Resonances for water waves over flows with piecewise constant vorticity. (February 2021) - Martin, Calin Iulian; Basu, Biswajit
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, λ 0 and λ 1 that represent the (relative) surface wave speed. We then confirm analytically that λ 1 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 λ 1 does not lead to four-wave resonances. In contrast to the previous conclusion λ 0 does not bring about three-wave resonances, but is able to generate four-wave resonances.