Hamiltonian formalism in wave turbulence problems

这个项目有助于更好地理解与海浪有关的非线性现象。特别令人感兴趣的是两个问题，它们涉及表面波与底层水流之间或表面波与浮动结构之间的复杂相互作用。这些耦合过程仍然没有得到很好的理解，并在海洋学和工程学中提出了具有挑战性的问题。波浪-流相互作用在许多情况下都起着关键作用，比如产生异常浪或在公海中运输污染物，以及在沿海地区驱动海滩侵蚀的沉积物运输机制。所有这些都对与航运、旅游、渔业或石油工业以及沿海基础设施有关的广泛的人类活动产生了深远的影响。波浪-结构相互作用的一个重要应用是通过浮动波能转换器提取波浪能。海浪作为一种可再生能源具有巨大的潜力，但也带来了许多科学挑战。波浪农场将波浪能量转换器阵列放置在扩展的海洋区域的几何配置中，被认为是一个严肃的选择。确定在各种波浪条件下的最佳配置对于在这种系统中最大化功率吸收是至关重要的。本研究为这些耦合现象开发了新的数学模型，这些现象迄今为止在业务波浪预报中表现不佳，但在气候变化和能源危机背景下具有很大的相关性。该项目也为研究生的参与和培训提供了机会。考虑了复杂环境中在大长度和时间尺度上发生非线性波相互作用的情况，这给它们的渐近分析和数值模拟带来了严重的困难。例子包括与涡旋流相互作用的海浪和与浮动波能量转换器阵列相互作用的海浪。在这两种情况下，都可以建立哈密顿公式来描述问题，因此哈密顿技术是正确分析问题的理想方法。然而，这些技术在非线性偏微分方程的背景下仍然不够先进。研究者为这种哈密顿形式主义构建了构建块，其中多个尺度的存在可以自然地适应于渐近分析，同时产生近似，保留重要的结构性质，如能量守恒。这一研究有助于复杂介质中弱波湍流理论的发展。采用确定性和统计两种观点，建立了波浪振幅和波谱长期演化的简化非线性模型。导出了这些模型方程与运动不变量相关的精确平衡解，并对更一般的非线性情况进行了数值模拟以补充理论预测。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project contributes to a better understanding of nonlinear phenomena related to ocean waves. Of special interest are two problems that involve complex interactions between surface waves and underlying currents or between surface waves and floating structures. These coupled processes are still not well understood and raise challenging questions in oceanography and engineering. Wave-current interactions play a key role in many circumstances like the generation of rogue waves or the transport of contaminants in the open ocean, as well as the mechanism of sediment transport which drives beach erosion in coastal areas. All these have far-reaching implications for a broad range of human activities related to the shipping, tourism, fishing or oil industries, and coastal infrastructure. An important application of wave-structure interactions is to wave power extraction by floating wave energy converters. Ocean waves have great potential as a source of renewable energy but entail many scientific challenges. Wave farms where arrays of wave energy converters are placed in a geometric configuration over extended maritime areas have been considered as a serious option. Determination of the optimal configuration under various wave conditions is crucial for maximizing power absorption in such a system. This research develops new mathematical models for these coupled phenomena that have so far been poorly represented in operational wave forecasting yet are of great relevance in the context of climate change and energy crisis. This project also provides opportunities for the participation and training of graduate students.Under consideration are situations where nonlinear wave interactions occur over a wide range of length and time scales in a complex environment, which poses serious difficulties for their asymptotic analysis and numerical simulation. Examples include ocean waves interacting with a vortical current and ocean waves interacting with an array of floating wave energy converters. In both cases, a Hamiltonian formulation can be established to describe the problem and therefore Hamiltonian techniques are ideal to properly analyze it. Such techniques however are still not sufficiently advanced in the context of nonlinear partial differential equations. The investigator constructs building blocks for this Hamiltonian formalism where the presence of multiple scales can be naturally accommodated in the asymptotic analysis while producing approximations that preserve important structural properties such as energy conservation. This research contributes to the development of the theory of weak wave turbulence in complex media. Both deterministic and statistical viewpoints are adopted to obtain reduced nonlinear models for the long-time evolution of the wave amplitude and wave spectrum. Exact equilibrium solutions of these model equations associated with invariants of motion are derived and numerical simulations for more general nonlinear cases are performed to complement the theoretical predictions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.