Coastal zone simulations with variational Boussinesq modelling

Abstract

The main challenge in deriving a Boussinesq model for water wave is to model accurately the dispersion and nonlinearity of waves. The dispersion is a depth-dependent relation between the wave speed and the wavelength. A Boussinesq-type model can be derived from the so-called variational principle in which the wave phenomena can be exactly described as a Hamiltonian system. The challenge is then to approximate the kinetic energy. The Variational Boussinesq Model (VBM) is derived by restricting the vertical flow in the kinetic energy into a subclass of fluid potentials: a sum of its value at the free surface and a linear combination of vertical profiles with spatially dependent functions as coefficients. The minimization property of the kinetic energy requires that these spatial functions have to satisfy a (linear) elliptic equation. The vertical profile is chosen a priori and determine completely the dispersive properties of the model. In this thesis, we use so-called vertical Airy profiles functions, which appear in the exact expression for harmonic waves of linear potential theory. Using these functions we can get flexibility to improve the dispersion. This improvement is based on a method to use in an optimal way the parameters (wavenumbers) in the vertical Airy profiles so that broad-band waves such as wind-waves can be dealt with. The optimal choice follows again by exploiting the minimization property of the kinetic energy. However, to become practically applicable, information from the initial state is needed. This has as consequence that each specific problem gets a tailor-made model, with dispersion that is sufficiently accurate for all waves under consideration. The underlying variational formulation of the model has been used to design a numerical Finite Element implementation; simple piecewise linear splines can be used since no higher than first order spatial derivatives appear in the positive definite Hamiltonian. The quality of our modeling is shown by results of simulations for various cases of broad-band waves. Simulations of irregular wind waves are compared for various cases with recent experiments by MARIN hydrodynamics laboratory. Finally we show simulations for realistic wind-waves in the complicated geometry and bathymetry of the Jakarta harbour.