Mathematical Sciences: Nonlinear Partial Differential Equations from Hydrodynamics

点击翻译按钮获取中文摘要

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.