Fully nonlinear higher-order model equations for long internal waves in a two-fluid system

Abstract

Fully nonlinear model equations, including dispersive effects at one-order higher approximation than in the model of Choi & Camassa (J. Fluid Mech., vol. 396, 1999, pp. 1–36), are derived for long internal waves propagating in two spatial horizontal dimensions in a two-fluid system, where the lower layer is of infinite depth. The model equations consist of two coupled equations for the displacement of the interface and the horizontal velocity of the upper layer at an arbitrary elevation, and they are correct to O(μ2) terms, where μ is the ratio of thickness of the upper-layer fluid to a typical wavelength. For solitary waves propagating in one horizontal direction, the two coupled equations reduce to a single equation for the elevation of the interface. Solitary wave profiles obtained numerically from this equation for different wave speeds are in good agreement with computational results based on Euler's equations. A numerical approach for the propagation of solitary waves is provided in the weakly nonlinear case.