FRG: Collaborative Research: Fully Nonlinear, Three-Dimensional, Surface Water Waves in Arbitrary Depth

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The overall objectives of this work are to develop a thorough understanding of three-dimensional water waves of finite amplitude, and ultimately to develop a practical model to describe these waves efficiently. A model that is both accurate and computationally efficient could have many practical applications. Specific problems to be addressed are: (1) the existence and stability of three-dimensional, doubly-periodic, traveling water-wave patterns, through the full range of depths; (2) the prevalence of hexagonal, rectangular or crescent-shaped waves (or other multiply periodic wave patterns) among ocean waves; (3) the long-wave and modulational descriptions of water waves, and the subsequent stability analyses that are feasible in these cases; (4) the design and implementation of algorithms to make practical use of exact solutions of asymptotic models in shallow and deep water; (5) the relation between the detailed dynamics of three-dimensional, nonlinear waves and some commonly used ocean-wave transport models; and (6) the impact of a detailed local description of nonlinear wave dynamics on these transport models, in the presence of large amplitude nonlinear waves or under conditions of nonlinear wave focusing. These problems will be studied using analysis, computation, asymptotics, and algebraic geometry, involving the full equations and approximate models, all in conjunction with state-of-the-art physical experiments.The destructive force of large-amplitude ocean waves is well known. Large-scale ocean waves have a major impact on the design of ocean-going ships, of off-shore oil platforms, and of other structures in a coastal environment. These waves also impact the scheduling and routing of shipping patterns, and they strongly affect air-sea transport processes. Yet most theoretical models of ocean waves now in use are based on waves of small amplitude. In this investigation we focus on developing a thorough understanding of large-amplitude waves. The ultimate goal is to develop a practical, mathematical model that may be used operationally in the applications listed above. In particular, the investigators plan to build on their recent work in which they have observed certain coherent patterns of large-amplitude waves. They have observed these patterns in laboratory experiments, as solutions to the well-known equations of water waves, and as solutions to other equations that are (more) approximate models of water waves. Their work involves a variety of mathematical and computational tools as well as state-of-the-art laboratory experiments. In the present work the investigators will combine all of their tools to understand and describe these coherent patterns and to use them as the building blocks for a practical model of ocean waves.