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Second-order Lagrangian description of tri-dimensional gravity wave interactions

Published online by Cambridge University Press:  30 April 2015

Frédéric Nouguier ,
Bertrand Chapron  and
Charles-Antoine Guérin
Frédéric Nouguier*
Affiliation:
Mediterranean Institute of Oceanography (MIO), UM 110, Université de Toulon, CNRS, Aix-Marseille Université, IRD, 83957 La Garde, France
Bertrand Chapron
Affiliation:
Laboratoire d’Océanographie Spatiale, Ifremer, 29280 Plouzané, France
Charles-Antoine Guérin
Affiliation:
Mediterranean Institute of Oceanography (MIO), UM 110, Université de Toulon, CNRS, Aix-Marseille Université, IRD, 83957 La Garde, France
*
†Email address for correspondence: [email protected]
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Abstract

We revisit and supplement the description of gravity waves based on perturbation expansions in Lagrangian coordinates. A general analytical framework is developed to derive a second-order Lagrangian solution to the motion of arbitrary surface gravity wave fields in a compact and vectorial form. The result is shown to be consistent with the classical second-order Eulerian expansion by Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480) and is used to improve the original derivation by Pierson (1961 Models of random seas based on the Lagrangian equations of motion. Tech. Rep. New York University) for long-crested waves. As demonstrated, the Lagrangian perturbation expansion captures nonlinearities to a higher degree than does the corresponding Eulerian expansion of the same order. At the second order, it can account for complex nonlinear phenomena such as wave-front deformation that we can relate to the initial stage of horseshoe-pattern formation and the Benjamin–Feir modulational instability to shed new light on the origins of these mechanisms.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 165 - 196
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Copyright
© 2015 Cambridge University Press 

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References

Aberg, S. 2007 Wave intensities and slopes in Lagrangian seas. Adv. Appl. Probab. 39 (4), 1020–1035.CrossRefGoogle Scholar
Aberg, S. & Lindgren, G. 2008 Height distribution of stochastic Lagrange ocean waves. Prob. Engng Mech. 23 (4), 359–363.Google Scholar
Annenkov, S. Yu. & Shrira, V. I. 1999 Sporadic wind wave horse-shoe patterns. Nonlinear Process. Geophys. 6 (1), 27–50.CrossRefGoogle Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.CrossRefGoogle Scholar
Buldakov, E. V., Taylor, P. H. & Taylor, R. E. 2006 New asymptotic description of nonlinear water waves in Lagrangian coordinates. J. Fluid Mech. 562 (1), 431–444.CrossRefGoogle Scholar
Clamond, D. 2007 On the Lagrangian description of steady surface gravity waves. J. Fluid Mech. 589, 433–454.CrossRefGoogle Scholar
Collard, F. & Caulliez, G. 1999 Oscillating crescent-shaped water wave patterns. Phys. Fluids 11 (11), 3195–3197.CrossRefGoogle Scholar
Creamer, D. B., Henyey, F., Schult, R. & Wright, J. 1989 Improved linear representation of ocean surface waves. J. Fluid Mech. 205, 135–161.CrossRefGoogle Scholar
Daemrich, K.-F. & Woltering, S.2008 How nonlinear are linear waves? In German Joint Symposium on Hydraulic and Ocean Engineering (JOINT 2008).Google Scholar
Elfouhaily, T., Thompson, D., Vandemark, D. & Chapron, B. 1999 Weakly nonlinear theory and sea state bias estimation. J. Geophys. Res. 104 (C4), 7641–7647.CrossRefGoogle Scholar
Fouques, S., Krogstad, H. E. & Myrhaug, D. 2006 A second order Lagrangian model for irregular ocean waves. J. Offshore Mech. Arctic Engng 128 (3), 177–183.Google Scholar
Fouques, S. & Stansberg, C. T. 2009 A modified linear Lagrangian model for irregular long-crested waves. In ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering, pp. 495–502. American Society of Mechanical Engineers.Google Scholar
Gerstner, F. J. 1809 Theorie der wellen. Ann. Phys. 32, 412–445.CrossRefGoogle Scholar
Gjosund, S. H. 2003 A Lagrangian model for irregular waves and wave kinematics. Trans. ASME 125, 94–102.Google Scholar
Hasselmann, K. 1962 On the nonlinear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12, 481–500.Google Scholar
Hsu, H.-C., Chen, Y.-Y. & Wang, C.-F. 2010 Perturbation analysis of short-crested waves in Lagrangian coordinates. Nonlinear Anal. 11 (3), 1522–1536.CrossRefGoogle Scholar
Hsu, H.-C., Ng, C.-O. & Hwung, H.-H. 2012 A new Lagrangian asymptotic solution for gravity–capillary waves in water of finite depth. J. Fluid Mech. 14 (1), 79–94.Google Scholar
Kenyon, K. E. 1969 Stokes drift for random gravity waves. J. Geophys. Res. 74 (28), 6991–6994.CrossRefGoogle Scholar
Kimmoun, O., Branger, H. & Kharif, C. 1999 On short-crested waves: experimental and analytical investigations. Eur. J. Mech. (B/Fluids) 18 (5), 889–930.Google Scholar
Kinsman, B. 1965 Wind Waves: Their Generation and Propagation on the Ocean Surface. Prentice-Hall.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press/Dover.Google Scholar
Leblanc, S. 2004 Local stability of Gerstner’s waves. J. Fluid Mech. 506, 245–254.CrossRefGoogle Scholar
Lindgren, G. 2006 Slepian models for the stochastic shape of individual Lagrange sea waves. Adv. Appl. Probab. 38 (2), 430–450.CrossRefGoogle Scholar
Lindgren, G. & Aberg, S. 2009 First order Lagrange model for asymmetric ocean waves. J. Offshore Mech. Arctic Engng 131 (3), 031602.Google Scholar
Lindgren, G. & Lindgren, F. 2011 Stochastic asymmetry properties of 3D Gauss–Lagrange ocean waves with directional spreading. Stoch. Models 27 (3), 490–520.Google Scholar
Longuet-Higgins, M. S. 1963 The effects of nonlinearities on statistical distributions in the theory of sea waves. J. Fluid Mech. 17, 459–480.Google Scholar
Longuet-Higgins, M. S. 1986 Eulerian and Lagrangian aspects of surface waves. J. Fluid Mech. 173, 683–707.Google Scholar
Longuet-Higgins, M. S. 1987 Lagrangian moments and mass transport in Stokes waves. J. Fluid Mech. 179, 547–555.CrossRefGoogle Scholar
Naciri, M. & Mei, C. C. 1992 Evolution of a short surface wave on a very long surface wave of finite amplitude. J. Fluid Mech. 235, 415–452.CrossRefGoogle Scholar
Nouguier, F., Guérin, C. A. & Chapron, B. 2009 ‘Choppy wave’ model for nonlinear gravity waves. J. Geophys. Res. 114 (C13), 09012.CrossRefGoogle Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Pierson, W. J.1961 Models of random seas based on the Lagrangian equations of motion. Tech. Rep.. New York Univ., Coll. of Eng. Res. Div., Dept. of Meteorology and Oceanography, prepared for the Office of Naval Research under contract NONR-285(03).CrossRefGoogle Scholar
Shemer, L. 2010 On Benjamin–Feir instability and evolution of a nonlinear wave with finite-amplitude sidebands. Nat. Hazards Earth Syst. Sci. 10 (11), 2421–2427.Google Scholar
Shrira, V. I., Badulin, S. I. & Kharif, C. 1996 A model of water wave ‘horse-shoe’ patterns. J. Fluid Mech. 318, 375–405.CrossRefGoogle Scholar
Socquet-Juglard, H., Dysthe, K., Trulsen, K., Krogstad, H. E. & Liu, J. 2005 Probability distributions of surface gravity waves during spectral changes. J. Fluid Mech. 542, 195–216.Google Scholar
Stoker, J. J. 1957 Water Waves: The Mathematical Theory with Applications. Interscience.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8 (441), 197–229.Google Scholar
Weber, B. L. & Barrick, D. E. 1977 On the nonlinear theory for gravity waves on the ocean’s surface. J. Phys. Oceanogr. 7 (1), 11–21.2.0.CO;2>CrossRefGoogle Scholar
Yakubovich, E. I. & Zenkovich, D. A. 2001 Matrix approach to Lagrangian fluid dynamics. J. Fluid Mech. 443, 167–196.CrossRefGoogle Scholar