Existence, Stability, and Qualitative Theory of Traveling Water Waves

行进水波的存在性、稳定性和定性理论

• 批准号: 1514910
• 负责人: Samuel Walsh
• 金额: $ 13.74万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1514910&HistoricalAwards=false
• 关键词: Existence; Stability; Qualitative; Theory; Traveling

Water waves are a fundamental subject of study in atmosphere and ocean science. They have been investigated for hundreds of years, yet many of their most basic features remain poorly understood mathematically. For instance, only very recently has significant progress been achieved in developing a rigorous existence theory for steady rotational water waves, that is, waves with vorticity in the bulk of the fluid. These types of waves are ubiquitous in nature: vorticity is generated by incoming currents, the wind blowing over the ocean, or the density stratification due to salinity. A deeper understanding of the mathematics underlying these phenomena would have many scientific and practical applications. For example, traveling waves in density-stratified water (internal waves) are a generic feature of coastal flows in the ocean; they are known to play a major role in driving the circulation of ocean. Vorticity in air is likewise strongly connected to the process of wind generation of water waves, a topic of great importance to ocean and climate modeling as well as ocean engineering. The investigator studies traveling waves in density-stratified water, traveling waves with localized vorticity, and the stability of several important classes of steady waves. He also studies the question of whether stratified waves depend continuously on their density distribution. This question has serious practical implications for how well layered-fluid models can reliably represent flows in the ocean. The specific goals of this project are to (i) expand the existence theory for steady stratified waves and rotational traveling waves; (ii) develop new tools for studying the qualitative properties of internal waves; and (iii) ascertain the stability/instability of several important classes of steady waves. The investigators and colleagues improve on the current existence theory for large-amplitude internal waves by allowing for a general velocity profile in the far field. Concurrently, they study the question of whether stratified waves depend continuously on their density distribution. Confirmation or denial of this fact has serious practical implications: if it is not true, then there is an unbridgeable gap between continuously stratified fluids and the layered-fluid systems that are commonly used to model them. Conversely, if the dependence is smooth, then the modulus of continuity gives bounds on the error arising from the layered approximation. The investigator studies traveling waves with localized vorticity. In earlier work, he and collaborators proved the existence of classes of traveling wave with compactly supported vorticity (including point vortices and vortex patches), but completely unexplored is the intermediate regime where the vorticity is localized but not compact. This important regime is attacked using techniques from bifurcation theory and semi-linear elliptic theory on unbounded domains. Using newly developed tools for determining stability/instability of bound states for non-canonical Hamiltonian systems, they also study the orbital stability of traveling waves with point vortices. Lastly, the problem of wind generation of water waves is considered. This entails studying the stability of small-amplitude steady waves in a two-fluid Euler system, which sheds light on the role of the air-sea interface in the transfer of energy from wind to water.

水波是大气和海洋科学研究的一个基本课题。它们已经被研究了数百年，但它们的许多最基本的特征在数学上仍然很难理解。例如，直到最近才在发展稳定旋转水波的严格存在理论方面取得了重大进展，稳定旋转水波是指在大部分流体中具有涡性的波。这些类型的波浪在自然界中无处不在：涡度是由流入的洋流、吹过海洋的风或由盐度引起的密度分层产生的。深入了解这些现象背后的数学原理将有许多科学和实际应用。例如，密度分层水中的行波（内波）是海洋中沿海流动的一般特征；众所周知，它们在推动海洋环流方面发挥着重要作用。空气中的涡度同样与水波的风力产生过程密切相关，这对海洋和气候建模以及海洋工程都是一个非常重要的话题。研究者研究了密度分层水中的行波、具有局部涡量的行波以及几类重要的定常波的稳定性。他还研究了分层波是否连续依赖于它们的密度分布的问题。这个问题对于层状流体模型如何可靠地代表海洋中的流动具有重要的实际意义。本课题的具体目标是：(1)扩展稳态分层波和旋转行波的存在理论；（ii）开发研究内波定性性质的新工具；(3)确定几类重要的定常波的稳定性/不稳定性。研究人员和同事们通过考虑远场的一般速度分布，改进了目前关于大振幅内波的存在理论。同时，他们研究分层波是否连续依赖于它们的密度分布的问题。确认或否认这一事实具有严重的实际意义：如果这不是真的，那么在连续分层流体和通常用于模拟它们的分层流体系统之间存在不可逾越的鸿沟。相反，如果依赖关系是光滑的，则连续模给出了由分层逼近引起的误差的界限。研究者研究具有局部涡度的行波。在早期的工作中，他和合作者证明了具有紧支撑涡度（包括点涡和涡斑）的行波的存在，但完全未被探索的是涡度局部但不紧致的中间区域。利用无界域上的分岔理论和半线性椭圆理论对这一重要区域进行了研究。使用新开发的工具来确定非正则哈密顿系统束缚态的稳定性/不稳定性，他们还研究了带点涡的行波的轨道稳定性。最后，考虑了水波的风力产生问题。这就需要研究双流体欧拉系统中小振幅稳定波的稳定性，从而揭示海气界面在从风到水的能量传递中的作用。