Hamiltonian formalism in wave turbulence problems

波湍流问题中的哈密顿形式主义

• 批准号: 2307712
• 负责人: Philippe Guyenne
• 金额: $ 22万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307712&HistoricalAwards=false
• 关键词: Hamiltonian; formalism; wave; turbulence; problems

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.

该项目有助于更好地理解与海浪相关的非线性现象。特别令人感兴趣的是两个问题，涉及表面波和底层电流之间或表面波和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。波流相互作用在许多情况下发挥着关键作用，例如汹涌波浪的产生或公海污染物的输送，以及驱动沿海地区海滩侵蚀的沉积物输送机制。所有这些都对与航运、旅游业、渔业或石油业以及沿海基础设施相关的广泛人类活动产生深远影响。波浪结构相互作用的一个重要应用是通过浮动波浪能转换器提取波浪能。海浪作为可再生能源具有巨大潜力，但也带来了许多科学挑战。在广阔的海域上以几何结构放置波浪能转换器阵列的波浪发电场被认为是一个严肃的选择。确定各种波浪条件下的最佳配置对于最大化此类系统的功率吸收至关重要。这项研究为这些耦合现象开发了新的数学模型，迄今为止，这些耦合现象在实际波浪预报中表现不佳，但在气候变化和能源危机的背景下具有很大的相关性。该项目还为研究生的参与和培训提供了机会。所考虑的是在复杂环境中在大范围长度和时间尺度上发生非线性波相互作用的情况，这给渐近分析和数值模拟带来了严重困难。示例包括海浪与涡流相互作用以及海浪与浮动波浪能量转换器阵列相互作用。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选择。然而，在非线性偏微分方程的背景下，此类技术仍然不够先进。研究人员为这种哈密顿形式主义构建了构建模块，其中多个尺度的存在可以自然地适应渐近分析，同时产生保留重要结构特性（例如能量守恒）的近似值。这项研究有助于复杂介质中弱波湍流理论的发展。采用确定性和统计的观点来获得波幅和波谱长期演化的简化非线性模型。推导了与运动不变量相关的这些模型方程的精确平衡解，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 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。