Nonlinear Dispersive Waves and Applications to Geophysical Fluid Flows

该项目围绕分层流体中内部重力波传播的两个主要主题进行组织：(A)非线性和色散之间的相互作用，以及(B)不稳定性和共振。在第一个主题中，长期目标是提供一个分层分层中的内波模型，该模型正确地解释色散和高度非线性，用于替代目前采用的非色散和弱非线性模型。其目的是保持这些理论相对于完整的欧拉(或纳维-斯托克斯)运动方程的简单性的优势，同时仍然保持描述实际感兴趣的波动动力学的能力。高度非线性的制度往往会产生不稳定。完全非线性内波的这些不稳定性的原因和演化是第二个主题中的焦点。特别是，基于Howard-Miles定理的定常剪切流的稳定性判据和在稳定的Richardson数下发生的参数不稳定性将在内重力波的背景下重新讨论。这项研究集中在一个具有广泛的大气和海洋应用的波传播领域，因此具有重大的社会意义。例如，上层海洋的风强迫产生的大内波可能对飓风的强度有很强的反馈作用。因此，基于能够准确表示大幅度波及其弥散行为的模型的数值代码可以在飓风预报中发挥重要作用，而相对于目前采用的流体静力代码几乎没有额外的成本。该奖项将支持有助于建立这种改进的计算代码的理论和技术基础的工作。

This project is organized around two leading themes in internal gravity wave propagation in stratified fluids: (a) interplay between nonlinearity and dispersion, and (b) instabilities and resonances. Within the first theme, the long term goal is to provide a model of internal waves in layered stratification that correctly accounts for dispersion and high nonlinearity, to be used in alternative to the currently adopted non-dispersive and weakly nonlinear models. The aim is to retain the advantage of the simplicity of these theories with respect to the full Euler (or Navier-Stokes) governing motion equations, while still maintaining the ability to describe wave dynamics of practical interest. Highly nonlinear regimes often generate instabilities. The causes and evolution of these instabilities for fully nonlinear internal waves are the focus within the second theme. In particular, stability criteria based on the Howard-Miles theorem for stationary shear flows and parametric instability occurring at stable Richardson's number will be revisited in the context of internal gravity waves.This research focuses on an area of wave propagation which has widespread atmospheric and oceanic applications, and, therefore, large societal implications. For instance, large internal waves generated by wind forcing of the upper ocean may have a strong feedback effect on the intensity of hurricanes. Thus, numerical codes based on models capable of representing accurately large amplitude waves and their dispersive behavior (responsible for the spreading of wave energy over increasingly larger regions) can play an important role in hurricane forecasts, at little additional cost with respect to the currently employed hydrostatic codes. The award will support work that will help establish the theoretical and technical foundations of such improved computational codes.