Toward a First Principle Understanding of Internal Waves, Eddies, and Their Interactions

尽管海洋非常复杂，但加勒特-蒙克谱(1972)给出的内波的光谱能量密度被认为是普遍存在的。一般认为，这种内波的光谱能量密度是波-波非线性相互作用的结果，而波浪湍流理论描述了这种系统中的光谱能量传递。波浪湍流对我们理解大气和海洋中的光谱能量传输有很大的贡献。然而，问题仍然是，为什么内波场频谱具有这样的普遍性。这项由该奖项资助的研究的目标是从理论上解释这种明显的普遍性。首先，我们将证明弱非线性共振波-波相互作用的传统波浪湍流假设被相互作用的内波所违背。有必要对传统的波浪湍流进行概括，包括不一定共振的强非线性波-波相互作用。海洋学家认为，仅考虑波与波的相互作用就足以描述内波频谱能量密度的形成。事实将证明，这种直觉在很大程度上是正确的。然而，波-波相互作用本身并不是形成内波光谱能量密度的唯一原因。利用波-波相互作用的直接数值模拟，将证明能量倾向于向越来越长的水平波方向级联。此外，还将考虑海洋中的大尺度涡旋及其与内波的相互作用。特别是，将研究准地转位涡湍流和内波的耦合。这种波-涡相互作用将描述内波是如何受到影响的，并影响大尺度的海洋涡旋。为了实现这些目标，将使用已为内波开发的新的哈密顿结构，并将使用由提出者最近开发的对波浪湍流理论的严格重新表述。海洋波场是一个大而复杂的系统。对于这类系统，描述基本动力的能谱往往是极其有用的：表面波纹中包含多少能量，横跨海洋的巨浪包含多少能量，潮流中包含多少能量，墨西哥湾流等大尺度洋流中包含多少能量，以及这些在巨大不同尺度上的特征如何相互影响。该奖项支持的工作重点是内波。对于一个简单的桌面插图，你应该想象一个玻璃容器，它的下半部分装满了水，上半部分是油。那么，波浪会在油水界面四处晃动，而油性表面却显得平静。夏季，由于太阳加热，在大陆架上方的海洋中，经常可以观察到不同密度流体之间的界面上的这种波，它们反过来又影响天气模式。据观察，海洋中的这种内波具有普遍存在的能量谱。这项研究工作将有助于从理论上理解这种普遍性。它还将有助于更好地了解海洋和大气中的过程，从而有助于气候和天气建模和预测的数学模型。

Despite the enormous complexity of the ocean, the spectral energy density of internal waves is believed to be universal, given by the Garrett-Munk spectrum (1972). It has been generally understood that such a spectral energy density of internal waves is the result of nonlinear wave-wave interaction, and the theory of wave turbulence describes the spectral energy transfers in such systems. Wave turbulence has contributed tremendously to our understanding of spectral energy transfers in atmosphere and ocean. However, the question remains why the internal wavefield spectrum possesses such a universality. The goal of the research that is funded by this award is to give a theoretical explanation of this apparent universality. To begin with, it will be demonstrated that the traditional wave turbulence assumptions of weak nonlinear resonant wave-wave interactions are violated by interacting internal waves. A generalization of traditional wave turbulence is necessary, including strongly nonlinear wave-wave interactions that are not necessarily resonant. Oceanographers believe that it is sufficient to consider only wave-wave interactions to describe the formation of the spectral energy density of internal waves. It will be demonstrated that this intuition is correct to a large extend. Yet wave-wave interactions alone are not exclusively responsible for the formation of the spectral energy density of internal waves. Using direct numerical simulation of wave-wave interactions, it will be demonstrated that energy tends to cascade towards longer and longer horizontal waves. Additional, large scale vortices in the ocean and their interactions with internal waves will be considered. In particular, the coupling of the quasi-geostrophical potential vorticity turbulence and of internal waves will be studied. Such wave-vortex interactions will describe how internal waves are influenced and influence large scale oceanic vortices. To achieve these goals, a novel Hamiltonian structure that has been developed for internal waves will be used, and a rigorous reformulation of wave turbulence theory that was developed recently by the proposer will be employed. The oceanic wavefield is a large and complex system. For such systems, it is often extremely useful to decribe the energy spectrum of the underlying dynamics: How much energy is contained in surface ripples, how much in swells travelling across the ocean, how much in tidal flows, how much in large scale currents such as the Gulf Stream, and how do these features at vastly different scales influence each other. The work supported by this award focuses on internal waves. For a simple table-top illustration, one should visualize a glass container whose bottom half is filled with water, with oil in the top half. Then it is possible that waves slosh around at the water-oil interface while the oily surface appears calm. Such waves at interfaces between fluids of different density are often observed in the ocean over the continental shelf during the summer, due to solar heating, and they in turn influence weather patterns. It has been observed that such internal waves in the ocean have an energy spectrum that is universal. This research work will contribute to a theoretical understanding of this universality. It will also lead to a better understanding of processes in the ocean and in the atmosphere, and thus contribute to mathematical models for climate and weather modeling and prediction.