The Wave painting by Ivan Aivazovsky
From BBC by Nic Fleming
For centuries sailors told stories of enormous waves tens of metres tall.
They were dismissed as tall tales, but in fact they are alarmingly common
TEN-storey high, near-vertical walls of frothing water.
Smashed portholes and flooded cabins on the upper decks.
Thirty-metre behemoths that rise up from nowhere to throw ships about like corks, only to slip back beneath the depths moments later.
Evocative descriptions of abnormally large "rogue waves" that appear out of the blue have been shared among sailors for centuries.
With little or no hard evidence, and the size of the waves often growing with each telling, there is little surprise that scientists long dismissed them as tall tales.
Until around half a century ago, this scepticism chimed with the scientific evidence.
According to scientists' best understanding of how waves are generated, a 30m wave might be expected once every 30,000 years.
However, we now know that they are no maritime myths.
The most familiar waves occur in water, but there are plenty of other kinds, such as radio waves that travel invisibly through the air.
Although a wave rolling across the Atlantic is not the same as a radio wave, they both work according to the same principles, and the same equations can be used to describe them.
A rogue wave is one that is at least twice the "significant wave height", which refers to the average of the third highest waves in a given period of time.
According to satellite-based measurements, rogue waves do not only exist, they are relatively frequent.
This led scientists to altogether more difficult questions.
Given that they exist, what causes rogue waves?
More importantly for people who work at sea, can they be predicted?
Until the 1990s, scientists' ideas about how waves form at sea were heavily influenced by the work of British mathematician and oceanographer Michael Selwyn Longuet-Higgins. In work published from the 1950s onwards, he stated that, when two or more waves collide, they can combine to create a larger wave through a process called "constructive interference".
According to the principle of "linear superposition", the height of the new wave should simply be the total of the heights of the original waves.
A rogue wave can only form if enough waves come together at the same point according to this view.
However, during the 1960s evidence emerged that things might not be so simple.
The key player was mathematician and physicist Thomas Brooke Benjamin, who studied the dynamics of waves in a long tank of shallow water at the University of Cambridge.
With his student Jim Feir, Benjamin noticed that while waves might start out with constant frequencies and wavelengths, they would change unexpectedly shortly after being generated.
Those with longer wavelengths were catching those with shorter ones.
This meant that a lot of the energy ended up being concentrated in large, short-lived waves.
At first Benjamin and Feir assumed there was a problem with their equipment.
However, the same thing happened when they repeated the experiments in a larger tank at the UK National Physical Laboratory near London.
What's more, other scientists got the same results.
For many years, most scientists believed that this "Benjamin-Feir instability" only occurred in laboratory-generated waves travelling in the same direction: a rather artificial situation.
However, this assumption became increasingly untenable in the face of real-life evidence.
At 3am on 12 December 1978, a German cargo ship called The München sent out a mayday message from the mid-Atlantic.Despite extensive rescue efforts, she vanished never to be found, with the loss of 27 lives.
A lifeboat was recovered.
Despite having been stowed 66ft (20m) above the water line and showing no signs of having been purposefully lowered, the lifeboat seemed to have been hit by an extreme force.
However, what really turned the field upside down was a wave that crashed into the Draupner oil platform off the coast of Norway shortly after 3.20pm on New Year's Day 1995.
Hurricane winds were blowing and 39ft (12m) waves were hitting the rig, so the workers had been ordered indoors.
No-one saw the wave, but it was recorded by a laser-based rangefinder and measured 85ft (26m) from trough to peak.
The significant wave height was 35.4ft (10.8m).
According to existing assumptions, such a wave was possible only once every 10,000 years.
The Draupner giant brought with it a new chapter in the science of giant waves.
When scientists from the European Union's MAXWAVE project analysed 30,000 satellite images covering a three-week period during 2003, they found 10 waves around the globe had reached 25 metres or more.
"Satellite measurements have shown there are many more rogue waves in the oceans than linear theory predicts," says Amin Chabchoub of Aalto University in Finland.
"There must be another mechanism involved."
In the last 20 years or so, researchers like Chabchoub have sought to explain why rogue waves are so much more common than they ought to be.
Instead of being linear, as Longuet-Higgins had argued, they propose that rogue waves are an example of a non-linear system.
A non-linear equation is one in which a change in output is not proportional to the change in input.
If waves interact in a non-linear way, it might not be possible to calculate the height of a new wave by adding the originals together.
Instead, one wave in a group might grow rapidly at the expense of others.
It turns out that certain non-linear version of the Schrödinger equation can be used to help explain rogue wave formation.
The basic idea is that, when waves become unstable, they can grow quickly by "stealing" energy from each other.
Researchers have shown that the non-linear Schrödinger equation can explain how statistical models of ocean waves can suddenly grow to extreme heights, through this focusing of energy.
In a 2016 study, Chabchoub applied the same models to more realistic, irregular sea-state data, and found rogue waves could still develop.
"We are now able to generate realistic rogue waves in the laboratory environment, in conditions which are similar to those in the oceans," says Chabchoub.
"Having the design criteria of offshore platforms and ships being based on linear theory is no good if a non-linear system can generate rogue waves they can't cope with."
Still, not everyone is convinced that Chabchoub has found the explanation.
"Chabchoub was examining isolated waves, without allowing for interference with other waves," says optical physicist Günter Steinmeyer of the Max Born Institute in Berlin.
"It's hard to see how such interference can be avoided in real-world oceans."
Instead, Steinmeyer and his colleague Simon Birkholz looked at real-world data from different types of rogue waves.
They looked at wave heights just before the 1995 rogue at the Draupner oil platform, as well as unusually bright flashes in laser beams shot into fibre optic cables, and laser beams that suddenly intensified as they exited a container of gas.
Their aim was to find out whether these rogue waves were at all predictable.
The pair divided their data into short segments of time, and looked for correlations between nearby segments.
In other words, they tried to predict what might happen in one period of time by looking at what happened in the periods immediately before.
They then compared the strengths of these correlations with those they obtained when they randomly shuffled the segments.
The results, which they published in 2015, came as a surprise to Steinmeyer and Birkholz.
It turned out, contrary to their expectations, that the three systems were not equally predictable.
They found oceanic rogue waves were predictable to some degree: the correlations were stronger in the real-life time sequence than in the shuffled ones.
There was also predictability in the anomalies observed in the laser beams in gas, but at a different level, and none in the fibre optic cables.
However, the predictability they found will be little comfort to ship captains who find themselves nervously eyeing the horizon as the winds pick up.
"In principle, it is possible to predict an ocean rogue wave, but our estimate of the reliable forecast time needed is some tens of seconds, perhaps a minute at most," says Steinmeyer.
"Given that two waves in a severe North Sea storm could be separated by 10 seconds, to those who say they can build a useful device collecting data from just one point on a ship or oil platform, I'd say it's already been invented.
It's called a window."
However, others believe we could foresee rogue waves a little further ahead.
The complexity of waves at sea is the result of the winds that create them.
While ocean waves are chaotic in origin, they often organise themselves into packs or groups that stay together.
In 2015 Themis Sapsis and Will Cousins of MIT in Cambridge, Massachusetts, used mathematical models to show how energy can be passed between waves within the same group, potentially leading to the formation of rogue waves.
The following year, they used data from ocean buoys and mathematical modelling to generate an algorithm capable of identifying wave groups likely to form rogues.
Most other attempts to predict rogue waves have attempted to model all the waves in a body of water and how they interact.
This is an extremely complex and slow process, requiring immense computational power.
Instead, Sapsis and Cousins found they could accurately predict the focusing of energy that can cause rogues, using only the measurements of the distance from the first to last waves in a group, and the height of the tallest wave in the pack.
"Instead of looking at individual waves and trying to solve their dynamics, we can use groups of waves and work out which ones will undergo instabilities," says Sapsis.
He thinks his approach could allow for much better predictions.
If the algorithm was combined with data from LIDAR scanning technology, Sapsis says, it could give ships and oil platforms 2-3 minutes of warning before a rogue wave formed.
Others believe the emphasis on waves' ability to catch other waves and steal their energy – which is technically called "modulation instability" – has been a red herring.
"These modulation instability mechanisms have only been tested in laboratory wave tanks in which you focus the energy in one direction," says Francesco Fedele of Georgia Tech in Atlanta.
"There is no such thing as a uni-directional stormy sea.
In real-life, oceans' energy can spread laterally in a broad range of directions."
In a 2016 study, Fedele and his colleagues argued that more straightforward linear explanations can account for rogue waves after all.
They used historic weather forecast data to simulate the spread of energy and ocean surface heights in the run up to the Draupner, Andrea and Killard rogue waves, which struck respectively in 1995, 2007 and 2014.
Their models matched the measurements, but only when they factored in the irregular shapes of ocean waves.
Because of the pull of gravity, real waves have rounded troughs and sharp peaks – unlike the perfectly smooth wave shapes used in many models.
Once this was factored in, interfering waves could gain an extra 15-20% in height, Fedele found.
"When you account for the lack of symmetry between crest and trough, and add it to constructive interference, there is an enhancement of the crest amplitudes that allows you to predict the occurrence observed in the ocean," says Fedele.
What's more, previous estimates of the chances of simple linear interference generating rogue waves only looked at single points in time and space, when in fact ships and oil rigs occupy large areas and are in the water for long periods.
This point was highlighted in a 2016 report from the US National Transportation Safety Board, written by a group overseen by Fedele, into the sinking of an American cargo ship, the SS El Faro, on 1 October 2015, in which 33 people died.
"If you account for the space-time effect properly, then the probability of encountering a rogue wave is larger," Fedele says.
Also in 2016, Steinmeyer proposed that linear interference can explain how often rogue waves are likely to form.
As an alternative approach to the problem, he developed a way to calculate the complexity of ocean surface dynamics at a given location, which he calls the "effective" number of waves.
"Predicting an individual rogue wave event might be hopeless or non-practical, because it requires too much data and computing power.
But what if we could do a forecast in the meteorological sense?" says Steinmeyer.
"Perhaps there are particular weather conditions that we can foresee that are more prone to rogue wave emergence."
Steinmeyer's group found that rogue waves are more likely when low pressure leads to converging winds; when waves heading in different directions cross each other; when the wind changes direction over a wide range; and when certain coastal shapes and subsea topographies push waves together.
They concluded that rogue waves could only occur when these and other factors combined to produce an effective number of waves of 10 or more.
Steinmeyer also downplays the idea that anything other than simple interference is required for rogue wave formation, and agrees that wave shape plays a role.
However, he disagrees with Fedele's view that sharp peaks can have a significant impact on wave height.
"Non-linearities have a role, but it's a minor one," he says.
"Their main role is that ocean waves are not perfect sine waves, but have more spikey crests and depressed troughs.
However, what we calculated for the Draupner wave is that the effect of non-linearities on wave height was in the order of a few tens of centimetres."
In fact, Steinmeyer thinks that Longuet-Higgins had it pretty much right 60 years ago, when he emphasised basic linear interference as the driver of large waves, rogue or otherwise.
But not everyone agrees.
In fact, the argument over exactly why rogue waves form seems set to rumble on for some time.
Part of the issue is that several kinds of scientists are studying them – experimentalists and theoreticians, specialists in optical waves and fluid dynamics – and they have not as yet done a good job of integrating their different approaches.
There is no sign that a consensus is developing.
But it is an important question to solve, because we will only be able to predict these deadly waves when we understand them.
For anyone sitting on an isolated oil rig or ship, watching the swell of the waves under a stormy sky, those few minutes of warning could prove crucial.
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