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

Abstract

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 ε 1 , ε 2 along the basis of wave vectors which satisfy the dispersion relation, following Iooss and Plotnikov (2009), travelling waves exist with an asymptotic expansion in powers of ε 1 , ε 2 , for nearly all pair of angles made by the basic wave vectors with the critical propagation direction, and for values of the couple ( ε 1 2 , ε 2 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. To cite this article: G. Iooss, P. Plotnikov, C. R. Mecanique 337 (2009).