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.

水波是大气科学和海洋科学的一个基本研究课题。 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知之甚少。 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流体中具有涡量的波。 这些类型的波在自然界中无处不在：涡度是由流入的水流、吹过海洋的风或由于盐度引起的密度分层产生的。 对这些现象背后的数学有更深入的理解，将有许多科学和实际应用。 例如，密度分层水中的行波（内波）是海洋中海岸流的一般特征;已知它们在驱动海洋环流中起主要作用。 空气中的涡旋同样与风产生水波的过程密切相关，这是一个对海洋和气候建模以及海洋工程非常重要的主题。 调查研究行波密度分层的水，行波与局部涡量，稳定的几个重要类别的稳定波。 他还研究了分层波是否连续依赖于其密度分布的问题。 这个问题对于分层流体模型如何可靠地表示海洋中的流动具有严重的实际意义。 该项目的具体目标是：㈠扩展稳定分层波和旋转行波的存在理论; ㈡开发研究内波定性性质的新工具; ㈢确定几类重要稳定波的稳定性/不稳定性。 研究人员及其同事通过允许远场的一般速度分布来改进目前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其密度分布的问题。 证实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着不可逾越的鸿沟。 相反，如果相关性是平滑的，则连续模给出了分层近似所产生的误差的界限。 研究人员研究具有局部涡量的行波。 在早期的工作中，他和合作者证明了具有紧支撑涡量的行波类的存在（包括点涡和涡斑），但完全没有探索的是涡量局部化但不紧的中间区域。 这一重要制度的攻击使用的技术从分歧理论和半线性椭圆理论的无界域。 使用新开发的工具来确定非正则哈密顿系统的束缚态的稳定性/不稳定性，他们还研究了点涡行波的轨道稳定性。 最后，考虑了水波的风力发电问题。 这需要研究两流体欧拉系统中小振幅稳态波的稳定性，这揭示了海-气界面在能量从风到水的转移中的作用。