Study On Interfacial Waves In N-layer Density-stratified Fluids

Interfacial waves, which travel on the interfaces between density- stratified fluids, are frequent wave phenomena inside the ocean and lakes. Investigation on them is very important not only on the phenomena related to themselves but also is helpful for us to understand the three-dimensional ocean internal waves, find out the actions of internal waves on the exchange process of mass, momentum, energy, etc in the ocean, and increase the knowledge about the interactions between internal waves and many other multi-scale ocean waves (for example: internal waves - surface waves, internal waves – current, internal waves – eddy, etc, especially the contribution for large scale processes), ocean engineering, military affairs and so on. On the research of interfacial waves, we usually adopt the two-layer model. But in the real ocean, the density-stratified phenomena sometimes are very complicated. So it is necessary to develop the relative study on the interfacial waves of multilayer density- stratified fluids. In this thesis, based on small amplitude wave theory and the assumption of irrotational, inviscid and incompressible on the water mass, the interfacial waves in N-layer density-stratified fluids are investigated by using a perturbation method in two-dimension and three-dimension. Firstly, the two-dimensional second-order Stokes wave solutions of the interfacial waves are presented without background current. The study shows that the first-order solutions are linear wave solutions, and the second-order solutions are determined by the first-order solutions, the second-order nonlinear modification and the second-order nonlinear interactions between the interfacial waves. The first-order solutions and second-order solutions depend on the density and the depth of each layer. Dispersion relation is a N ?1-order polynomial of angular frequency ω 2, and its solutions are corresponding to the N ?1-movement modes of the interfacial waves. Secondly, the two dimensional second-order asymptotic solutions of the interfacial waves are derived with background current, and the correction of background current to the first and the second Stokes wave solutions are studied. Lastly, the three dimensional second-order random wave solutions of the interfacial waves are deduced without background current. The results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies. They also show that the second-order solutions consist of two parts: the first one is the first-order solutions, and the second one is the solutions of the second- order asymptotic equations which describe the second-order nonlinear modification and the second-order wave-wave interactions among both the wave components on same density interfaces and the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. Some existing results can regard as the particular case of the present work in this thesis.