Preprint E8/2007
André Nachbin | Ruiz de Zárate, Ailín
Keywords: Internal waves | inhomogeneous media | asymptotic theory
In the context of ocean dynamics, a reduced strongly nonlinear one-dimensional model for the evolution of internal waves over an arbitrary seabottom with submerged structures is derived. The reduced model is aimed at obtaining an efficient numerical method for this two-dimensional problem. Two layers containing inviscid, immiscible, irrotational fluids of different densities are defined. The upper layer is shallow compared with the characteristic wavelength at the interface of the two-fluid system, while the bottom region’s depth is comparable to the characteristic wavelength. The non-linear evolution equations describe the behaviour of the internal wave elevation and mean upper-velocity for this water configuration. The system is a generalization of the one proposed by Choi and Camassa for the flat bottom case in the same physical settings. Due to the presence of topography a variable coefficient accompanies each space derivative. These Boussinesq-type equations contain the Intermediate Long Wave (ILW) equation and the Benjamin-Ono (BO) equation when restricted to the unidirectional wave regime. We intend to use this model to study the interaction of the wave with the bottom profile. The dynamics include wave scattering, dispersion and attenuation among other phenomena. The research is relevant in oil recovery in deep ocean waters, where salt concentration and differences in temperature generate stratification in such a way that internal waves can affect offshore operations and submerged structures. Important properties of the model will be discussed. A hierarchy of onedimensional models is derived from this strongly nonlinear model by considering the different regimes (linear, weakly nonlinear or strongly nonlinear) as well as the flat or corrugated bottom cases. Numerical schemes based on the method of lines for all of them will be described. The numerical results from the Matlab implementations will be shown including periodic topography experiments and solitary waves solutions.