Kinematics of fluid particles on the sea surface. Hamiltonian theory

Francesco Fedele (1) , Cristel Chandre (2, 3) , Mohammad Farazmand (4)

1 ECE GeorgiaTech - School of Electrical and Computer Engineering - Georgia Insitute of Technology
2 CPT - Ex E8 Dynamique non linéaire
3 CPT - Centre de Physique Théorique - UMR 7332
4 School of Physics [Atlanta]

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.

Domaines: Dynamique Chaotique [nlin.CD]; Mécanique des fluides [physics.class-ph]

Francesco Fedele, Cristel Chandre, Mohammad Farazmand. Kinematics of fluid particles on the sea surface. Hamiltonian theory. Journal of Fluid Mechanics, 2016, 801, pp.260. ⟨10.1017/jfm.2016.453⟩. ⟨hal-01207149⟩