Mixing and Mixtures in Geo- and Biophysical Flows: A Focus on Mathematical Theory and Numerical Methods

Vorticity in shallow water flows: from wave current interactions to turbulent bores

David Lannes

IMB Bordeaux


Motivated by the study of nonlinear wave-current interactions (such as rip-currents) we study the influence of vorticity on surface water waves. We first derive a generalization of the classical hamiltonian Zakharov-Craig- Sulem formulation of irrotational water waves that takes into account the effects of the vorticity. It allows to keep track of the influence of vorticity on the flow. We show the local well-posedness of these equations, and establish their stability with respect to shallow water limits. Based on the bounds thus obtained, we turn to derive shallow water asymptotic models. The big difference with irrotational flows is that the dynamics of the vorticity is fully three dimensional, while shallow water models are typically two-dimensional (through vertical averaging). We show however that the vorticity contribution can be reduced to two-dimensional equations; the idea, based on an analogy with turbulence theory, is that the vorticity contributes to the averaged momentum equation through a Reynolds-like tensor. A cascade of equations is then derived for this tensor, but contrary to standard turbulence theory, closure of the equations is obtained after a finite number of steps. The equations thus obtained will be discussed, and the problem of creation of vorticity during wave breaking (reproduced by numerical simulations) will also be addressed.