A Combined Theoretical and Experimental Approach for Internal Wave Dynamics: Coupling to Free Surface and Instabilities

内波是海洋和大气中密度分层的常见表现形式，其动力学是太阳能转化为从局部到全球范围影响地球的运输、混合和耗散现象的关键组成部分。由于仪器的改进和人类活动的增加，这种波在广泛的地球物理环境中无处不在，并且在野外和实验室实验中都很容易观察到，这些波中有一类重要的波，即几乎孤立的、相干的大振幅结构，最近得到了新的关注。在前一种情况下，收集超出原位传感器合理负担范围的数据的主要工具之一是对内波表面表达的卫星观测，内波被视为在平静的海洋中移动的波涛汹涌的水域。深入研究这种表面-内部波相互作用的机制，以及对这类波传播的非线性动力学建模的数学框架中的一些分支，是本研究项目的主要推力。有了精确的模型，就可以解决仅从海面观测反演大孤立波作用下海洋内部运动状态的逆问题。当与卫星和遥感能力相结合时，这将允许对这种波浪运动进行全球测绘，特别是对于世界海洋的大片区域，如南中国海和安达曼海，这些大波浪经常被观察到，并影响到人类的利益，如石油钻探和航行。这项工作围绕三个相互重叠的工作来组织，针对本课程的突出问题：(i)自由表面/大振幅内波耦合动力学的渐近模型的推导，目标是保持对母系统（如分层欧拉方程）足够的定量保真度，同时保持对大规模模拟的分析和数值工具的可访问性；（ii）严格评估这些模型的有效性（一致性、存在时间尺度和紧密性），特别关注与表面张力相关的稳定性问题；（iii）在受控的实验室环境中实施第一次表面/内波动力学耦合实验研究，进一步关注示踪剂的输送和混合。实现这些目标依赖于新的数学技术的应用和实施：在(i)新的渐近工具中，消除经典的弱非线性假设将不得不在这个方向上的最新进展中得到改进；因为（ii）随着表面张力和内波诱导流作用下表面动力学的分析估计的发展，经典水波问题的自由表面公式的相应进展必须适应于内部分层流体动力学；在（iii）中，结合使用表面波发生器和表面活性剂来探索相关的物理参数，如表面群速度和内波相速度，包括毛细管/重力波相互作用的制度，通过控制实验验证和测试表面-内部耦合动力学的理论和数值进展。

Internal waves are a common manifestation of density stratification in the ocean and atmosphere, and their dynamics is a key component of solar energy conversion into transport, mixing, and dissipation phenomena that affect our planet from local to global scales. An important class of these waves, that of nearly solitary, coherent large amplitude structures, has recently been given renewed attention with the realization, thanks to instrumentation improvements and increased human activity, that such waves are ubiquitous in a wide range of geophysical settings and are easily observed in both field and laboratory experiments. In the former case, one of the main tools for collecting data over scales beyond what in-situ sensors can reasonably afford is the satellite observation of the surface expression of internal waves, seen as bands of choppy waters moving in otherwise calm seas. The in-depth investigation of the mechanisms responsible for this surface-internal wave interaction, as well as some of the ramifications in the mathematical framework for modeling the nonlinear dynamics of this class of wave propagations, is the main thrust of this research project. With accurate models, it becomes feasible to solve the inverse problem of reconstructing solely from surface observations the internal state of motion of the ocean due to large solitary waves. When combined with satellite and remote sensing capabilities, this would allow a global mapping of such wave motion, especially for the large swaths of world ocean, such as the South China and Andaman Seas, where these large waves are frequently observed and affect human interests such as oil-drilling and navigation. The work is organized around three mutually overlapping efforts that target outstanding issues in this class: (i) the derivation of asymptotic models for the coupled dynamics of free surface/large amplitude internal waves, with the goal of maintaining sufficient quantitative fidelity with respect to parent systems such as stratified Euler equations, while maintaining accessibility to analysis and numerical tools for large-scale simulations; (ii) the rigorous assessment of validity of these models (consistency, existence time scales and closeness), with particular attention to stability issues coupled with surface tension, and (iii) the implementation of the first experimental investigations in a controlled lab environment of coupled surface/internal wave dynamics, with further attention to transport and mixing of tracers. Achieving these goals relies on the application as well as the implementation of novel mathematical techniques: in (i) new asymptotic tools removing the classical weak nonlinearity assumption would have to be refined beyond those obtained in recent advancements in this direction; for (ii) the corresponding advances in the free surface formulation of the classical water wave problem have to be adapted to the internal stratified fluid dynamics, with the development of analytical estimates for surface dynamics under surface tension and internal-wave induced currents; in (iii), the combined use of surface wave generators and surfactants is implemented to explore the relevant physical parameters such as surface group velocity and internal wave phase speed, including regimes with capillary/gravity wave interactions, to validate and test the theoretical and numerical advances in coupled surface-internal dynamics with controlled experiments.