Hamiltonian formalism in wave turbulence problems

该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，这两个问题涉及表面波和底层流之间或表面波和浮动结构物之间的复杂相互作用。这些相互耦合的过程仍然没有被很好地理解，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，如在公海产生无赖海浪或污染物的运输，以及促使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、渔业或石油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-结构相互作用的一个重要应用是利用浮式波能转换器提取波能。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。波浪养殖场将波浪能量转换器阵列放置在延伸海域的几何形状中，已被认为是一个重要的选择。在不同的波浪条件下，确定最优配置对于最大化这种系统中的功率吸收是至关重要的。这项研究为这些耦合现象开发了新的数学模型，到目前为止，这些耦合现象在业务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重要意义。该项目也为研究生的参与和培训提供了机会。所考虑的情况是在复杂的环境中发生了大范围长度和时间尺度的非线性波相互作用，这给其渐近分析和数值模拟带来了严重的困难。例如，海浪与涡流相互作用，海浪与浮动波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方法。然而，在非线性偏微分方程组的背景下，这种技术仍然不够先进。研究人员为这种哈密顿形式构造了积木，其中多个尺度的存在可以自然地适应渐近分析，同时产生保持重要结构属性(如能量守恒)的近似。这一研究为发展复杂介质中的弱波湍流理论做出了贡献。采用确定性和数理统计相结合的观点，得到了波幅和波谱的长时间演化的简化非线性模型。这些与运动不变量相关的模型方程的精确平衡解被推导出来，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。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.