Nonlinear Dispersive Water Waves in Multiscale Interaction Problems

该项目有助于更好地了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背景的问题：(I)面波与内波相互作用；(Ii)面波与粗糙地形相互作用。这两个问题具有重要的物理意义。在情况(I)中，内波是大幅度波，在许多海洋过程中发挥关键作用，如混合和能量耗散，这些过程影响污染物和营养物质的传输和扩散。与之相关的强流也会影响海洋声学，并对近海和水下结构物构成潜在危险。在情况(2)中，海岸波浪动力学的特征已知是非常复杂的，可能导致诸如波浪破裂等极端现象。反过来，这些波浪现象又影响到许多其他海岸过程，如洋流产生和泥沙输送，最终导致沙洲形成和海滩侵蚀。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，可帮助解决从理论到更实际的各种问题(例如，改进内波遥感、波浪预报模型的参数化以及沿海地区波浪能场的设计)。因此，该项目在海洋学、海洋生物学、海岸工程和气候学等最终影响人类活动的领域产生了更广泛的影响，并对解决与目前有相似之处的材料科学等不同领域的问题具有深远的应用。研究生参与了该项目的工作。研究人员研究了波-波和波-底相互作用在非常不同的尺度上发生，这给他们的渐近分析和直接的数值模拟带来了严重的挑战。更具体地说，在情况(I)中，注意长内波与较小的表面波包的共振耦合，在情况(II)中，注意在快速变化的地形上传播的面波。这种相互作用产生了复杂的动力学，如波的局部化和散射，更好地理解具有重要的物理和技术意义。到目前为止，几乎没有人致力于分析这两个问题，尤其是它们仍然缺乏一个数学上合理的渐近理论。研究人员通过推导描述这些基于大尺度分离的非线性色散波系统的基本特征的新的简化模型，为这样的理论发展了基础。为此，他从哈密顿系统和齐次化理论出发，发展了新的分析方法来处理多尺度。特别令人感兴趣的是表面波的调制机制，其中的解表现出两个尺度的依赖，允许快速和缓慢的动力学。后者可以用一个演化偏微分方程组(或一组这样的方程组)来描述，而前者可以由例如共振条件来确定，它们的影响可以通过这个方程中的有效系数来再现。随着光学和成像技术的迅速进步，这种模型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-底相互作用的次网格参数化。研究生都参与了这个项目的工作。

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.