Rogue waves, also called freak waves, are abnormal waves whose amplitude is significantly larger than the background sea-state. Recently, a vor-NLSE has been derived to investigate, in particular, the influence of a background constant vorticity on the linear stability of weakly modulated Stokes waves. For an unstable modulation, which always exists in the self-focussing case (abnormal dispersion regime), it has been suggested that the resulting instability, also called the Benjamin-Feir modulational instability, can be seen as an important mechanism of rogue waves formation. In the present work we take the opportunity to use the vor-NLS equation in order to describe the potential effects of a background (constant) vorticity on this mechanism. As is well known, some exact breather- type solutions have been suggested by several authors as rogue waves models (see Onorato et al., 2013, Physics Reports 528, 47–89, and references therein). From scaled forms of the vor-NLS equation, exact and approximate analytical results describing the long-term evolution of the Benjamin-Feir instability are presented and used to examine the influence of vorticity on the characteristics of the Benjamin-Feir instability. These are tested against results from numerical simulations of vor-NLS equation in the deep water limit, for which the effect of background vorticity on the wave-induced mean flow is more than the affect of the nonlinear coefficient by the shear strength, as the second-order wave-induced mean flow does not vanish like in the situation with no shear currents.

Acknowledgments: This work was supported by the Direction Générale de l'Armement and funded by the ANR project #ANR-13-ASTR-0007.