Highly Nonlinear Internal Solitary Waves with a Free Surface

A new mathematical model is derived for internal wave propagation in a two-layer fluid domain with a free surface in which the lower layer is of finite depth. The mathematical model consists of three coupled nonlinear equations for displacement of the interface and velocity potentials of the two layers, each of which is expressed as the convex combination of their values at two arbitrary depths. Model equations are correct up to O(?4) terms, where ? is the ratio of harmonic mean of the depths of the two layers to double of a typical wavelength. Finite amplitude internal solitary wave profiles as well as free surface wave profiles are obtained numerically for different wave speeds, density ratios and depth ratios. Interfacial solitary wave profiles obtained here are in very good agreement with recent experimental observations of Zhao et al. (2020). The figures presented here demonstrate how the interfacial wave profile predicted by the free surface model differs from predictions of rigid-lid two-layer fluid models.