Kinematics of fluid particles on the sea surface. Hamiltonian theory

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. 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 traveling waves we show that particle velocities remain bounded at all times, ruling out the possibility of the finite-time blowup of solutions.

References in corpus (6)

• Time-Reversal Generation of Rogue Waves
• Linking reduced breaking crest speeds to unsteady nonlinear water wave group behavior
• Lagrangian kinematics of steep waves up to the inception of a spilling breaker
• Geometric phases of water waves
• On the local properties of highly nonlinear unsteady gravity water waves. Part 1. Slowdown, kinematics and energetics
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Cited by in corpus (3)

• Crest speeds of unsteady surface water waves
• Surface waves enhance particle dispersion
• Wave-current interaction on a free surface