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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: [email protected]
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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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