Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train

We show that a free surface water flow of constant nonzero vorticity beneath a wave train and above a flat bed has to be two-dimensional and the vorticity must have only one nonzero component which points in the horizontal direction orthogonal to the direction of wave propagation. The obtained results are of relevance to studies of resonant wave train interactions in flows of constant nonzero vorticity: in contrast to irrotational flows, all wave trains have to propagate in the same direction. An important practical aspect of these considerations lies in that wave trains of constant vorticity model the interaction of swell with tidal currents, negative vorticity being appropriate for the flood current and positive vorticity for the ebb current.