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Kinematics of fluid particles on the sea surface: Hamiltonian theory

Published online by Cambridge University Press:  21 July 2016

F. Fedele ,
C. Chandre Open the ORCID record for C. Chandre [Opens in a new window]  and
M. Farazmand
F. Fedele*
Affiliation:
School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30322, USA School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30322, USA
C. Chandre
Affiliation:
CNRS-Aix–Marseille University, CPT UMR7332, 13009 Marseille, France
M. Farazmand
Affiliation:
Center for Nonlinear Sciences, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
*
†Email address for correspondence: fedele@gatech.edu
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Abstract

We derive the John–Sclavounos equations, describing the motion of a fluid particle on the sea surface, from first principles using Lagrangian and Hamiltonian formalisms applied to the motion of a frictionless particle constrained on an unsteady surface. This framework leads to a number of new insights into the particle kinematics. The main result is that vorticity generated on a stress-free surface vanishes at a wave crest when the horizontal particle velocity equals the crest propagation speed, which is the kinematic criterion for wave breaking. If this holds for the largest crest, then the symplectic two-form associated with the Hamiltonian dynamics reduces instantaneously to that associated with the motion of a particle in free flight, as if the surface did not exist. Further, exploiting the conservation of the Hamiltonian function for steady surfaces and travelling waves, we show that particle velocities remain bounded at all times, ruling out the possibility of the finite-time blowup of solutions.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Hamiltonian theory Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 801 , 25 August 2016 , pp. 260 - 288
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Copyright
© 2016 Cambridge University Press 

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References

Banner, M. L., Barthelemy, X., Fedele, F., Allis, M., Benetazzo, A., Dias, F. & Peirson, W. L. 2014 Linking reduced breaking crest speeds to unsteady nonlinear water wave group behavior. Phys. Rev. Lett. 112, 114502.CrossRefGoogle ScholarPubMed
Barthelemy, X., Banner, M. L., Peirson, W. L., Dias, F., Fedele, F. & Allis, M.2015a On the local properties of highly nonlinear unsteady gravity water waves. Part 1. Slowdown, kinematics and energetics. J. Fluid Mech. (submitted) arXiv:1508.06001.Google Scholar
Barthelemy, X., Banner, M. L., Peirson, W. L., Fedele, F., Allis, M. & Dias, F.2015b On the local properties of highly nonlinear unsteady gravity water waves. Part 2. Dynamics and onset of breaking. J. Fluid Mech. (submitted) arXiv:1508.06002.Google Scholar
Bridges, T. J. 2004 Superharmonic instability, homoclinic torus bifurcation and water–wave breaking. J. Fluid Mech. 505, 153–162.CrossRefGoogle Scholar
Bridges, T. J., Hydon, P. E. & Reich, S. 2005 Vorticity and symplecticity in Lagrangian fluid dynamics. J. Phys. A 38 (6), 1403–1418.CrossRefGoogle Scholar
Cartan, É. 1922 Lessons on Integral Invariants. A. Hermann.Google Scholar
Chabchoub, A. & Fink, M. 2014 Time-reversal generation of rogue waves. Phys. Rev. Lett. 112, 124101.CrossRefGoogle ScholarPubMed
Chandre, C. 2013 Casimir invariants and the Jacobi identity in Dirac’s theory of constrained Hamiltonian systems. J. Phys. A 46, 375201.Google Scholar
Chandre, C., de Guillebon, L., Back, A., Tassi, E. & Morrison, P. J. 2013 On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets. J. Phys. A 46, 125203.Google Scholar
Dirac, P. A. M. 1950 Generalized Hamiltonian dynamics. Canad. J. Math. 2, 129.CrossRefGoogle Scholar
Dirac, P. A. M. 1958 Generalized Hamiltonian dynamics. Proc. R. Soc. Lond. A 246, 326.Google Scholar
Duncan, J. H., Qiao, H., Philomin, V. & Wenz, A. 1999 Gentle spilling breakers: crest profile evolution. J. Fluid Mech. 379, 191–222.CrossRefGoogle Scholar
Eyink, G. L.2007 ‘Turbulence theory’ course notes. Tech. Rep. The Johns Hopkins University.Google Scholar
Fedele, F. 2014 Geometric phases of water waves. Europhys. Lett. 107 (6), 69001.CrossRefGoogle Scholar
John, F. 1953 Two-dimensional potential flows with a free boundary. Commun. Pure Appl. Maths 6 (4), 497–503.CrossRefGoogle Scholar
Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech. 272, 1–20.CrossRefGoogle Scholar
Kurnia, R. & van Groesen, E. 2014 High order Hamiltonian water wave models with wave-breaking mechanism. Coast. Engng 93, 55–70.CrossRefGoogle Scholar
Littlejohn, R. G. 1979 A guiding center Hamiltonian: a new approach. J. Math. Phys. 20, 2445.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A 360 (1703), 471–488.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J Fluid Mech. 240, 659–679.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1998 Vorticity and curvature at a free surface. J. Fluid Mech. 356, 149–153.CrossRefGoogle Scholar
Lundgren, T. & Koumoutsakos, P. 1999 On the generation of vorticity at a free surface. J. Fluid Mech. 382, 351–366.CrossRefGoogle Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.CrossRefGoogle Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45 (1), 115–145.CrossRefGoogle Scholar
Qiao, H. & Duncan, J. H. 2001 Gentle spilling breakers: crest flow-field evolution. J. Fluid Mech. 439, 57–85.CrossRefGoogle Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Lond. A 331 (1622), 735–800.Google Scholar
Saket, A., Peirson, W. L., Banner, M. L., Barthelemy, X. & Allis, M.2015 Wave breaking onset of two-dimensional deep-water wave groups in the presence and absence of wind. arXiv:1508.07702.CrossRefGoogle Scholar
Sclavounos, P. D. 2005 Nonlinear particle kinematics of ocean waves. J. Fluid Mech. 540, 133–142.CrossRefGoogle Scholar
Shemer, L. 2013 On kinematics of very steep waves. Nat. Hazards Earth Syst. Sci. 13 (8), 2101–2107.CrossRefGoogle Scholar
Shemer, L. & Ee, B. K. 2015 Steep unidirectional wave groups – fully nonlinear simulations versus experiments. Nonlinear Process. Geophys. Discussions 2 (4), 1159–1195.Google Scholar
Shemer, L. & Liberzon, D. 2014 Lagrangian kinematics of steep waves up to the inception of a spilling breaker. Phys. Fluids 26 (1), 016601.CrossRefGoogle Scholar
Song, J.-B. & Banner, M. L. 2002 On determining the onset and strength of breaking for deep water waves. Part I: unforced irrotational wave groups. J. Phys. Oceanogr. 32 (9), 2541–2558.CrossRefGoogle Scholar
Stansell, P. & MacFarlane, C. 2002 Experimental investigation of wave breaking criteria based on wave phase speeds. J. Phys. Oceanogr. 32 (5), 1269–1283.2.0.CO;2>CrossRefGoogle Scholar
Tulin, M. P. 2007 On the transport of energy in water waves. J. Engng Maths 58 (1), 339–350.CrossRefGoogle Scholar
Wu, J. Z. 1995 A theory of three dimensional interfacial vorticity dynamics. Phys. Fluids 7 (10), 2375–2395.CrossRefGoogle Scholar
Xu, H., Pumir, A., Falkovich, G., Bodenschatz, E., Shats, M., Xia, H., Francois, N. & Boffetta, G. 2014 Flight-crash events in turbulence. Proc. Natl Acad. Sci. 111 (21), 7558–7563.CrossRefGoogle ScholarPubMed
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194.CrossRefGoogle Scholar
Zakharov, V. E. 1999 Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. Eur. J. Mech. (B/Fluids) 18 (3), 327–344.CrossRefGoogle Scholar