Mathematical Sciences: Nonlinear Partial Differential Equations from Hydrodynamics

数学科学：流体动力学的非线性偏微分方程

• 批准号: 8721771
• 负责人: Robert E Turner
• 金额: $ 5.11万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-05-15 至 1991-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8721771&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Partial; Differential

Work to be done on this project will concentrate on nonlinear problems arising in hydrodynamics. Particular emphasis will be placed on the propagation of waves in stratified media. These waves, arising from density stratification due to changes in salinity or temperature have been investigated analytically for several decades. Much of the internal wave phenomena which has been observed experimentally, remains to be clarified. Mathematical models for stratified wave motion are given by nonlinear partial differential equations for the pseudostream function. Computing this, together with the wave speed, is one of the primary goals of the project. Current work has focused on equations with smooth densities in which there is a gradual density change in an ocean thermocline. In cases where there are abrupt changes, work will proceed on the question of how much streamlines steepen along solution branches. Other work has considered surges in two-fluid systems. No rigorous analytic studies of this type of internal wave have been presented in the literature before. A dynamical systems approach to the corresponding elliptic problem and a center manifold to model the behavior of small waves has now been developed. The question of global behavior of surges is quite open and attention will be given to this issue by using global bifurcation methods. Numerical studies on the multitude of internal wave structures will also be carried out. Initial work was carried out on periodic progressing gravity waves between immiscible fluids. Present investigations are also analyzing a new phenomenon of overhanging waves. In addition, a study is being continued on the limiting behavior of solitary waves in a two-fluid system with upper and lower fixed horizontal walls. This models ocean flows under ice caps. Applications of this research to the understanding of important wave phenomena are manifest.

在这个项目中要做的工作将集中在流体力学中出现的非线性问题。将特别强调波在分层介质中的传播。这些波浪是由于盐度或温度变化引起的密度分层引起的，人们对它们进行了几十年的分析研究。许多已在实验中观察到的内波现象仍有待澄清。利用伪流函数的非线性偏微分方程给出了分层波动的数学模型。计算这一点，连同波速，是该项目的主要目标之一。目前的工作集中在具有平滑密度的方程上，其中在海洋温跃层中存在逐渐的密度变化。在有突然变化的情况下，将继续研究沿解分支的流线变陡多少的问题。其他工作则考虑了双流体系统中的激增。在以前的文献中没有对这种类型的内波进行严格的分析研究。目前已经发展了一种求解相应椭圆问题的动力系统方法和一种模拟小波行为的中心流形。浪涌的全局行为问题是一个非常开放的问题，我们将利用全局分岔方法来研究这个问题。对大量内波结构的数值研究也将进行。初步工作是对不混相流体间的周期性递进重力波进行的。目前的调查也在分析悬挑波的新现象。此外，还对上下固定水平壁双流体系统中孤立波的极限行为进行了进一步的研究。这个模型模拟了冰盖下的海洋流动。这一研究在理解重要波浪现象方面的应用是显而易见的。