# Water waves: computational approaches for complex problems (13w5069)

Arriving in Banff, Alberta Sunday, June 30 and departing Friday July 5, 2013

## Organizers

Walter Craig (McMaster University)

J. Nathan Kutz (University of Washington)

Paul Milewski (University of Bath)

Andre Nachbin (Instituto Nacional de Matematica Pura e Aplicada)

## Objectives

The water wave problem is a computationally difficult yet crucially important problem in applications. The difficulties arise mostly from its free boundary (the water surface) nature. The boundary is unknown a priori, and the boundary conditions for the fluid equations are nonlinear. The workshop shall focus on the inviscid problem (which is often physically reasonable) and which renders the equations conservative (Hamiltonian). This simplification also poses additional numerical issues, since, ideally, the computational methods should also reflect the original conservation properties of the equations. We have chosen six ``themes" around which the workshop will be organized. The principal objective will be the cross-fertilization of ideas between computational issues within and amongst these themes.

1. Three-dimensional initial value problem. This is perhaps the most ``fundamental" problem in water wave computations. In two-dimensions, there are several methods that can quickly and accurately solve the equations of motion, the most efficient of which are based on either boundary integral methods or conformal mappings. In three-dimensions, the situation is much less clear: most methods are based on reducing the problem to a nonlocal equation for the evolution of the boundary, but issues of stability, accuracy and efficiency abound.

2. Coherent structures in free surface flows. Solitary waves, the simplest of ``coherent structures" in the water wave problem, have by themselves spawned a large literature spanning from engineering to pure mathematics. Beyond the famous Korteweg-de Vries solitary wave, the water wave problem encompasses many more surprising structures: from three-dimensional localized wave-packet solitary waves to possibly travelling periodic localized solutions. The stability of these structures is also important as it affects whether or not the structure will be an observable part of the wave field.

3. The Faraday-droplet quantum mechanics duality. There have been several intriguing experiments recently on the coupled problem between Faraday waves (parametrically excited surface waves arising from the vertical shaking of a container) and a bouncing droplet of fluid which both generates waves and moves, its path directed by them. Most surprisingly this coupled problem exhibits several features of quantum mechanics, even providing a macroscopic analogue of the double slit particle-wave duality diffraction experiment.

4. Waves with complex surface conditions. The simplest surface waves are those subject to the restoring forces of gravity alone. For problems involving smaller scales, surface tension effects are also included. There are however, a large number of more complex effects that can enter into physically realistic modelling. Some examples are surfactants and surface diffusion effects, elastic surface effects to model waves under continuous ice sheets, and waves under the action of electromagnetic fields.

5. Waves over topography. When the wavelength of surface waves is comparable to the depth of the fluid, the waves are affected by the bottom topography. This is what, for example, makes waves refract so that, in most cases, they impinge on beaches head on. Tsunamis are an extreme case: their horizontal length scales are so large that even in the deepest ocean their propagation is controlled by the topography. The interaction of waves with complex bottom topographies is both important and requires special attention in numerical simulations.

6. Extreme waves. Extreme waves, also called ``freak" or ``rogue" waves are unusually large wave events localized in both space and time. They have the potential to be extremely destructive when they collide with ships or other man-made ocean structures (oil wells etc...). Both their theoretical underpinning and their computation is a current subject of intensive research.

We mention three further topics which are not explicitly included above but which pose serious computational challenges and could be also included in the program: (i) wind-wave interaction, (ii) wave breaking, and (iii) sloshing (such as the movement of liquified gas in ocean tankers). These topics add further difficulties - two fluids, turbulence and fluid-structure interaction that render their computations even more complex.

We believe a workshop of this kind is overdue. To our knowledge there has not been a mathematically oriented workshop centred on computation of surface wave phenomena recently (although many other conferences on waves include various elements of the topics discussed in this proposal) and therefore such a workshop would allow for focussed attention on the problem. Furthermore, the proposal fits within the 2013 Mathematics of Planet Earth initiative, given the broad impact that water waves have on both the environment and on human activity.

1. Three-dimensional initial value problem. This is perhaps the most ``fundamental" problem in water wave computations. In two-dimensions, there are several methods that can quickly and accurately solve the equations of motion, the most efficient of which are based on either boundary integral methods or conformal mappings. In three-dimensions, the situation is much less clear: most methods are based on reducing the problem to a nonlocal equation for the evolution of the boundary, but issues of stability, accuracy and efficiency abound.

2. Coherent structures in free surface flows. Solitary waves, the simplest of ``coherent structures" in the water wave problem, have by themselves spawned a large literature spanning from engineering to pure mathematics. Beyond the famous Korteweg-de Vries solitary wave, the water wave problem encompasses many more surprising structures: from three-dimensional localized wave-packet solitary waves to possibly travelling periodic localized solutions. The stability of these structures is also important as it affects whether or not the structure will be an observable part of the wave field.

3. The Faraday-droplet quantum mechanics duality. There have been several intriguing experiments recently on the coupled problem between Faraday waves (parametrically excited surface waves arising from the vertical shaking of a container) and a bouncing droplet of fluid which both generates waves and moves, its path directed by them. Most surprisingly this coupled problem exhibits several features of quantum mechanics, even providing a macroscopic analogue of the double slit particle-wave duality diffraction experiment.

4. Waves with complex surface conditions. The simplest surface waves are those subject to the restoring forces of gravity alone. For problems involving smaller scales, surface tension effects are also included. There are however, a large number of more complex effects that can enter into physically realistic modelling. Some examples are surfactants and surface diffusion effects, elastic surface effects to model waves under continuous ice sheets, and waves under the action of electromagnetic fields.

5. Waves over topography. When the wavelength of surface waves is comparable to the depth of the fluid, the waves are affected by the bottom topography. This is what, for example, makes waves refract so that, in most cases, they impinge on beaches head on. Tsunamis are an extreme case: their horizontal length scales are so large that even in the deepest ocean their propagation is controlled by the topography. The interaction of waves with complex bottom topographies is both important and requires special attention in numerical simulations.

6. Extreme waves. Extreme waves, also called ``freak" or ``rogue" waves are unusually large wave events localized in both space and time. They have the potential to be extremely destructive when they collide with ships or other man-made ocean structures (oil wells etc...). Both their theoretical underpinning and their computation is a current subject of intensive research.

We mention three further topics which are not explicitly included above but which pose serious computational challenges and could be also included in the program: (i) wind-wave interaction, (ii) wave breaking, and (iii) sloshing (such as the movement of liquified gas in ocean tankers). These topics add further difficulties - two fluids, turbulence and fluid-structure interaction that render their computations even more complex.

We believe a workshop of this kind is overdue. To our knowledge there has not been a mathematically oriented workshop centred on computation of surface wave phenomena recently (although many other conferences on waves include various elements of the topics discussed in this proposal) and therefore such a workshop would allow for focussed attention on the problem. Furthermore, the proposal fits within the 2013 Mathematics of Planet Earth initiative, given the broad impact that water waves have on both the environment and on human activity.