Collaborative Research: Nonlinear Interactions between Surface and Internal Gravity Waves in the Ocean

风对海洋表面的作用传递动量，是湍流和波浪的动能来源，是海洋表面边界层结构的控制因素。当风吹过海洋时，海洋会以两种重要的方式做出反应。第一种是粘性剪应力通过湍流动量传递传递到海面以下。二是海面具有起伏特征，表面波前后端面的压差通过形式阻力传递动量。虽然海洋表面的起伏特征对船上的海洋科学家来说是显而易见的，但在标准的海洋表面边界层模式中，忽略了与表面重力波破裂有关的混合、表面波切变的不稳定性以及面波的耦合？S·斯托克斯漂移和风致切变形成朗缪尔环流。这些模型还缺少相对较高频率的内波，这些内波很容易在混合层下方破裂。该项目力求扩大对海洋表面边界条件和上层海洋混合之间的联系的了解。该项目将开发的高度非线性的多层模式以及数值和分析工具可用于研究上层海洋的各种物理问题，包括剪切不稳定性和朗缪尔涡等相干特征。还必须强调的是，热带海洋内部混合系数的微小变化可以对海表面温度产生巨大反馈，从而对具有气候学意义的其他物理量，包括对流和降水产生巨大反馈。这个项目将为应用数学的研究生和博士后提供关于非线性波及其相互作用的第一原理理解方面的培训。通过积极参与这一跨学科合作的研究培训将为他们提供一个独特的机会，以扩大他们在物理海洋学方面的研究经验，并增进他们对这两个学科之间相互作用的了解。海洋观测和基本的物理考虑指向一种模式，即在高风速时大大加强传输，在浮力频率转折点困在上层海洋，允许非线性平衡过程和与促进增强内部波消散的低频切变的相互作用。这种范式需要考虑一些比共振分析更复杂的东西。该项目的目的是了解引起内部重力波非线性激发的表面重力波的作用，并评估混合上层海洋的内波倾向。提出了三种可能的参数制度。在低风速时，传输往往是从背景内重力波(IGW)场到表面重力波(SGW)场。在高风速下，目前对SGW-IGW转移率的理论预测与风速成正比，即对风速非常敏感。传输反符号，能量从SGW场转移到IGW场。标志的变化表示从低风向大风的转变，与大风情况相吻合。将这种相关性外推到大风强度上，更不用说飓风强度了，这使得非线性理论的有效性失效。第三个参数制度可能与这一理论的崩溃不谋而合。拟议的研究包括三项协调一致的努力。第一项是观测性研究，目的是记录相对于标准混合层方案的上层湍流耗散的垂直结构和SGW-IGW转移率的估计，并记录高频内波变率与风浪条件的关系。其中第二项工作是建立一个高度非线性的多层系统模型，主要针对三层(上层混合良好、过渡层较薄、下层较深)的情况，在波长尺度上没有任何限制，并进行数值研究，研究有限振幅下SGW-IGW的共振和非共振相互作用。将讨论内波破裂和过渡层混合的开始问题。第三个工作是使用所提出的分层公式来构造非线性SGW-IGW相互作用的自洽有限振幅解析描述。提出的第三种方法是利用模型的正则哈密顿结构，研究IGW场与SGW变率的平衡以及这种平衡在有限振幅下是如何变化的。然后，对数值研究和分析研究进行交叉验证，并与海洋观测结果进行比较。

The action of the wind on the ocean surface transfers momentum, serves as a source of kinetic energy for both turbulence and waves and is a controlling factor of the structure of the ocean surface boundary layer. When wind blows over the ocean, the ocean responds in two significant ways. The first is that viscous shear stresses are communicated below the sea surface via transfer of turbulent momentum. The second is that the sea surface acquires an undulatory character as the pressure differences across the leading and trailing faces of surface waves transfer momentum via form drag. While the undulatory character of the ocean surface can be painfully obvious to the ocean scientist onboard a ship, effects such as mixing associated with surface gravity wave breaking, instabilities of the surface wave shear, and coupling of the surface wave?s Stokes drift with wind driven shear to form Langmuir circulations are ignored in standard ocean surface boundary layer models. Missing as well from these models is a representation of relatively high frequency internal waves that could be prone to breaking immediately below the mixed layer. This project seeks to broaden understanding of the connectivity between ocean surface boundary conditions and upper ocean mixing. The highly nonlinear multi-layer model and the numerical and analytical tools to be developed in this project could be useful for a wide range of physical problems in the upper ocean including shear instabilities and coherent features such as Langmuir vortices. It is also important to stress that small changes in the interior mixing coefficients in the tropical oceans can have an immense feedback on Sea Surface Temperature and, therefore, other physical quantities, including convection and precipitation, of climatological significance. This project will provide training in first principles understanding of nonlinear waves and their interactions to a graduate student and a post-doc in applied mathematics. Research training through their active participation in this cross-disciplinary collaboration will provide them a unique opportunity to broaden their research experience in physical oceanography and improve their understanding of the interplay between the two disciplines. Ocean observations and basic physical considerations point towards a paradigm of greatly enhanced transfers at high wind speeds, trapping in the upper ocean at buoyancy frequency turning points that allows a nonlinear equilibration process and interaction with lower frequency shear that promotes enhanced internal wave dissipation. This paradigm demands consideration of something more sophisticated than a resonant analysis. The objective of this project is to understand the role of surface gravity waves resulting in the nonlinear excitation of internal gravity waves and assessing the internal waves propensity for mixing the upper ocean. Three possible parameter regimes are proposed. At low wind speeds, transfers tend to be from the background Internal Gravity Wave (IGW) field to the Surface Gravity Wave (SGW) field. At high wind speeds, current theoretical predictions of SGW-IGW transfer rates are proportional to wind speed, i.e. a very sensitive dependence upon wind speed. Transfers reverse sign and energy is transferred from the SGW field to the IGW field. The change in sign denoting the transition from low-wind to high-wind conditions coincides with gale force wind conditions. Extrapolating such dependencies to gale force, let alone hurricane force, invalidates the validity of the nonlinear theory. A likely third parameter regime coincides with the breakdown of this theory. The proposed research consists of three coordinated efforts. The first is an observational study with the objectives of documenting the vertical structure of upper ocean turbulent dissipation relative to standard mixed-layer schemes and estimates of SGW-IGW transfers rates, and documenting the relationship of high frequency internal wave variability to wind and wave conditions. The second of the three efforts is to develop a highly nonlinear model for a multi-layer system, focusing on the three-layer (well-mixed upper, relatively thin transitional, and deep lower layers) case, without any limitations on wavelength scales, and to perform a numerical study, to investigate both resonant and non-resonant SGW-IGW interactions at finite amplitude. Questions of the onset of internal wave breaking and transition layer mixing will be addressed. The third effort is to construct a self-consistent finite amplitude analytic description of nonlinear SGW-IGW interactions using the proposed layered formulation. The proposed third approach is, by taking advantage of the canonical Hamiltonian structure of the model, to investigate the equilibration of the IGW field with SGW variability and how this equilibration changes at finite amplitude. Then, the numerical and analytic studies will be cross-validated and compared with the ocean observations.