We present a Hamiltonian formulation for water waves over a variable bottom based on potential flow theory. In this formulation, the problem is reduced to a lower-dimensional one involving boundary variables alone. This is accomplished by introducing the Dirichlet-Neumann operator which expresses the normal fluid velocity at the free surface in terms of the velocity potential there, and in terms of the surface and bottom variations which determine the fluid domain. A Taylor series expansion of the Dirichlet-Neumann operator in homogeneous powers of the surface and bottom variations is proposed. This formulation has implications for the convenience of perturbation calculations and numerical simulations. Using Hamiltonian perturbation theory and multiple-scale analysis, we derive model equations for long waves over random topography. Asymptotic solutions will be discussed and preliminary numerical simulations will be shown.
This is joint work with Anne de Bouard (Ecole Polytechnique), Walter Craig (McMaster University), Oliver Diaz-Espinosa (SAMSI & Duke) and Catherine Sulem (University of Toronto).