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换为传输、混合和耗散现象的关键组成部分，这些现象从局部到全球影响我们的星球。最近，由于仪器的改进和人类活动的增加，这种波的一种重要类别--几乎孤立的相干大振幅结构--得到了新的关注，人们认识到，这种波在广泛的地球物理环境中无处不在，在野外和实验室实验中都很容易观察到。在前一种情况下，收集超出现场传感器合理承受能力范围的数据的主要工具之一是卫星观测内波的表面表现，内波被视为在平静的海面上移动的波带。深入研究这种表面波-内波相互作用的机制，以及在模拟这类波传播的非线性动力学的数学框架中的一些结果，是本研究项目的主要内容。有了精确的模型，就可以解决仅根据表面观测重建由于大孤波而引起的海洋内部运动状态的反问题。与卫星和遥感能力相结合，这将使全球能够绘制这种波动的地图，特别是在世界大片海洋，如南中国和安达曼海，这些大浪经常被观测到，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努力组织起来，这些努力针对这类悬而未决的问题：(1)自由表面/大振幅内波耦合动力学的渐近模型的推导，目的是保持对分层欧拉方程等母系统的足够的定量保真度，同时保持对大规模模拟的分析和数值工具的可及性；(2)严格评估这些模型的有效性(一致性、存在时间尺度和封闭性)，特别注意与表面张力有关的稳定性问题；(3)在表面/内波动力学耦合的受控实验室环境中进行第一次实验调查，并进一步注意示踪剂的传输和混合。实现这些目标依赖于新的数学技术的应用和实施：(I)去除经典弱非线性假设的新的渐近工具必须在这方面的最新进展的基础上加以改进，因为(Ii)经典水波问题的自由面公式中的相应进展必须适应内部分层流体动力学，并发展表面张力和内波诱导流下的表面动力学的解析估计；在(III)中，采用表面波发生器和表面活性剂的联合使用来探索相关的物理参数，如表面群速度和内波相速，包括毛细/重力波相互作用的区域，以对照实验验证和检验表面-内部动力学耦合的理论和数值进展。

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.