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A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1

Published online by Cambridge University Press:  03 September 2018

H. C. Hsu ,
C. Kharif Open the ORCID record for C. Kharif [Opens in a new window] ,
M. Abid  and
Y. Y. Chen
H. C. Hsu
Affiliation:
Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung, 801, Taiwan
C. Kharif*
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384, Marseille, France
M. Abid
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384, Marseille, France
Y. Y. Chen
Affiliation:
Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung, 801, Taiwan
*
†Email address for correspondence: kharif@irphe.univ-mrs.fr
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Abstract

A nonlinear Schrödinger equation for the envelope of two-dimensional gravity–capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity–capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703–714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity–capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity–capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 854 , 10 November 2018 , pp. 146 - 163
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Copyright
© 2018 Cambridge University Press 

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