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Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation

Part of: Incompressible inviscid fluids Rotating fluids Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  19 July 2018

Benjamin Melinand
Benjamin Melinand*
Affiliation:
Institut de Mathématiques Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405 Talence, France
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Abstract

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

Keywords

water wave equationsasymptotic modelsCoriolis forcingshallow waters

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 76B03: Existence, uniqueness, and regularity theory 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76U05: Rotating fluids
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 6 , December 2018 , pp. 1201 - 1237
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

*

Present address: Department of Mathematics, Indiana University, 831 East 3rd Street, Bloomington, IN 47405, USA (bmelinan@indiana.edu).