Kinematics of fluid particles on the sea surface. Hamiltonian theory

TL;DR

This paper derives equations for fluid particle motion on the sea surface using Hamiltonian mechanics, revealing conditions for wave breaking and showing particle velocities stay bounded, thus preventing finite-time singularities.

Contribution

It introduces a Hamiltonian framework for particle kinematics on the sea surface and links vorticity behavior to wave breaking criteria from first principles.

Findings

Vorticity vanishes at wave crests when particle velocity matches crest speed.

Hamiltonian dynamics simplify at wave crests, resembling free particle motion.

Particle velocities are proven to remain bounded over time, avoiding blowup.

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.