Mathematical Sciences: Analysis on Waves in Stratified Fluids of Infinite Depth

数学科学：无限深度分层流体中的波分析

• 批准号: 9623060
• 负责人: Shu-Ming Sun
• 金额: $ 5.44万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623060&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analysis; Waves; Stratified

9623060 Sun The purpose of this project is to present a mathematical and numerical study of solitary and periodic waves in stratified fluids of infinite depth subject to the gravitational force. The proposal consists of two parts. The first part intends to give a rigorous answer to an open question whether there exist solitary internal waves with algebraic decay at infinity in a continuously stratified fluid that is bounded only by a rigid bottom and has a constant density except for a layer with continuous density stratification. The proposed research will give a rigorous justification of the existence of solitary waves derived from a formal model equation, called the Benjamin-Ono equation. The second part deals with the existence of periodic waves of large amplitude in a two-layer fluid without boundaries. A new formulation of the problem will be introduced, which transforms the governing equations into a single integral equation. Then the solutions of the integral equation will be studied numerically and theoretically for any density ratio without restrictions on the amplitude of solutions. In this project, the main thrust is to show that the fully nonlinear governing equations for stratified fluids have solutions of finite and large amplitude and give the various properties of the solutions using numerical and theoretical approaches. An interplay of theories in differential equations and functional analysis will be essential to obtaining the rigorous justifications, while numerical computation gives some crucial information on the solutions. %%% The gravity waves of large amplitude in a fluid of density variation, also called a stratified fluid, with great depth are of considerable geophysical interest. Large amplitude internal wave disturbances are common features in the oceans as well as in the lower atmosphere. In particular, a solitary wave, whose form is a localized single hump, and a periodic wave, which always has a same form after a certain distance, are relevant to various oceanic and atmospherical phenomena. The solitary waves of large amplitude have been associated with the formation of tornados in the atmosphere and the enormous transport of momentum and energy within the oceans. This work focuses on obtaining the conditions under which the solitary or periodic waves can exist and predicting how large the amplitude of the waves can be if they exist. It will also try to capture the qualitative features of these waves using numerical computations. The results obtained from this research may provide some theoretical explanation of the formation of these waves and give more understanding of certain wave motions in the oceans and atmosphere so that these waves may be either utilized or avoided in different physical applications. ***

9623060太阳这个项目的目的是介绍无限深度分层流体中孤立和周期性波的数学和数值研究。该提案由两个部分组成。第一部分打算对一个开放的问题给出严格的答案，是否存在孤立的内部波，在无穷大的代数衰减中，在连续分层的流体中仅存在代数衰变，该波动仅由刚性底部限制，并且具有恒定密度，除了具有连续密度分层的层。 拟议的研究将为源自正式模型方程（称为benjamin-ono方程）的孤立波的存在提供严格的理由。第二部分涉及在没有边界的两层流体中存在大幅度的周期性波。将引入问题的新公式，将管理方程转换为单个积分方程。 然后，对于任何密度比，将在数值和理论上研究积分方程的溶液，而无需限制溶液的振幅。在该项目中，主要的推力是表明分层流体的完全非线性管理方程具有有限和大幅度的解决方案，并使用数值和理论方法提供了解决方案的各种特性。 在微分方程和功能分析中的理论相互作用对于获得严格的理由至关重要，而数值计算给出了有关解决方案的一些重要信息。 %% %%在密度变化的流体（也称为分层液体）中，重力幅度很大，深度很大。大幅度内波动扰动是海洋和下部大气中的常见特征。特别是，一个孤立的波是其局部的单个驼峰的形式，而周期性波（在一定距离之后总是具有相同形式）与各种海洋和大气现象有关。 大幅度的孤立波与大气中龙卷风的形成以及海洋中动量和能量的巨大运输有关。这项工作着重于获得孤立波或周期性波的存在的条件，并预测波浪幅度是否存在的大小。 它还将尝试使用数值计算捕获这些波的定性特征。从这项研究中获得的结果可能会对这些波的形成提供一些理论上的解释，并对海洋和大气中的某些波动运动有更多了解，以便可以在不同的物理应用中使用或避免使用这些波。 ***