Two-dimensionality of gravity water flows governed by the equatorial f-plane approximation

We show that gravity wave trains governed by the equatorial f-plane approximation propagate at the free surface of a rotational water flow of constant vorticity vector over a flat bed only if the flow is two-dimensional. Owing to the presence of Coriolis effects, our result is also true even if the vorticity vector vanishes. This represents a striking difference when compared with the cases without geophysical effects discussed in Constantin (Europhys Lett 86:29001, 2009, Eur J Mech 30:12-16; 2011) and Martin (J Math Fluid Mech 2016. doi:10.1007/s00021-016-0306-1 RefTarget Address="10.1007/s00021-016-0306-1" TargetType="DOI"), where the conclusion about the two-dimensionality of the flow was possible under the assumption of constant nonvanishing vorticity vector. Another upshot is that the only nonzero component of the vorticity that may not vanish is , that is, the one pointing in the horizontal direction orthogonal to the direction of wave propagation.