High-order unidirectional model with adjusted coefficients for large-amplitude long internal waves

Abstract

To describe large amplitude internal solitary waves in a two-layer system, we consider the high-order unidirectional (HOU) model that extends the Korteweg–de Vries equation with high-order nonlinearity and leading-order nonlinear dispersion. While the original HOU model is valid only for weakly nonlinear waves, its coefficients depending on the depth and density ratios are adjusted such that the adjusted model can represent the main characteristics of large amplitude internal solitary waves, including effective wavelength, wave speed, and maximum wave amplitude. It is shown that the solitary wave solution of the adjusted HOU (aHOU) model agrees well with that of the strongly nonlinear Miyata–Choi–Camassa (MCC) model up to the maximum wave amplitude, which cannot be achieved by the original HOU model. To further validate the aHOU model, numerical solutions of the aHOU model are presented for the propagation and interaction of solitary waves and are shown to compare well with those of the MCC model. The aHOU model is further extended to the case of variable bottom and is solved numerically. In comparison with the MCC model for variable bottom, it is found that the aHOU model is a simple, but reliable theoretical model for large amplitude internal solitary waves, which would be useful for practical applications.