Modelling Rogue Waves

BBC-FreakWave

Rogue wave [image from BBC Horizons, 2002]

There are many eyewitness accounts by mariners of gigantic waves – almost vertical walls of water towering over ocean-going ships – that appear from nowhere and do great damage, sometimes destroying large vessels completely. Oceanographers, who have had no way of explaining these ‘rogue waves’, have in the past been dismissive of these reports [TM090, or search for “thatsmaths” at irishtimes.com].

Rogue Waves Measured

But now these enormous mountains of water have been observed and measured, yielding unambiguous instrumental records. In 1985 the Fastnet Lighthouse off Ireland’s south coast was struck by a wave reported to be 48 metres, about the height of a 14 storey building. The first scientifically verified rogue wave was recorded at the Draupner oil platform in the North Sea on New Year’s Day, 1995. It was about 26 metres high. And in January 2014, a rogue wave of over 29 metres was measured off Killard Point in Co. Clare in water 40 metres deep.

Draupner-Wave

Wikimedia Commons, teh free media repository

Oceanographers describe the sea state with a statistical measure called the significant wave height. This is the average height of the one-third highest waves. A rogue wave is one that is more than double the significant wave height. Rogue waves are sometimes called freak waves, but this is misleading: although they are rare and extreme, rogue waves are part of the normal behaviour of the oceans.

Mathematicians and engineers have studied wave motion for centuries using linear models. With such models, two waves can reinforce each other, producing a wave with amplitude double that of the individual components. Wave focusing can occur but eventually the focused waves become nonlinear. Linear models cannot simulate rogue waves realistically.

Nonlinear Models

In nonlinear models, wave components do not simply add up: they interact with each other and can trigger instabilities that produce extreme – if transient – peaks in wave height. The resulting rogue waves can appear suddenly and disappear again just as suddenly. When waves interact nonlinearly, some parts of the wave train can grow at the expense of others, focusing the energy in a sharp peak that is far higher than surrounding crests.

NLS

The nonlinear schrödinger equation

One popular equation for modelling rogue waves is the nonlinear Schrödinger equation, an elaborate analogue of the equation underlying quantum mechanics. With nonlinear models, oceanographers can simulate the behaviour of the ocean realistically, and predict the risk of heavy seas. This allows mariners to avoid dangerous areas when planning their voyages. Such route planning can result in savings of thousands of Euros per day in fuel consumption for a single vessel.

Rogue waves appear to have more than a single cause, depending on the initial sea state, wind speed and direction, water depth, continental shelving and other factors. Their precise origin remains a mystery and they are a subject of active research. Following the Killard wave, a research project, led by Prof Frédéric Dias of UCD, took place in early 2015 to measure extreme waves off the west coast of Ireland. A large database was assembled and several extreme waves were detected. Analysis of this data suggests that the standard explanation of rogue waves in terms of what is called modulational instability is probably invalid.

Our understanding of ocean waves continues to grow through fundamental research into the processes governing the ocean surface. As our knowledge improves, so will the accuracy of wave predictions, resulting in saving of money and of lives.


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