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This article is part of a series celebrating the 20th birthday of the Isaac Newton Institute in Cambridge. The Institute is a place where leading mathematicians from around the world can come together for weeks or months at a time to indulge in what they like doing best: thinking about maths and exchanging ideas without the distractions and duties that come with their normal working lives. And as you'll see in our articles, what starts out as abstract mathematics scribbled on the back of a napkin can have a major impact in the real world.
Few things in nature are as dramatic, and potentially dangerous, as ocean waves. The impact they have on our daily lives extends from shipping to the role they play in driving the global climate. From a theoretical viewpoint water waves pose rich challenges: solutions to the equations that describe fluid motion are elusive, and whether they even exist in the most general case is one of the hardest unanswered questions in mathematics. (See Plus article How maths can make you rich and famous to find out more.)
Isaac Newton, in his Principia, was the first to attempt a mathematical theory of water waves, so it was fitting that the 2001 Surface Water Waves programme should be held at the Isaac Newton Institute. The programme was the longest-ever academic meeting to focus entirely on water waves. It brought together 66 researchers, comprising theorists and applied scientists from 17 countries, and was the first to unite researchers from the West with scientists from the former USSR. The result is described by co-organiser Stephen Belcher as an "extremely fruitful exchange of ideas", which inspired both theorists and applied scientists.
Freak waves and hurricanes
Few things in nature are as dramatic, and potentially dangerous, as ocean waves.
Several strands of research initiated at the programme have found sophisticated applications in what's probably the most visible aspect of fluid mechanics: weather and wave forecasting. An example is a method for predicting "freak" waves, a brainchild of Peter Janssen of the European Centre for Medium-Range Weather Forecasts (ECMWF). "During the programme I came up with the idea for a theoretical result, which makes it possible to predict the probability that freak waves will occur," says Janssen. The result is based on a mathematically rigorous understanding of the statistical aspects of wave behaviour. It eventually developed into a working algorithm, which is now used by the ECMWF in predictions issued to European countries to provide guidance for shipping. "Users can contact the main office and one or two hours later they get a routing advice, which includes the information on wave height," says Janssen.
Ocean waves are intimately related to another formidable force of nature: hurricanes. As Belcher explains, "the interaction of hurricanes with ocean waves has become a very hot topic." It's a topic of particular interest to US scientists. "A clear benefit of the programme was that we had several US scientists visiting and this led to some very strong collaborations." One of these collaborations involved Belcher himself, Tetsu Hara from the University of Rhode Island and, later on, the US hurricane specialist Isaac Ginis, also from Rhode Island. "The work we started at the Institute has now been implemented by Ginis in a hurricane forecast model," says Belcher. This model is now in use at the US National Hurricane Center.
The importance of the meeting did not escape the UK’s own forecasting agency, the Met Office. "The Met Office now have a wave forecasting tool, called WaveWatch III," says Belcher. "It's based on a US system, but it's fair to say that the programme was influential in the Met Office seeing the value and benefits of wave forecasting and where it had got to as a science."
Breaking waves and global climate
Waves don't just determine local conditions, they also act right at the boundary between the two great driving forces of global climate – the Earth's atmosphere and its oceans. "One problem that has absolutely exploded in the science world in the last few years is the mixing that occurs in the upper ocean layer," says Belcher.
The Gulf Stream is an Atlantic ocean current that influences European climate. North America is shown in black and dark blue (cold), the Gulf Stream in red (hot). Image: NASA.
This mixing – the dynamics of the ocean surface layer – is an important driver of global climate. It is influenced by waves, in particular by the way they break. "One of the goals of the programme was to understand [wave breaking] more clearly," explains Belcher. "This was an aspect of ocean waves that was wide open at the time." The programme helped refine the measurements that needed to be made out on the ocean to better understand how waves break. "One of the US participants, Ken Melville [from the Scripps Institution of Oceanography, University of California, San Diego], then went out to make these measurements. He has since made fantastic progress in quantifying wave breaking."
The improved understanding of how ocean waves influence global climate has set directions for the future. One major initiative whose roots can be traced back to the programme is the UK National Environment Research Council's recent call for a £3.6 million research project exploring the ocean surface boundary layer. The aim of the project is "to build on recent UK scientific and technological advances to develop a fundamental improvement in weather and climate predictions on timescales from a few days to centuries."
East meets West
In view of our changing climate, it is hard to understate the importance of accurate forecasting and global climate science. But the programme has had another, more subtle, impact, which has helped shape the future of wave science in the UK. The programme was the first to bring together researchers from two schools of thought on waves and turbulence, one that had originated in the US and Europe and one that had its roots in the former USSR. "Ever since the programme the two schools have come together in a much more meaningful way," says Belcher. A result of this closer interaction is that several eminent wave scientists from the former Eastern bloc countries are now working in Britain. "This has had an enormous benefit for UK mathematics."
The academic impact of the programme can be measured by the number of subsequent publications by its participants. These include two recent books on water waves ( and ), as well as a host of papers in prestigious journals including Nature, the Journal of Fluid Mechanics, and the Proceedings of the Royal Society of London A. Participants who were brought together by the programme continue to meet at international research events. One of them, Vladimir Zakharov, was awarded the 2003 Dirac medal for his work on weak turbulence theory in water waves.
The power to attract participants of high academic calibre is an important feature of the Institute. By enabling them to combine their scientific prowess, the Institute continues its famous eponym's legacy, producing results and applications that impact on our lives every day, and possibly for centuries to come.
 Rogue Waves in the Ocean: Observations, Theories and Modelling, C Kharif, E Pelinovsky & A Slunyaev, Springer 2009
 Gravity-capillary free-surface flows, JM Vanden-Broeck, CUP 2010