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Pré-Publication, Document De Travail Année : 2017

Whitham approach for the study of nonlinear waves on a vertically sheared current in shallow water

Résumé

Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyper-bolic equations for the surface elevation and the horizontal velocity are derived. Using Riemann invariants of these equations, that are obtained analytically, a closed-form nonlinear evolution equation for the surface elevation is derived. A dispersive term is added to this equation using the exact linear dispersion relation. With this new single first-order partial differential equation, vorticity effects on undular bores are studied. Within the framework of weakly nonlinear waves, a KdV-type equation and a Whitham equation with constant vorticity are derived from this new model and the effect of vorticity on solitary waves and periodic waves is considered. Futhermore, within the framework of the new model and the KdV and Whitham equations a study of the effect of vorticity on the breaking time of dispersive waves and hyperbolic waves as well is carried out.
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Dates et versions

hal-01580419 , version 1 (01-09-2017)

Identifiants

  • HAL Id : hal-01580419 , version 1

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Christian Kharif, Malek Abid. Whitham approach for the study of nonlinear waves on a vertically sheared current in shallow water. 2017. ⟨hal-01580419⟩
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