Resonances for water waves over flows with piecewise constant vorticity

Abstract

We investigate three and four-wave resonances of capillary–gravity water waves arising as the free surface of water flows exhibiting piecewise constant vorticity. More precisely, our type of flow has a jump in the vorticity distribution that separates a rotational layer at the top (commonly generated by wind-shear) from another rotational layer adjacent to the sea-bed, region that accommodates strong sheared currents. Instrumental in deriving our findings is the dispersion relation for such flows having a jump in the distribution of the vorticity. In general, for rotational flows of non-constant vorticity, the dispersion relation is very intricate. However, we show that a disentanglement occurs in the case of capillary–gravity water waves for which the ratio “thickness of the near-surface vortical layer/the wavelength of the surface wave” is sufficiently large. More precisely, we find explicitly two solutions, λ0 and λ1 that represent the (relative) surface wave speed. We then confirm analytically that λ1 gives rise to three-wave resonances for capillary–gravity water waves with wavelengths not exceeding 2 cm. However, for significant wavelength range, we establish that λ1 does not lead to four-wave resonances. In contrast to the previous conclusion λ0 does not bring about three-wave resonances, but is able to generate four-wave resonances.