Existence of a directional Stokes drift in asymmetrical three-dimensional travelling gravity waves

Gérard Iooss (1, 2) , Pavel Plotnikov (3)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 IUF - Institut universitaire de France
3 LIH - Lavrentyev Institute of Hydrodynamics

We consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid. The pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non resonant situation. Defining a couple of amplitudes ε1, ε2 along the basis of wave vectors which satisfy the dispersion relation, following [8], travelling waves exist with an asymptyotic expansion in powers of ε1, ε2, for nearly all pair of angles made by the basic wave vectors with the critical propagation direction, and for values of the couple (ε1^2, ε2^2) in a subset of the plane, with asymptotic full measure at the origin. We prove the remarkable property that on the free surface, observed in the moving frame, the propagation direction of the waves differs from the asymptotic direction taken by fluid particles, by a small angle which is computed.

Mots clés: potential perfect fluid flow; nonlinear water waves; travelling gravity waves; bifurcation theory; asymmetric 3D waves

Domaines: Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]; Formation de Structures et Solitons [nlin.PS]; Systèmes dynamiques [math.DS]

Gérard Iooss, Pavel Plotnikov. Existence of a directional Stokes drift in asymmetrical three-dimensional travelling gravity waves. Comptes Rendus Mécanique, 2009, ⟨10.1016/j.crme.2009.09.001⟩. ⟨hal-01265171⟩