Nonlinear Dispersive Water Waves in Multiscale Interaction Problems

这个项目有助于更好地理解与水波有关的各种现象。特别令人感兴趣的是两个超越均匀介质经典设置的问题：(i)表面波与内波的相互作用和（ii）表面波与粗糙地形的相互作用。这两个问题在物理上很重要。在情形(i)中，内波是振幅较大的波，在许多海洋过程中起关键作用，如混合和能量耗散，影响污染物和营养物的输送和扩散。与之相关的强流也会影响海洋声学，并对近海和水下建筑构成潜在危害。在第（ii）种情况下，海岸波浪动力学的特征是非常复杂的，并可能导致极端现象，如波浪破碎。反过来，这些波浪现象影响许多其他海岸过程，如水流的产生和沉积物的输送，最终驱动沙洲的形成和海滩的侵蚀。由于所涉及的物理机制的复杂性，对这两个问题的详细描述需要相当大的数学挑战。该项目开发了分析和数值工具，可以帮助解决广泛的问题，从理论到更实际的问题（例如，改善内波的遥感，波浪预报模型的参数化，以及沿海地区波浪能农场的设计）。因此，该项目在海洋学、海洋生物学、海岸工程和气候学等领域具有更广泛的影响，最终影响人类活动，并且在材料科学等与当前领域有相似之处的不同领域的波浪问题也具有深远的应用。研究生也参与了该项目的工作。研究者研究了发生在完全不同尺度上的波-波和波-底相互作用，这对它们的渐近分析和直接数值模拟提出了严峻的挑战。更具体地说，在情况(i)中，关注与较小表面波包共振耦合的长内波，以及在情况（ii）中传播的表面波在快速变化的地形上。这种相互作用产生了复杂的动力学，如波的局部化和散射，更好的理解具有重要的物理和技术意义。到目前为止，对这两个问题进行分析研究的努力很少，特别是它们仍然缺乏数学上证明的渐近理论。研究者通过推导新的简化模型来建立这样一个理论的基础，这些模型描述了基于大尺度分离的非线性色散波系统的基本特征。为此，他从哈密顿系统和均质化理论出发，发展了新的多尺度分析方法。特别感兴趣的是表面波的调制制度，其中的解决方案表现出两尺度依赖，允许快速和缓慢的动力学。后者可以用一个进化偏微分方程（或一个这样的方程系统）来描述，而前者可以由共振条件来决定，它们的影响可以通过方程中的有效系数来再现。随着光学和成像技术的迅速发展，这种模式有可能改善内波遥感技术的性能，以及迄今为止在业务预报中表现不佳的波底相互作用的亚网格参数化。研究生也参与了该项目的工作。

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.