Dynamics of Three-dimensional Nonlinear Internal Waves Over Topography

自然界中产生重力内波的一种常见机制是在地形上流动。这项工作的目的是提高我们对这一产生机制的理论理解，特别是对三维非线性扰动的研究。为此，将发展一个渐近理论来描述在三维地形上的近均匀分层流动产生的有限幅度扰动，这种三维地形在展向上比在流向上更细长。提出的理论使探索完全非线性波的某些特征成为可能-例如，波浪破碎如何受到三维效应的影响-尽管具有主要的地球物理兴趣，但到目前为止还没有得到理论上的处理。此外，这里采用的数学方法不同于以往的研究，也有望从数学的角度推进我们对非线性内波传播的理解。内重力波是海洋、湖泊和大气中的常见特征。它们的存在是由于温度的变化以及盐度(对于海洋)和压力(对于大气)的变化而在这些自然流体体内存在的密度分层。众所周知，地球物理内波具有巨大的尺度，从公里到数百公里，因此它们是决定当地天气模式和气候动力学的重要因素也就不足为奇了。这项拟议的工作将提供一种数学描述，描述在地形上流动产生的内波，模拟风在大气中的山脉上的作用或洋流在海底基床上的流动。重点放在特定的流动条件下，在这种条件下，内波的产生最剧烈，预计将在地球物理应用中发挥重要作用(例如，雷暴的发展)。

0072145AkylasA common mechanism by which internal gravity waves are generated in nature is flow over topography. The goal of the proposed work is to improve our theoretical understanding of this generation mechanism with particular emphasis on three-dimensional nonlinear disturbances. To this end, an asymptotic theory will be developed to describe the generation of finite-amplitude disturbances by nearly uniformly stratified flow over three-dimensional topography that is more elongated in the spanwise than in the streamwise direction. The proposed theory makes it possible to explore certain features of fully nonlinear waves --- for instance, how wave breaking is influenced by three-dimensional effects --- that, although of primary geophysical interest, so far have defied theoretical treatment. Moreover, the mathematical approach taken here is distinct from prior studies and promises to advance our understanding of nonlinear internal wave propagation from a mathematical standpoint as well.Internal gravity waves are common features in oceans, lakes and in the atmosphere. They owe their existence to the density stratification present in these natural fluid bodies due to variations in temperature and changes in the salinity (in the case of oceans) and pressure (in the case of the atmosphere). Geophysical internal waves are known to possess enormous scales, on the order of kilometers to hundreds of kilometers, so it is not surprising that they are important factors in determining local weather patterns and climate dynamics. The proposed work will provide a mathematical description of the generation of internal waves by flow over topography, modeling the action of wind over mountain ranges in the atmosphere or the flow of currents over a sill in the ocean bottom. Emphasis is placed on particular flow conditions under which internal-wave generation is most violent and is expected to play an important part in geophysical applications (e.g., the development of thunderstorms).