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

¹  Faculty of Mathematics, University of Vienna - 1090 Vienna, Austria, EU
²  RISC, J. Kepler University - Linz 4040, Austria, EU

Corresponding authors: adrian.constantin@univie.ac.at lena@risc.uni-linz.ac.at

Received: 27 March 2009
Accepted: 31 March 2009

Abstract

The influence of an underlying current on three-wave interactions of capillary water waves is studied. The fact that in irrotational flow resonant three-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 and 4) the magnitude of a positive constant vorticity triggering a resonance cannot be too small.