Seminar

The problem of surface water waves over an uneven bottom is a classical problem of fluid mechanics. The governing equations are the Euler equations in the presence of a free surface and varying bathymetry. A rigorous theory of the solution of this problem is extremely complex due not only to the fact that the water-wave problem is a classical free-boundary problem, but also because the boundary conditions are strongly nonlinear. From a mathematical viewpoint, the surface water wave equations pose surprisingly deep and subtle challenges for rigorous analysis and numerical simulation. An important direction of inquiry in the general field of nonlinear waves is the development and application of simplified model equations. There are many asymptotic scaling regimes of interest, including long-waves and long scale variations in the variable bottom fluid boundary. To measure the effect of large variations in the depth, i.e. for scales where the dispersive effects should be included, there are no widely accepted or used simple dispersive models. Their numerical simulation also requires sophisticated numerical methods that are suited to the mathematical structure. The main task consists of evaluating the Dirichlet-Neumann (DN) operator for the Laplacian in the fluid domain. In this talk, I will introduce a Hamiltonian non-local shallow water wave model for flow over variable topography using the Hamiltonian formulation of the free surface potential flow introduced by Zakharov-Craig-Sulem. The model proposed in this talk is a simple approximation to the DN operator that involves a pseudo-differential operator which has the same structural features as the DN operator for variable depth. This leads to a bidirectional nonlocal  “Whitham-Boussinesq-type” shallow water wave model for variable topography that can be used for 2-D and 3-D flow. Whitham-Boussinesq models are of considerable current interest as nonlocal extensions of well-studied dispersive shallow water wave models, such as the KdV and Boussinesq equations.