Water Waves with Vorticity and Asymptotic Models
Speaker: David Lannes, University of Bordeaux
Location: Warren Weaver Hall 605
Date: Friday, April 24, 2015, 11 a.m.
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. The canonical variables for this formally Hamiltonian generalization are the surface elevation, the "gradient component" of the horizontal component of the tangential vector field, and the vorticity. It allows therefore 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.