Preprint C107/2010
Estabilidade de um trem de ondas sobre um fundo marinho altamente variável
Ana Maria Luz
Keywords: water waves | Nonlinear Schrödinger Equation | asymptotic theory
We derive a nonlinear Schrödinger equation for the envelope of slowly modulated waves propagating over a large amplitude topography at intermediate depth. To obtain such a model, we consider the Euler equations for inviscid fluids. We perform the asymptotic simplification of the nonlinear potential theory equations by a method of multiple scales resulting in a reduced model. Such model is called reduced because it simplifies the equations of the nonlinear potential theory from two spatial dimensions to a one dimensional model at the free boundary. Regarding the geometry of the problem, we consider a mean-zero large amplitude topography at the bottom and/or the presence of submarine structures together with a free boundary at the surface of the sea. For such a geometry we use curvilinear coordinates. Through a conformal mapping we transform our cartesian system into curvilinear coordinates, namely mapping the original physical domain into a simpler domain (a uniform strip). At this stage we use Matlab's Schwarz-Christoffel Toolbox to help us gain intuition. From the mathematical point of view, and computational simulations, the derived model is simpler than the problem in its original formulation. By means of an efficient model one can get important information in coastal regions and address questions like viability of oil and gas recovery in these regions. The model includes more general topographies than Pihl et al. and Mei and Hancock, that required restrictions on the smoothness of the small amplitude bathymetry. Here we will not require such restrictions. The Schrodinger equation derived contains information about the topography in more than one coefficient, including the cubic nonlinear term. This allowed us to make a study on how the topography affects the focusing/defocusing properties of Stoke's waves.