Nonlinear Analysis of Three-Dimensional Water-Wave Patterns via Exponential Asymptotics

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Three-dimensional water-wave patterns due to moving surface and submerged sources, such as ships and submarines, or due to currents flowing over seamounts and underwater ridges, are familiar from everyday experience. Apart from their aesthetic appeal, these wave phenomena also are fundamental to applications in various scientific fields, including geophysical fluid dynamics, ship hydrodynamics, oceanography, and meteorology. This research project will develop a novel mathematical methodology for analyzing three-dimensional water-wave patterns. In contrast to all previous analytical treatments, the new approach will not be restricted to waves of small steepness and aims to explain theoretically several striking features revealed by recent numerical simulations of water waves induced by currents over uneven ocean bottom topography. The project will also involve graduate students in the research. The generation of steady three-dimensional (3D) water-wave patterns by moving surface and submerged sources is a fundamental topic of fluid dynamics with applications to geophysical fluid modeling, oceanography, and ship hydrodynamics. The usual way of tackling these problems analytically is based on the linearized water-wave equations, assuming infinitesimally small disturbances. Departing from such linear analysis, this project will devise an analytical (asymptotic) technique that will enable nonlinear treatment of steady 3D free-surface gravity wave patterns due to various types of wave sources. Motivation comes from the recent discovery (by the PI's group) of a nonlinear mechanism that controls the amplitude of 2D forced gravity waves for a broad range of flow conditions, limiting severely the validity of linear analysis. This mechanism depends on all orders of the disturbance amplitude (nonlinearity parameter) and can be captured by perturbation expansions that go beyond all orders in the nonlinearity parameter. This project will develop such a beyond-all-orders (exponential asymptotics) technique suitable for 3D wave patterns. This new methodology will be used to advance understanding of nonlinear effects in 3D forced gravity water waves in two distinct flow regimes: (i) waves on water of finite depth due to a stream over fully localized topography or due to a localized pressure patch moving on the free surface; and (ii) deep-water waves due to flow past a point source or doublet submerged at finite depth from the free surface or due to a pressure patch moving on the free surface. These problems model (i) oceanic flows over seamounts and underwater ridges as well as ship wakes in shallow water; and (ii) ship wave patterns on deep water. The asymptotic analysis will aim to explain theoretically several striking nonlinear features revealed by recent numerical simulations of these mathematical models and to shed light on novel nonlinear aspects that as yet have not been captured numerically. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.