# Modélisation mathématique et analyse des vagues extrêmes / Mathematical modeling and analysis of extreme sea waves

Colloque organisé par Denys Dutykh (L.A.M.A., Université de Savoie, détaché à l’Université College Dublin) et Paul Milewski (University of Winsconsin-Madison, USA) du 8 au 13 octobre 2012

**Participants**

Ricardo Barros, Oliver Bühler, Miguel Bustamante, Wooyoung Choi, Didier Clamond, Frédéric Dias, Angel Duran Martin, Denys Dutykh, Francesco Fedele, Serge Guillas, Christian Kharif, Chiang Mei, Paul Milewski, Marie Hai Yen Nguyen, Themistoklis Stefanakis, Esteban Tabak, Jon Wilkening

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**Abstract**

The following is a summary of a weeklong seminar at the Fondation des Treilles held October 8 – 13, 2012. The subject was Extreme Sea Waves, meaning waves that are unusual, and may lead to material or human losses. The research presentations combined numerical, modelling, stochastic and analytical techniques, mainly focussed on “rogue” waves, tsunamis and internal waves.

The productive exchange of ideas generated new insights and focussed attention on important questions.

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**Compte-rendu**

The workshop covered a wide variety of issues regarding “extreme” waves. Tsunamis and rogue waves were the main types of waves discussed, but there were also presentations and discussions of less commonly studied rare large events, for example breaking internal waves, standing waves in Faraday experiments and flexural waves which model elastic wave propagation under an ice-sheet. The type of work presented ranged from the fundamental modelling aspects of extreme waves to computational methods and challenges.

Tsunamis essentially have four important phases in their life cycle: the generation by landslides or earthquakes, followed by propagation in the deep ocean, amplification due to shoaling past the continental shelf and, finally, runup. In each of these phases research has three fundamental goals: theoretical elucidation of physical issues in idealised problems, modelling and computation of more realistic situations, and, lastly, operational issues such as risk estimation and uncertainty quantification. All of these aspects were covered during presentations and discussions and some fundamental open problems were identified.

Rogue gravity water waves, defined as waves whose height is a multiple (commonly 2 or 3) of the current significant wave height, are now known to exist in a variety of circumstances, but the various mechanisms that give rise to them are still being argued and validated. There appear to be at least four effects that can generate or enhance these sometimes devastating waves: interaction of the swell with currents, strong winds, the modulational instability of uniform gravity wave trains and shoaling. Often, combinations of these effects are necessary to generate rogue waves. The presentations and discussions centered on all of these issues and served to focus attention on research directions.

Most of the presentations can be categorised into (M) mathematical and physical modelling, (N) numerical computation, or (S) statistical or probabilistic methods. We highlight below some of the results presented and open directions of research that were identified. Despite the fact that surface and interfacial waves are a well studied subject, the wealth of complexity and the mathematical difficulties of the problem still spawn fundamental questions in mathematical modelling. Denys Dutykh and Didier Clamond broadened the prospect for new models by introducing “nonstandard” variational approaches to the water wave problem (MN). They posed fundamental questions about the contrast between the exact imposition of commonly used paradigms (for example, irrotationality and impermeability of boundaries) versus various a posteriori approximations such as “long” waves and small amplitude. The discussions were lively and productive and the ideas could be transformative in the field. Ricardo Barros presented three dimensional models of internal waves, discussing the role of vertical and horizontal vorticity (M). The important issues raised still require understanding, particularly as to what are the least restrictive vorticity assumptions in internal interfacial models and whether the resulting evolution preserve these assumptions. Esteban Tabak and Paul Milewski tackled the problem of mixing (MN). Extreme internal waves break, and when they do, the stratified fluid mixes. Conservation-law based models can provide insight in various situations and in some limits provide exact mixing closures. Computational methods for these new conservation laws that do not rely on Riemann problems have to be devised. There are many directions that require further research with the ultimate goal of mathematically predicting breaking, mixing and energy dissipation in continuously stratified flows. Oliver Bühler presented studies that modelled the lifecycle of topographically generated internal waves in the ocean with particular emphasis on the vertical distribution of dissipation (MS). Themis Stefanakis and Fr´ed´eric Dias presented idealised models for the initial generation of waves due to topographical motions (MS). Particularly interesting was the effect of pre-existing topography. The problem is complex and clearly important for better operational predictions. Christian Kharif presented experimental and modelling results on the effect of wind on rogue waves and new modulational models of surface gravity waves in the presence of shear currents (MN). Paul Milewski discussed problems at the interface of full surface wave computations and two-dimensional modulational theories (MN). These studies point to the need for mathematical exploration beyond the Ma-Peregrine one-dimensional nonlinear Schrödinger model for rogue waves. Chiang C. Mei provided modelling directions for the interaction of long waves with arrays of objects, modelling forests, mangroves, etc…(M) These problems are particularly challenging and of crucial importance in tsunami runup calculations. Frédéric Dias, Themistoklis Stefanakis and Denys Dutykh presented results on the amplification of run-up along beaches (MN). The occurrence of resonances leading to large runup was demonstrated via analytical theory, numerical and experimental results. Serge Guillas focussed on statistical models particularly for tsunami problems, based on experience from other geophysical and climate contexts (S). One of the important problems discussed was searching through a high dimensional parameter space for cases of large tsunami runup. Angel Duran presented rigorous results on Petviashvilli-type numerical schemes for computation of solitary waves (N). Jon Wilkening presented results on large amplitude space-time periodic surface wave solutions (N). From the classic deep water gravity standing wave conjecture of Penney and Price to travelling breathers, high accuracy solutions provide a bewildering array of water wave dynamics. Presentations by Marie Nguyen (MN), Miguel Bustamante (M) and Francesco Fedele (MN) focussed on more general geophysical and fluid problems of, respectively, internal lens vortices, Rossby wave triads and transition to the turbulence in axisymmetric pipe flows.

Conclusions and perspectives. Recurring questions raised during the workshop point to future research. (i) Rogue Waves. What are the appropriate envelope models for fully two dimensional ocean surface possibly with the inclusion of currents? What is the role of crossing seas? What can be learned from these methods beyond the dynamic “proof-of-principle” of rogue wave generation? Should individual crest dynamics be included in some way in NLS models? How can probabilistic methods combine with dynamical ones? (ii) Fundamental wave modelling. Are there new classes of models to be derived by relaxing some of the standard assumptions through variational formulations that can explain physical regimes not appropriately addressed by more classical modelling? Can the two approaches be reconciled in some cases? (iii) Internal waves. Can one predict where waves will break in continuously stratified flows and the subsequent momentum transfer and mixing in the fluid? What is the role of vorticity in the fully three dimensional problem? (iv) Tsunamis. What are appropriate rheological models for subaerial landslides? Should one use wave models or “tidal current” models in the nearshore zone? How is vertical vorticity shed from sharp boundaries (whirlpool generation)?

The format of workshops at the Fondation des Treilles and the helpfulness of the support staff created an ideal environment for the free exchange of scientific ideas. The participants and the organisers thank the Foundation for this opportunity.