Communications in Mathematical Sciences

Volume 6 (2008)

Number 2

A reduced model for internal waves interacting with topography at intermediate depth

Pages: 385 – 396



A. Nachbin

A. Ruiz de Zarate


A reduced strongly nonlinear model is derived for the evolution of one-dimensional internal waves over an arbitrary bottom topography. Two layers containing inviscid, immiscible, irrotational fluids of different densities are defined. The upper layer is shallow compared with the characteristic wavelength at the interface, while the bottom region's depth is comparable to the wavelength. The nonlinear evolution equations obtained are in terms of the internal wave elevation and the mean upper-velocity for the configuration described. The system is a generalization of the one proposed by Choi and Camassa for the flat bottom case in the same physical settings. Due to the presence of a topography a variable coefficient system of partial differential equations arises. These Boussinesq-type equations contain the Intermediate Long Wave (ILW) equation and the Benjamin- Ono (BO) equation when restricted to the unidirectional propagation regime. We intend to use this model to study the interaction of waves with the bottom profile. The dynamics include wave scattering, dispersion and attenuation, among other phenomena.


internal waves, inhomogeneous media, asymptotic theory

2010 Mathematics Subject Classification

35Q35, 76B07, 76B15, 76B55

Published 1 January 2008