Fully nonlinear long-wave models in the presence of vorticity

We study here Green-Naghdi type equations (also called fully nonlinear Boussinesq, or Serre equations) modelling the propagation of large-amplitude waves in shallow water without a smallness assumption on the amplitude of the waves. The novelty here is that we allow for a general vorticity, thereby 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 two-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.