Resonant interactions of rotational water waves in the equatorial f-plane approximation

Abstract

We investigate here the resonance phenomenon in periodic unidirectional water waves in flows of constant vorticity governed by the equatorial f-plane approximation. The relevance of such water waves displaying a one dimensional wave vector is also underlined in the paper—in the context of equatorial capillary-gravity water waves—and serves as the basis for the resonance analysis which is carried out by means of dispersion relations for equatorial water waves that were quite recently derived [see the work of Constantin, Differ. Integr. Equations 26(3-4), 237–252 (2013) and Martin, Nonlinear Anal.: Theory, Methods Appl. 96, 1–17 (2014)]. We show that, while gravity water waves do not exhibit three-wave resonance, the four-wave resonance occurs irrespective of the vorticity.