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

尽管海洋非常复杂，但内波的光谱能量密度被认为是普遍的，由Garrett-Munk谱（1972）给出。人们普遍认为，这种内波的频谱能量密度是非线性波-波相互作用的结果，波湍流理论描述了这种系统中的频谱能量传递。波浪湍流极大地促进了我们对大气和海洋中光谱能量转移的理解。然而，问题仍然是为什么内波场频谱具有这样的普遍性。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。首先，将证明相互作用的内波违反了弱非线性共振波-波相互作用的传统波动湍流假设。传统波浪湍流的推广是必要的，包括不一定共振的强非线性波-波相互作用。海洋学家认为，仅考虑波与波之间的相互作用就足以描述内波的频谱能量密度的形成。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本身并不是形成内波光谱能量密度的唯一原因。通过对波与波相互作用的直接数值模拟，可以证明能量趋向于向越来越长的水平波级联。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。重点研究了准地转位涡湍流与内波的耦合关系。这种波涡相互作用将描述内波如何受到影响以及如何影响大尺度海洋涡旋。为了实现这些目标，我们将使用一种已经为内波开发的新型哈密顿结构，并采用作者最近开发的波湍流理论的严格重新表述。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，有多少能量包含在跨越海洋的涌浪中，有多少能量包含在潮汐流中，有多少能量包含在像墨西哥湾流这样的大尺度洋流中，以及这些特征在不同尺度上是如何相互影响的。该奖项支持的工作重点是内波。对于一个简单的桌面插图，人们应该想象一个玻璃容器，它的下半部分装满了水，上半部分装满了油。那么就有可能波浪在水-油界面上晃动，而油表面看起来很平静。在夏季，由于太阳的加热作用，经常在大陆架上方的海洋中观察到不同密度流体之间的界面上的这种波，它们反过来影响天气模式。据观察，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。它还将使人们更好地了解海洋和大气中的过程，从而有助于建立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.