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，也是海洋表面边界层结构的控制因素。当风吹过海洋时，海洋以两种重要的方式做出反应。第一种是粘性剪切应力通过湍流动量的传递传递到海面以下。第二个是，海面获得起伏字符作为压力差的领先和落后的面表面波通过形状阻力传递动量。虽然海洋表面的波动特征对船上的海洋科学家来说是显而易见的，但诸如与表面重力波破碎相关的混合、表面波剪切的不稳定性和表面波耦合等效应？在标准的海洋表面边界层模式中，Stokes漂移和风驱动的切变形成朗缪尔环流被忽略。这些模型中也缺少相对高频的内波的表示，这些内波可能倾向于在混合层下方立即破裂。该项目旨在扩大对海洋表面边界条件和上层海洋混合之间的联系的理解。本项目将开发的高度非线性多层模型和数值及分析工具可用于解决海洋上层的各种物理问题，包括剪切不稳定性和朗缪尔涡旋等相干特征。同样重要的是要强调，热带海洋内部混合系数的微小变化可以对海洋表面温度产生巨大的反馈，因此，其他物理量，包括对流和降水，具有气候学意义。 本计画将提供应用数学研究生与博士后对非线性波及其交互作用的第一原理理解的训练。通过他们积极参与这种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，并提高他们对两个学科之间相互作用的理解。海洋观测和基本的物理考虑指向一个范例，大大增强了传输在高风速，捕获在上层海洋浮力频率转折点，允许非线性平衡过程和相互作用的低频剪切，促进增强内波耗散。这种范式需要考虑比共振分析更复杂的东西。该项目的目的是了解表面重力波的作用，导致内部重力波的非线性激发，并评估内部波混合上层海洋的倾向。提出了三种可能的参数制度。在低风速下，传输倾向于从背景内部重力波（IGW）场到表面重力波（SGW）场。在高风速下，SGW-IGW传输速率的当前理论预测与风速成比例，即对风速的非常敏感的依赖性。传输反向符号，能量从SGW字段传输到IGW字段。表示从低风速到高风速条件的转变的符号变化与大风条件一致。将这种依赖关系外推到烈风力，更不用说飓风力，使非线性理论的有效性失效。一个可能的第三个参数制度与这一理论的崩溃相吻合。拟议的研究包括三个协调的努力。第一个是观测研究，其目的是记录相对于标准混合层方案的上层海洋湍流耗散的垂直结构和SGW-IGW传输速率的估计，并记录高频内波变化与风和波浪条件的关系。三个努力的第二个是开发一个高度非线性模型的多层系统，专注于三层（混合良好的上层，相对较薄的过渡层，和较深的下层）的情况下，没有任何限制的波长尺度，并进行数值研究，调查共振和非共振SGW-IGW相互作用在有限的振幅。内波破碎和过渡层混合的开始问题将得到解决。第三个努力是构建一个自洽的有限振幅的非线性SGW-IGW相互作用的解析描述，使用所提出的分层配方。建议的第三种方法是，通过利用该模型的正则哈密顿结构，研究IGW场与SGW变化的平衡以及这种平衡在有限振幅下如何变化。然后，数值和分析研究将进行交叉验证，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.