Variational Boussinesq modelling of surface gravity waves over bathymetry

In this thesis a method is presented to construct a variational model with an always positive Hamiltonian (sum of kinetic and potential energy). In this methodology the fluid motion beneath the surface is approximated, by making assumptions on the vertical structure of the flow velocities, in a fashion as first applied by Joseph Valentin Boussinesq (1842–1929) for the description of fairly-long surface waves in shallow water. The subsequent integrations over the total water depth result in simplified models: instead of a three-dimensional description, the result is a twodimensional model in the horizontal plane, denoted as the propagation space. The thesis presents the methodology resulting in an approximate and positive Hamiltonian, as well as the (linear) propagation and reflection characteristics of the associated wave models. These models conserve depth-integrated mass and energy. And in case of a horizontal sea bed, also depth-integrated horizontal momentum is conserved. Besides, wave action is conserved as a direct consequence of the variational description of the flow. The properties of the fully non-linear model – without assumptions regarding the relative wave height – are studied through numerical simulations. Comparison with the results from other models, as well as from laboratory experiments, show the nonlinear capacities of the variational Boussinesq modelling. Besides, model simulations are all numerically stable, which may be attributed to the guaranteed positivity of the wave energy (Hamiltonian density).