  We study here Green-Naghdi type equations (also called fully nonlinear
Boussinesq, or Serre equations) modeling the propagation of large amplitude
waves in shallow water. The novelty here is that we allow for a general
vorticity, hereby allowing complex interactions between surface waves and
currents. We show that the a priori 2+1-dimensional dynamics of the vorticity
can be reduced to a finite cascade of two-dimensional equations: with a
mechanism reminiscent of turbulence theory, vorticity effects contribute to the
averaged momentum equation through a Reynolds-like tensor that can be
determined by a cascade of equations. Closure is obtained at the precision of
the model at the second order of this cascade. We also show how to reconstruct
the velocity field in the 2 + 1 dimensional fluid domain from this set of
2-dimensional equations and exhibit transfer mechanisms between the horizontal
and vertical components of the vorticity, thus opening perspectives for the
study of rip currents for instance.
