Periodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface waves

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Bridges Thomas J. 1996Periodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface wavesPhil. Trans. R. Soc. A.354533–574http://doi.org/10.1098/rsta.1996.0019SectionRestricted accessArticlePeriodic patterns, linear instability symplectic structure and mean-flow dynamics for three-dimensional surface waves Thomas J. Bridges Google Scholar Find this author on PubMed Search for more papers by this author Thomas J. Bridges Google Scholar Find this author on PubMed Search for more papers by this author Published:15 March 1996https://doi.org/10.1098/rsta.1996.0019AbstractSpace and time periodic waves at the two-dimensional surface of an irrotational inviscid fluid of finite depth are considered. The governing equations are shown to have a new formulation as a generalized Hamiltonian system on a multisymplectic structure where there is a distinct symplectic operator corresponding to each unbounded space direction and time. The wave-generated mean flow in this framework has an interesting characterization as drift along a group orbit. The theory has interesting connections with, and generalizations of, the concepts of action, action flux, pseudofrequency and pseudowavenumber of the Whitham theory. The multisymplectic structure and novel characterization of mean flow lead to a new constrained variational principle for all space and time periodic patterns on the surface of a finite-depth fluid. With the additional structure of the equations, it is possible to give a direct formulation of the linear stability problem for three-dimensional travelling wavesFootnotesThis text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Hu W, Xiao C and Deng Z (2023) Multi-symplectic Method for an Infinite-Dimensional Hamiltonian System Geometric Mechanics and Its Applications, 10.1007/978-981-19-7435-9_3, (89-201), . Clamond D and Dutykh D (2012) Practical use of variational principles for modeling water waves, Physica D: Nonlinear Phenomena, 10.1016/j.physd.2011.09.015, 241:1, (25-36), Online publication date: 1-Jan-2012. Bridges T and Donaldson N (2010) VARIATIONAL PRINCIPLES FOR WATER WAVES FROM THE VIEWPOINT OF A TIME‐DEPENDENT MOVING MESH, Mathematika, 10.1112/S0025579310001233, 57:1, (147-173), Online publication date: 1-Jan-2011. Sedletsky Y (2006) A new type of modulation instability of Stokes waves in the framework of an extended NSE system with mean flow, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/39/31/L03, 39:31, (L529-L537), Online publication date: 4-Aug-2006. Sedletsky Y (2005) The modulational instability of Stokes waves on the surface of finite-depth fluid, Physics Letters A, 10.1016/j.physleta.2005.04.076, 343:4, (293-299), Online publication date: 1-Aug-2005. Laine-Pearson F and Bridges T (2004) Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves, SIAM Journal on Applied Mathematics, 10.1137/S0036139903423753, 64:6, (2096-2120), Online publication date: 1-Jan-2004. Bridges T (2003) Benjamin Memorial Lecture: Stability of Solitary Waves: Geometry, Symplecticity and Three-Dimensionality Wind over Waves II, 10.1533/9780857099532.178, (178-188), . Bridges T (1999) A new framework for studying the stability of genus-1 and genus-2 KP patterns, European Journal of Mechanics - B/Fluids, 10.1016/S0997-7546(99)80044-2, 18:3, (493-500), Online publication date: 1-May-1999. Dias F and Kharif C (1999) NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES, Annual Review of Fluid Mechanics, 10.1146/annurev.fluid.31.1.301, 31:1, (301-346), Online publication date: 1-Jan-1999. Bridges T (1999) Toral-equivariant partial differential equations and quasiperiodic patterns, Nonlinearity, 10.1088/0951-7715/11/3/005, 11:3, (467-500), Online publication date: 1-May-1998. Knobloch E and Pierce R (2009) On mean flows associated with traveling water waves, Fluid Dynamics Research, 10.1016/S0169-5983(97)00030-0, 22:2, (61-71), Online publication date: 1-Feb-1998. Bridges T (1997) A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453:1962, (1365-1395), Online publication date: 8-Jul-1997. This Issue15 March 1996Volume 354Issue 1707 Article InformationDOI:https://doi.org/10.1098/rsta.1996.0019Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Manuscript received20/09/1994Manuscript accepted07/02/1995Published online01/01/1997Published in print15/03/1996 License:Scanned images copyright © 2017, Royal Society Citations and impact