A nonlinear Schrödinger equation for the envelope of two dimensional surface water waves on finite depth with non-zero constant vorticity is derived, and the influence of this constant vorticity on the well-known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order, we have shown the importance of the nonlinear coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth, respectively.
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December 2012
Research Article|
December 13 2012
A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity
R. Thomas;
R. Thomas
a)
1
Institut de Recherche sur les Phénomènes hors Équilibre
, 49, rue F. Joliot Curie B.P. 146 13384 Marseille Cedex 13, France
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C. Kharif;
C. Kharif
2
École Centrale Marseille
, 38, rue Frédéric Joliot-Curie 13451 Marseille Cedex 20, France
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M. Manna
M. Manna
3
Université de Montpellier II
, Place Eugène Bataillon 34095 Montpellier, France
and CNRS
, Laboratoire Charles Coulomb UMR 5221, F-34095, Montpellier, France
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a)
Electronic mail: thomas@irphe.univ-mrs.fr.
Physics of Fluids 24, 127102 (2012)
Article history
Received:
January 22 2012
Accepted:
November 01 2012
Citation
R. Thomas, C. Kharif, M. Manna; A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity. Physics of Fluids 1 December 2012; 24 (12): 127102. https://doi.org/10.1063/1.4768530
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