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Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

Published online by Cambridge University Press:  03 June 2019

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

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

JFM classification

Waves/Free-surface Flows: Capillary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 871 , 25 July 2019 , pp. 1028 - 1043
Copyright
© 2019 Cambridge University Press 

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References

Ablowitz, J. M. & Segur, H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92 (4), 691–715.Google Scholar
Akers, B. & Milewski, P. A. 2009 A model equation for wavepacket solitary waves arising from capillary-gravity flows. Stud. Appl. Maths 122, 249–274.Google Scholar
Akers, B. & Milewski, P. A. 2010 Dynamics of three-dimensional gravity–capillary solitary waves in deep water. SIAM J. Appl. Maths 70 (7), 2390–2408.Google Scholar
Akylas, T. R. 1993 Envelope solitons with stationary crests. Phys. Fluids A 5, 789–791.Google Scholar
Boyd, J. P. 1989 Chebyshev and Fourier Spectral Methods. Springer.Google Scholar
Deconinck, B., Pelinovsky, D. E. & Carter, J. D. 2006 Transverse instabilities of deep-water solitary waves. Proc. R. Soc. Lond. A 462, 2039–2061.Google Scholar
Dias, F., Menasce, D. & Vanden-Broeck, J. M. 1996 Numerical study of capillary-gravity solitary waves. Eur. J. Mech. (B/Fluids) 15 (1), 17–36.Google Scholar
Dorio, J. D., Cho, Y., Duncan, J. H. & Akylas, T. R. 2011 Resonantly forced gravity–capillary lumps on deep water. Part 1. Experiments. J. Fluid Mech. 672, 268–287.Google Scholar
Hsu, H.-C., Kharif, C., Abid, M. & Chen, Y. Y. 2018 A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1. J. Fluid Mech. 854, 146–163.Google Scholar
Iooss, G. & Kirchgässner, K. 1990 Bifurcation d’ondes solitaires en présence d’une faible tension superficielle. C. R. Acad. Sci. Paris I 311, 265–268.Google Scholar
Kim, B. 2012 Long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves. J. Engng Maths 74, 19–28.Google Scholar
Kim, B. & Akylas, T. R. 2005 On gravity–capillary lumps. J. Fluid Mech. 540, 337–351.Google Scholar
Kim, B. & Akylas, T. R. 2007 Transverse instability of gravity–capillary solitary waves. J. Engng Maths 58, 167–175.Google Scholar
Lang, S. 1991 Real and Functional Analysis. Springer.Google Scholar
Longuet-Higgins, M. S. 1989 Capillary–gravity waves of solitary type on deep water. J. Fluid Mech. 200, 451–470.Google Scholar
Longuet-Higgins, M. S. 1993 Capillary–gravity waves of solitary type and envelope solitons on deep water. J. Fluid Mech. 252, 703–711.Google Scholar
Longuet-Higgins, M. S. & Zhang, X. 1997 Experiments on capillary-gravity waves of solitary type on deep water. Phys. Fluids 9, 1963–1968.Google Scholar
Masnadi, N. & Duncan, J. H. 2017a The generation of gravity–capillary solitary waves by a pressure source moving at a trans-critical speed. J. Fluid Mech. 810, 448–474.Google Scholar
Masnadi, N. & Duncan, J. H. 2017b Observation of gravity–capillary lump interactions. J. Fluid Mech. 814, R1.Google Scholar
Milewski, P. A. 2005 Three-dimensional localized solitary gravity–capillary waves. Commun. Math. Sci. 3 (1), 89–99.Google Scholar
Milewski, P. A. & Wang, Z. 2014 Transversally periodic gravity–capillary solitary waves. Proc. R. Soc. Lond. A 470, 20130537.Google Scholar
Parau, E. I., Vanden-Broeck, J. M. & Cooker, M. J. 2005 Nonlinear three-dimensional gravity–capillary solitary waves. J. Fluid Mech. 536, 99–105.Google Scholar
Park, B. & Cho, Y. 2016 Experimental observation of gravity–capillary solitary waves generated by a moving air suction. J. Fluid Mech. 808, 168–188.Google Scholar
Rypdal, K. & Rasmussen, J. J. 1989 Stability of solitary structures in the nonlinear Schrödinger equation. Phys. Scr. 40, 192–201.Google Scholar
Saffman, P. G. & Yuen, H. C. 1978 Stability of a plane soliton to infinitesimal two-dimensional perturbations. Phys. Fluids 21, 1450–1451.Google Scholar
Tang, T. 1993 The Hermite spectral method for Gaussian-type functions. SIAM J. Sci. Comput. 14, 594–606.Google Scholar
Vanden-Broeck, J. M. & Dias, F. 1992 Gravity–capillary solitary waves on water of infinite depth and related free surface flows. J. Fluid Mech. 240, 549–557.Google Scholar
Wang, Z. & Milewski, P. A. 2012 Dynamics of gravity–capillary solitary waves in deep water. J. Fluid Mech. 708, 480–501.Google Scholar
Weideman, J. A. & Reddy, S. C. 2000 A matlab differentiation matrix suite. ACM Trans. Math. Softw. (TOMS) 26 (4), 465–519.Google Scholar
Zakharov, V. E. & Rubenchik, A. M. 1974 Instability of wave guides ans solitons in nonlinear media. J. Expl Theor. Phys. 38, 494–500.Google Scholar
Zhang, X. 1995 Capillary-gravity and capillary waves generated in a wind-wave tank: observations and theories. J. Fluid Mech. 289, 51–82.Google Scholar
Zhang, X. 1999 Observations on waveforms of capillary and gravity–capillary waves. Eur. J. Mech. (B/Fluids) 18, 373–388.Google Scholar
Zhang, X. & Cox, C. S. 1994 Measuring the two-dimensional structure of a wavy water surface optically: a surface gradient detector. Exp. Fluids 17, 225–237.Google Scholar