Coupled and scalar asymptotic models for internal waves over variable topography

Abstract

The Green–Naghdi type model in the Camassa–Holm regime derived in [Comm. Pure Appl. Anal. 14(6) (2015) 2203–2230], describe the propagation of medium amplitude internal waves over medium amplitude topography variations. It is fully justified in the sense that it is well-posed, consistent with the full Euler system and converges to the latter with corresponding initial data. In this paper, we generalize this result by constructing a fully justified coupled asymptotic model in a more complex physical case of variable topography. More precisely, we are interested in specific bottoms wavelength of characteristic order λb=λ/α where λ is a characteristic horizontal length (wave-length of the interface). We assume a slowly varying topography with large amplitude (βα=O(μ), where β characterizes the shape of the bottom). In addition, our system permits the full justification of any lower order, well-posed and consistent model. We apply the procedure to scalar models driven by simple unidirectional equations in the Camassa–Holm and long wave regimes and under some restrictions on the topography variations. We also show that wave breaking of solutions to such equations occurs in the Camassa–Holm regime with slow topography variations and for a specific set of parameters.