Communications in Mathematical Sciences

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

A. Ruiz de Zarate and A. Nachbin

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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.

Article information

Commun. Math. Sci., Volume 6, Number 2 (2008), 385-396.

First available in Project Euclid: 1 July 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 76B55: Internal waves 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30] 76B07: Free-surface potential flows 35Q35: PDEs in connection with fluid mechanics

internal waves inhomogeneous media asymptotic theory


Ruiz de Zarate, A.; Nachbin, A. A reduced model for internal waves interacting with topography at intermediate depth. Commun. Math. Sci. 6 (2008), no. 2, 385--396.

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