Three-Dimensional Nonlinear Internal Wave Beams: Mathematical Models and Laboratory Experiments

由于浮力的恢复力，密度分层流体支持内部重力波。虽然不像自由表面波那样熟悉，但实际上，内波在海洋、湖泊和大气中无处不在。在这些自然环境中，分层是由温度变化以及（在海洋中）盐度和（在大气中）压力的变化引起的。内波运动是固有的各向异性，因为重力提供了首选方向。结果，能量以波束的形式传输，这是各向同性介质中熟悉的圆形波前的类似物。内波束在地球物理学上具有很大的意义，因为它们构成了海洋内潮的主干；此外，在大气中，它们经常因雷暴而产生。与大多数先前的研究相反，这些研究严格假设二维扰动，该项目关注具有三维变化的内部波束，正如在该领域所期望的那样。动机来自于研究者小组和其他人最近的发现，这些发现表明，内部波束在三维空间的传播与二维空间的传播有着根本的不同。除了对应用数学做出基础贡献外，该项目还有望揭示海洋内潮在动量沉积、物质运输和深海混合中的作用。这些过程影响气候动力学、污染物扩散和养分循环。该项目包括一名研究生。这项研究工作推进了理论建模，以获得对三维非线性内波束的基本理解，特别是伴随的与诱导平均流的共振非线性相互作用，这是三维波束动力学的核心。利用渐近方法，推导了考虑强迫、非均匀背景、耗散和地球自转影响的三维波束传播的非线性演化方程。为了从数学和物理的角度理解梁-平均流非线性相互作用，本文对这些新的数学模型进行了解析和数值分析。该理论研究得到了利用三维流动可视化最新进展的实验室实验的支持。

Density-stratified fluids support internal gravity waves due to the restoring force of buoyancy. Although not as familiar as free-surface waves, internal waves in fact are ubiquitous in oceans, lakes, and the atmosphere. In these natural settings, stratification is caused by temperature variations as well as changes in salinity (in oceans) and pressure (in the atmosphere). Internal wave motion is inherently anisotropic because gravity provides a preferred direction. As a result, energy is transported in the form of wave beams, which are the analogues of the familiar circular wave fronts in isotropic media. Internal wave beams are of great geophysical interest because they form the backbone of the oceanic internal tide; moreover, in the atmosphere, they frequently arise due to thunderstorms. In contrast to most prior studies, which assume strictly two-dimensional disturbances, this project is concerned with internal wave beams featuring three-dimensional variations, as expected in the field. Motivation comes from recent findings, by the investigator's group and others, that suggest that the propagation of internal wave beams in three dimensions differs fundamentally from its two-dimensional counterpart. Apart from making basic contributions to applied mathematics, this project is expected to also shed light on the role of the oceanic internal tide in momentum deposition, material transport, and deep-ocean mixing. These processes affect climate dynamics, pollutant dispersal, and nutrient cycles. A graduate student is included in the project.This research effort advances theoretical modeling to gain a fundamental understanding of three-dimensional nonlinear internal wave beams, particularly the attendant resonant nonlinear interaction with induced mean flows that is central to three-dimensional wave beam dynamics. Using asymptotic methods, nonlinear evolution equations are derived for three-dimensional beam propagation under conditions of geophysical interest, taking into account the effects of forcing, non-uniform background, dissipation, and the Earth's rotation. These new mathematical models are analyzed by analytical and numerical means in order to understand beam-mean-flow nonlinear interactions from a mathematical and a physical perspective. The theoretical study is supported by companion laboratory experiments that exploit recent advances in three-dimensional flow visualization.