Anyone who has spent time at a beach after a big storm knows that large waves are a powerful phenomenon. They change coastlines and generate rip currents that sweep swimmers out to sea. In the open ocean freak waves destroy boats. Surface waves also have a role in determining the Earth's climate: they are the means by which the winds drive much of the ocean circulation, while breaking waves dramatically increase air-sea gas exchange. There are many other types of waves that affect our lives. Breaking internal waves allow deep water to well up in the ocean and drive air currents in the upper atmosphere. The high and low pressure systems that effect our weather are associated with atmospheric Rossby waves. Large breaking Rossby waves play a role in the formation of blocking patterns which result in unusual weather events such as the heat wave that effected Europe in the summer of 2003.
For all these reasons, understanding and modelling breaking waves is important. The behaviour of waves in fluids is governed by a complicated set of nonlinear partial differential equations. Exact analytic solutions are generally impossible to obtain. Instead, numerical models must be used to find approximate solutions. The large range of length scales involved, from the wave length of the dominant wave down to the smallest flow features, make it a computationally demanding problem. Breaking surface waves, such as that shown in the image accompanying this text, present the additional problem of tracking the complex shape of the interface between air and water. Numerical simulations of all the details of even a single breaking wave are currently impossible. A difficult balance must then be struck: modelling enough of the details to enable sufficiently accurate predictions of the phenomenon under study, while not modelling so many details that the computation becomes intractable. An understanding of the mathematical structure of the equations and of the physical phenomena they describe, together with expertise the theory of computational mathematics, is essential to the development of accurate, robust numerical methods for these simulations.