by Peter Janssen and Jean-Raymond Bidlot of the European Center for Medium-Range Weather Forecasts.

In this post I thought I would take a look at the phenomenon of freak waves, the physical origins of said waves, and methods that physicists have used to create electromagnetic versions of them in the laboratory.

First, the natural question: what is the definition of a freak wave?  They may be informally described as isolated waves whose heights are much, much larger than expected for the current sea state.  More technically**, a freak wave is any wave which satisfies the equation,

,

where is the wave height (distance from the trough of the wave to the peak) and is the so-called significant wave height: 4 times the standard deviation of the surface elevation.  Loosely speaking, this defines a rogue wave as a wave twice as tall as the largest one expected for a given sea state.

The curious thing about this definition is that it seems almost a tautology: we define a freak wave as one that is freakishly large!  The definition does not encompass the origin of freak waves; as we will see, the physics of such waves is still not completely understood.

Via Wikipedia, a huge wave approaches a merchant ship in the Bay of Biscay.  Is it a freak wave?  Hard to say without knowing the sea state.

It is interesting to note that there is nothing in traditional wave physics that precludes waves of freakish height.  Because wave amplitudes combine when the individual waves cross paths, there is, in principle, always a chance that a group of waves could come together at the right place and the right time to create a monster.   A statistical analysis using wave physics shows that freak waves should be so unlikely that they would never, ever, be seen; however, modern measuring devices located on ships, oil platforms, and buoys have shown that they occur distressingly often.  This in turn suggested to researchers that freak waves represent a phenomenon unexplained by traditional wave physics.

Freak waves represent one of those rare scientific discoveries that was well-known outside the scientific community before its formal recognition***.  For centuries, mariners had described monster “walls of water” that could appear without warning, even on mostly calm seas.  These walls were often preceded by a deep trough, like a “hole in the sea”.  It was not until the 1960s and 1970s that the scientific community started to investigate such phenomena, and it wasn’t until 1995 that the first non-anecdotal data was taken and recognized†.

On New Year’s Day, 1995, on the Draupner natural gas platform off of the coast of Norway, a laser sensor recorded the following wave event (picture via Wikipedia):

The crest of the Draupner wave was estimated to be 18.5 meters, and the significant wave height was 11.8 meters.  Damage done to the platform confirmed that the event was not a measurement error.

The existence and origin of freak waves is not simply a question of academic interest; between 1969 and 1994 at least 22 supertankers are thought to have been sunk by freak waves, with many associated fatalities**.

Is this a video of a freak wave?  It acts somewhat like one, but again it’s hard to say for certain.

So what is the physics of freak waves?  The phenomenon represents a special challenge for researchers, because of (a) the rarity of the events, (b) the paucity of experimental data, and (c) the inability to study them directly in the lab.

A number of causes have been suggested for freak waves, and we discuss them each in turn below; it is to be noted that any given freak wave may be the result of one or more of these factors; the jury is still out, so to speak, on the importance of each of them.

***

1.  Focusing. When traveling in relatively shallow water, the speed of waves depends on the depth of the water.  This means that the wave fronts of water waves can be refracted or otherwise distorted by an appropriate subsurface structure, including focusing like a lens.  An illustration of the focusing of wavefronts (red) by a lens (gray) that slows the waves passing through it is shown below:

In fact, artificial focusing of waves has long been proposed to increase the efficiency of wave farms, which extract usable energy from ocean waves.

Though it is not expected that there are natural underwater structures that are perfect lenses, subsurface variations could produce “hot spots” of wave activity that dramatically increase the likelihood of unusually high amplitudes.  This explanation cannot account for all freak waves, however, as a significant percentage of them occur in deep water.

2.  Current compression. When automobile traffic enters an area of lower speed limit, the density of cars increases; cars coming from the “faster” zone catch up with cars in the “slower” zone:

Similarly, waves can travel from a region where they are flowing mostly with the current into a region where they are flowing mostly against the current; when the move in the latter region, they slow in speed and bunch together, simultaneously increasing in amplitude (just as the density of cars increases):

An example of an area where this is likely a factor is on the Southern tip of Africa, where the west-going Agulhas Current (AGC) bumps into east-going currents (SAC) in the Atlantic (picture source):

This region has a history of significant ship damage attributed to freak waves.

Because refraction is caused by an appropriate change in wave speed, inhomogeneous currents can also have a lensing effect, drawing waves together to current-induced “hot spots”.

3.  Dispersive effects. In general, waves of different frequencies travel at different speeds, a phenomenon known as dispersion;  I have talked about dispersion in a previous basics post.  Dispersion in water waves provide another mechanism which can bring together different waves to form giants.

Let us suppose that wind and weather conditions are such that short wavelength waves are produced early in a storm, followed by longer wavelength waves later.  Due to dispersion, the short wavelength waves move slower than the long, and the long wavelength waves will catch up to the shorter ones, producing large and potentially freak waves when they combine.  This is roughly illustrated by the following figure:

One of the problems of this hypothesis for me is that it replaces one problem with another.  Instead of asking, “How do freak waves occur?”, we are left asking, “How can we get weather conditions that produce waves of increasing wavelength?”

4.  Nonlinear effects.  The most interesting and likely explanation for the origin of freak waves, and also unfortunately the most difficult to explain, is the presence of nonlinear effects.  We have seen that, due to dispersion, the speed of a wave can depend on frequency; when a wave amplitude becomes large enough, the speed can also depend on the amplitude of the wave.

To see how this could produce a freak wave, we consider a simple nonlinear effect where the speed of the wave is directly proportional to the amplitude.  Suppose we start with a single-humped wave subject to this effect:

The central part of the wave, with the highest amplitude, will move the fastest.  It will overtake the tail of the wave to the right and gobble it up, taking it along for the ride.  After enough time has elapsed, the wave will have gobbled up everything to its right, and take on an extreme profile roughly depicted as follows:

We end up with a “wall of water”!  This simple model is for illustration purposes only, and the actual nonlinear effects believed to be behind freak waves are much more complicated.  The specific nonlinear effect thought to be closest to the freak wave effect is the so-called Benjamin-Feir instability, which I will not endeavor to describe here.

The important thing about nonlinear effects is that they typically require a significantly large amplitude in order to become important.  It seems reasonable to expect that one of the other mechanisms mentioned may serve as a “trigger” by pushing the wave amplitude to a height where nonlinear effects take over.

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So that, as I understand it, is more or less the current understanding of freak waves in water, and brings us back to a discussion of optics.  Freak waves are a tempting target for optical scientists because (a) optics experiments can “shed light” on the physics of the phenomenon in a laboratory setting, something not really possible with actual freak waves, (b) freaks represent a novel nonlinear optical phenomenon.

The first paper investigating the possibility of optical freak waves appeared in 2007 in Nature††, with an emphasis on testing the role of nonlinear effects in their creation.  The authors used an effect called supercontinuum generation, in which narrowband light propagating in a special optical fiber is converted into broadband (multicolor) light through a complicated nonlinear optical process.   The result can make a wonderful optical demonstration: when light of low intensity is passed through the fiber, it comes out unchanged, but when the intensity is increased past a certain threshold, the light suddenly includes all the colors of the rainbow (image source):

The equations that describe light propagation just below the threshold intensity for supercontinuum generation are comparable to those used to model freak waves.  The researchers indeed found anomalously large, freak-like waves in their system, with intensities some 30-40 times the average intensity!

However, this research is not the end of the story, because it is possible that freak waves can be produced by each of the different mechanisms described, or some combination of them.  The research described in the Physical Review Letter mentioned at the beginning of this post investigates whether linear effects alone can produce freak-style events.

The researchers implemented a study using microwaves in a quasi-two-dimensional system.  A collection of metal cones were sandwiched between a pair of metal plates, and a microwave source was placed on one side of this “cone sandwich”.  The top plate has a small detector attached to it, and is free to move horizontally;  this allows one to measure the strength of microwaves as a function of position.

The system of cones may be considered a crude analogy to the possibility of seafloor or current variations in the ocean; these scatterers can bounce microwaves around, and under the right conditions focus them into “hot spots”.

What did the experiments show?  When microwaves were sent into the system from a given direction and at a given frequency, the presence of a number of hot spots in the system was verified, and it appears that extreme wave events occur at these hot spots at a rate that is higher than what is expected according to traditional multiple scattering theory.  When microwaves  were sent into the system from multiple directions and at multiple frequencies, sharp intense microwave “spikes” were observed that appear very similar to freak waves.  The results seem to verify that  linear effects can produce freaks just as nonlinear ones can, or at the very least can act as a trigger for the true nonlinear freaks.

This research clearly still doesn’t mark the end of the story, but rather barely the beginning!  As one can see from this post, studying and understanding freak waves is tricky business.  Hopefully the study of them in the optical regime will result in a better understanding of their oceanic cousins.

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*The alternate title I considered for this post: Going rogue: optical rogue waves in the laboratory.

** A lot of my information on freak waves comes from the article by K. Dysthe, H.E. Krogstad and P. Müller, “Oceanic rogue waves,” Annu. Rev. Fluid Mech. 40 (2008), 287-310.

*** Another example that I can think of are giant squid, which were reported by sailors long before any specimens were acquired and studied by scientists.

† It seems that more spectacular measurements were taken at the Gorm field in the North Sea in 1985, but were not analyzed until much later.

†† D.R. Solli, C. Ropers, P. Koonath and B. Jalali, “Optical rogue waves,” Nature 450 (2007), 1054-1057.

Höhmann, R., Kuhl, U., Stöckmann, H., Kaplan, L., & Heller, E. (2010). Freak Waves in the Linear Regime: A Microwave Study Physical Review Letters, 104 (9) DOI: 10.1103/PhysRevLett.104.093901