Fully nonlinear long-waves models in presence of vorticity

Abstract: 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.