Generally, in coastal and ocean waters, the velocity profiles are typically established by bottom friction and surface wind stress, and so are varying with depth. Currents generate shear at the bed of the sea or of a river. For example, ebb and flood currents due to the tide may have an important effect on waves and wave packets. In any region where the wind is blowing, there is a surface drift of the water and water waves are particularly sensitive to the velocity in the surface layer.
I will present different model equations governing the dynamics of gravity waves on a vertically sheared current of constant vorticity. Firstly, the effect of vorticity on the modulational instability of gravity wave trains will be considered. Then, within the framework of shallow water theory, I will use a new approximate model from which the effect of vorticity on solitary waves, cnoidal waves and undular bores can be investigated. Finally, I will show a result on the effect of vorticity on rogue wave amplification