  We study here the propagation of long waves in the presence of vorticity. In
the irrotational framework, the Green-Naghdi equations (also called Serre or
fully nonlinear Boussinesq equations) are the standard model for the
propagation of such waves. These equations couple the surface elevation to the
vertically averaged horizontal velocity and are therefore independent of the
vertical variable. In the presence of vorticity, the dependence on the vertical
variable cannot be removed from the vorticity equation but it was however shown
in [?] that the motion of the waves could be described using an extended
Green-Naghdi system. In this paper we propose an analysis of these equations,
and show that they can be used to get some new insight into wave-current
interactions. We show in particular that solitary waves may have a drastically
different behavior in the presence of vorticity and show the existence of
solitary waves of maximal amplitude with a peak at their crest, whose angle
depends on the vorticity. We also show some simple numerical validations.
Finally, we give some examples of wave-current interactions with a non trivial
vorticity field and topography effects.
