Freak waves – using maths to understand weather conditions
When German cargo vessel, the MS München, disappeared beneath the ocean waves on a voyage to the US in 1978 not much was known about rogue waves. Investigations into the sinking of the ship theorised that it was caused by the occurrence of one or more abnormally large and powerful waves – also known as freak waves – that seemed to come out of nowhere and disappear just as quickly.
For decades scientists questioned the existence of rogue waves. They were scientifically improbable and — unlike tsunamis — there did not seem to be any specific underlying cause like an earthquake or volcanic eruption that would displace huge volumes of water.
“It is always difficult to define a rogue wave because nobody can agree on a definition, but it is a very large wave which is both localised in space and time. This means that it will occupy a specific area of, for example, one kilometres squared,” explains Professor Frédéric Dias, an applied mathematician the UCD School of Mathematical Sciences who specialises in ocean waves and hydrodynamics.
“These waves appear to come from nowhere but it depends on the sea state at the time. Usually these waves come in threes but sometimes alone or in a group of four and there is mathematics behind why this is so.”
This means that if you’re unlucky enough to be in a boat in the area of a rogue wave and there is another boat only 5km away, the observers on that boat will not even be able to see the wave. So not only are these waves rare, but seeing one in action is even more uncommon, as they are localised in space and the event will only last a minute or so.
There is, however, the Peregrine soliton: a nonlinear mathematical solution proposed by applied mathematician Howell Peregrine over 25 years ago and which theoretically accounted for the formation of these rogue waves. A “soliton” is a term in mathematics and physics describing a type of wave.
This beautiful yet simple nonlinear Schrödinger equation described a wave that was very large and localised both in space and time. The problem was that this wave had never been observed. Never observed, that is, until now.
Professor Dias is one of the authors of a paper published in Nature Physics that described a breakthrough moment whereby the Peregrine soliton was translated in reality for the very first time in lab conditions. This work, funded through schemes of the French Agence Nationale de la Recherche, the Academy of Finland, the Cyprus Research Promotion Foundation and the Australian Research Council, revealed the rogue wave as seen in the lab.
This area of applied mathematics specialises in ocean waves and hydrodynamics and sees Dias working with physicists in the field of non linear optics.
Interestingly the discovery was made not by observing waves in the ocean or a wave tank in the lab but by carrying out experiments with optic waves.
“A few years ago some researchers observed similar extreme phenomena in optics and the origin of this amplification is the same type of instability as is found in ocean waves,” says Dias.
“Several teams reproduced this experiment including a team I know in France. So then we decided to work together – applied mathematicians and physicists – to try to better understand the other conditions that create this big wave.”
It became really exciting for Dias and his co-researchers because the advantage of optics meant that they could perform experiments relatively easily in a smaller lab space and parameters within the experiments were easier to control.
But what about the mathematics behind these experiments? The Peregrine soliton accounts for a huge wave, which by its very nature is volatile and of a fleeting existence, but importantly the work being carried out by Dias and the other co-authors of the paper means that rogue waves do not have to be unpredictable.
Studying the nature, formation and causation of these waves may lead to accurate forecasting of where they might hit next.
As an applied mathematician Dias does not solely work on research surrounding rogue waves but also on the mathematical modelling and prediction of tsunamis. While there has been much work done on this area not all elements are well understood and the modelling of sediment as a contributory factor to tsunamis is an emerging field showing the vast opportunity in applying advanced maths to the world’s oceans in order to better understand the forces of nature.
“The equations we use to model ocean waves are very complicated. We always look for simpler models and one of these is the nonlinear Schrödinger equation. Essentially this model equation can reproduce some effects in ocean waves, optics and plasma physics as well as some other fields,” he explains.
This equation describes the evolution of wave amplitude as a function of space and time and it turns out that this equation has a very simple solution, which is called the Peregrine soliton.
Beginning at a finite amplitude the wave will amplify over time, give a very high peak and then disappear. Before the Nature Physics paper— although known to mathematicians—the Peregrine soliton was merely a mathematical object. “It had no connection with reality,” explains Dias.
If this mathematical solution is applied to ocean waves it describes not the wave itself but the envelope of the wave, which is the curve that joins the crest of one wave to another. So if the Peregrine soliton is the envelope then beneath this are the oscillations with the central peak being the rogue wave and, as has been observed, these waves often come in three with two smaller ones surrounding the highest wave.
The next step for Dias and his co-researchers is to extrapolate this work in optics to ocean waves. However, as Dias says, there is very little data, as the occurrence of these dangerous rogue waves is – fortunately for seafarers – quite rare, as are the chances of a scientist seeing one in action.