Nonlinear Dispersive Water Waves in Multiscale Interaction Problems

University Of Delaware, Newark DE

Investigators

Abstract

This project contributes to a better understanding of various phenomena related to water waves. Of special interest are two problems that go beyond the classical setting of a homogeneous medium: (i) surface waves interacting with internal waves and (ii) surface waves interacting with rough topography. These two problems are of physical importance. In case (i), internal waves are large-amplitude waves that play a key role in many oceanic processes like mixing and energy dissipation, which impact the transport and diffusion of contaminants and nutrients. The strong currents associated with them also affect ocean acoustics, and present a potential hazard to offshore and submerged structures. In case (ii), the character of coastal wave dynamics is known to be very complex and can lead to extreme phenomena such as wave breaking. In turn, these wave phenomena influence many other coastal processes such as current generation and sediment transport which eventually drive sandbar formation and beach erosion. A detailed description of these two problems entails considerable mathematical challenges due to the complexity of the physical mechanisms involved. This project develops analytical and numerical tools that can help address a wide range of questions, ranging from theoretical to more practical ones (e.g., to improve the remote sensing of internal waves, the parameterization of wave forecasting models, and the design of wave energy farms in coastal regions). The project thus has broader impacts in oceanography, marine biology, coastal engineering, and climatology, which ultimately affect human activities, and also has far-reaching applications to wave problems in such diverse areas as material science that share similarities with the present ones. Graduate students are involved in the work of the project. The investigator studies wave-wave and wave-bottom interactions occurring at substantially disparate scales, which poses serious challenges to their asymptotic analysis and direct numerical simulation. More specifically, attention is paid to long internal waves resonantly coupled with smaller surface wavepackets in case (i), and to surface waves propagating over rapidly varying topography in case (ii). Such interactions produce complex dynamics such as wave localization and scattering, and a better understanding has important physical and technological implications. So far little effort has been devoted to examining these two problems analytically and in particular they still lack a mathematically justified asymptotic theory. The investigator develops building blocks for such a theory by deriving new reduced models that describe essential features of these nonlinear dispersive wave systems based on the large separation of scales. For this purpose, he develops new analytical methods from Hamiltonian systems and homogenization theory to deal with the multiple scales. Of special interest is the modulational regime for surface waves, in which the solution exhibits a two-scale dependence allowing for fast and slow dynamics. The latter can be described by an evolutionary partial differential equation (or a system of such equations), while the former can be determined by e.g. resonance conditions and their effects can be reproduced via effective coefficients in this equation. Together with the rapid progress in optical and imaging technologies, such models have the potential to improve the performance of remote sensing techniques for internal waves, and the subgrid parameterization of wave-bottom interactions that have so far been poorly represented in operational forecasting. Graduate students are involved in the work of the project.