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，其动力学是太阳能转换为从局部到全球范围影响我们星球的传输，混合和耗散现象的关键组成部分。这些波的一个重要类别，近孤立的，相干的大振幅结构，最近得到了新的关注与实现，由于仪器的改进和人类活动的增加，这种波是无处不在的，在广泛的地球物理环境中，很容易在现场和实验室实验中观察到。在前一种情况下，收集超出现场传感器合理承受能力的尺度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洋中移动的波涛汹涌的沃茨带。负责这种表面内部波相互作用的机制，以及一些分支的数学框架建模的非线性动力学这类波传播的深入调查，是本研究项目的主要推力。有了精确的模型，它成为可行的解决逆问题的重建完全从表面观测的内部运动状态的海洋，由于大孤立波。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，特别是对世界海洋的大片区域，如南中国海和安达曼海，这些巨浪经常被观察到，并影响到人类的利益，如石油钻探和导航。本课程围绕三个相互重叠的工作来组织，这些工作针对的是本课程中的突出问题：（i）自由表面/大振幅内波耦合动力学的渐近模型的推导，其目标是保持与父系统（如分层欧拉方程）相关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访问性;（二）严格评估这些模型的有效性（一致性、存在时间尺度和接近程度），特别注意与表面张力有关的稳定性问题，（三）在受控实验室环境中对耦合表面/内部波动动力学进行首次实验研究，进一步注意示踪剂的运输和混合。实现这些目标依赖于新的数学技术的应用和实施：（i）新的渐近工具，消除了经典的弱非线性假设将不得不超越最近在这个方向上取得的进展;对于（ii）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，随着对表面张力和内波感应电流下的表面动力学的分析估计的发展;在（iii）中，实施表面波发生器和表面活性剂的组合使用以探索相关的物理参数，例如表面群速度和内部波相速度，包括具有毛细管/重力波相互作用的状态，通过受控实验验证和测试耦合表面-内部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.