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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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