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

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.