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

尽管海洋非常复杂，但内波的谱能量密度被认为是普遍的，由Garrett-Munk谱（1972）给出。一般认为，内波的这种谱能量密度是非线性波-波相互作用的结果，并且波湍流理论描述了这种系统中的谱能量传递。波浪湍流对我们理解大气和海洋中的光谱能量传输有着巨大的贡献。然而，问题仍然是为什么内波场谱具有这样的普适性。该奖项资助的研究目标是对这种明显的普遍性给出理论解释。开始，它将被证明，弱非线性共振波-波相互作用的传统波湍流假设是违反相互作用的内波。传统的波湍流的推广是必要的，包括强烈的非线性波-波相互作用，不一定是共振。海洋学家认为，只考虑波-波相互作用就足以描述内波谱能量密度的形成。这将证明这种直觉在很大程度上是正确的。然而，波与波之间的相互作用并不是形成内波能量谱密度的唯一原因。使用波-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级联。此外，海洋中的大尺度涡旋及其与内波的相互作用也将被考虑。特别是，准地转位涡湍流和内波的耦合将被研究。这种波涡相互作用将描述内波如何受到影响，并影响大尺度海洋涡旋。为了实现这些目标，一个新的哈密顿结构，已开发的内波将被使用，和一个严格的重新制定波湍流理论，最近开发的提议者将被采用。海洋波场是一个庞大而复杂的系统。对于这样的系统，描述潜在动力学的能量谱通常是非常有用的：有多少能量包含在表面波纹中，有多少能量存在于穿越海洋的涌浪中，有多少能量存在于潮汐流中，有多少能量存在于大规模的海流中，比如墨西哥湾流，以及这些特征在不同的尺度上如何相互影响。该奖项支持的工作重点是内波。对于一个简单的桌面插图，我们应该想象一个玻璃容器，其下半部分装满水，上半部分是油。那么，当油表面看起来平静时，波浪可能在水-油界面上四处晃动。由于太阳能加热，夏季在大陆架上方的海洋中经常观察到不同密度流体之间界面处的这种波，它们反过来影响天气模式。已经观察到，海洋中的这种内波具有普遍的能谱。这项研究工作将有助于从理论上理解这种普遍性。它还将导致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.