Influence of surface tension and bottom topography on internal waves

Abstract

Following the global strategy introduced recently by Bona, Lannes and Saut in Ref. 7, we derive here in a systematic way, and for a large class of scaling regimes, asymptotic models for the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, the presence of surface tension and with uneven bottoms. The full (Euler) model for this situation is reduced to a system of evolution equations posed spatially on ℝd, d = 1, 2, which involve two nonlocal operators. The different asymptotic models are obtained by expanding the nonlocal operators and the surface tension term with respect to suitable small parameters that depend variously on the amplitude, wavelengths and depth ratio of the two layers. Furthermore, the consistency of these asymptotic systems with the full Euler equations is established. © 2009 World Scientific Publishing Company.