Open Collections

A Whitham-Boussinesq long-wave model for variable topography

Banff International Research Station for Mathematical Innovation and Discovery

We study the propagation of water waves in a channel of variable depth using the long-wave asymptotic regime. We use the Hamiltonian formulation of the problem in which the non-local Dirichlet-Neumann (DN) operator appears explicitly in the Hamiltonian and due to the complexity of the expressions of the asymptotic expansion associated with this operator in the presence of a non-trivial bottom topography. We perform an ad-hoc modification of these terms using a pseudo differential operator (PDO) associated with the bottom topography. In this talk we propose a Whitham-Boussinesq model for bidirectional wave propagation in shallow water that involves a PDO that consider explicitly the expression for the depth profile. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth. We will introduce an accurate and efficient numerical method that has been developed to compute this PDO. We present the results for the normal modes and eigen-frequencies of the linearized problem for families of different topographies. We also present some experiments of the evolution of some initial wave profiles over different topographies. Due to the ad-hoc nature of this simplified model we present some comparisons between the full expression of the first term of the asymptotic expansion of the DN operator given by Craig, Guyenne, Nicholls, and Sulem and our PDO approach for some specific topographies.

Mathematics; Fluid mechanics; Partial differential equations; Fluid dynamics

video/mp4

Author affiliation: Universidad Nacional Autónoma de México

2017-05-03

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61458

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/