Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves

TL;DR

This paper investigates how a constant non-zero vorticity in underlying currents enables three-wave resonances in capillary water waves, which are otherwise impossible in irrotational flows, revealing conditions for resonance occurrence.

Contribution

It demonstrates that positive constant vorticity can induce three-wave resonances in capillary water waves, detailing the conditions and limitations for such resonances.

Findings

Resonant 3-wave interactions occur only in flows with constant non-zero vorticity.

Only positive vorticities can trigger three-wave resonances.

The number of positive vorticities that induce resonance is countable.

Abstract

The influence of an underlying current on 3-wave interactions of capillary water waves is studied. The fact that in irrotational flow resonant 3-wave interactions are not possible can be invalidated by the presence of an underlying current of constant non-zero vorticity. We show that: 1) wave trains in flows with constant non-zero vorticity are possible only for two-dimensional flows; 2) only positive constant vorticities can trigger the appearance of three-wave resonances; 3) the number of positive constant vorticities which do trigger a resonance is countable; 4) the magnitude of a positive constant vorticity triggering a resonance can not be too small.