Description:

The Variational Boussinesq Model (VBM) for gravity surface waves on a layer of ideal fluid conserves mass, momentum, energy, and contains decreased dimensionality compared to the full problem. It is derived from a Hamiltonian formulation via an approximation of the kinetic energy, and has related approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive properties of the 1-D linear model over flat or uneven bottom. Implementations with finite element and spectral methods are used to investigate the accuracy and robustness.

In [1] we describe a novel kinetic-energy optimization principle, which is used to find optimal parameters of the model. Within the VBM framework for the signalling problem over a flat bottom the simulations have been compared with experimental data for quite complex test cases for which the wave spectra are very broad.

Now we are investigating the reflection properties from a sloping bottom profile. This includes ageneralization of the VBM and applications of WKB theory.

Fig. 1. Wave elevations, maximal crest height and minimal trough depth, as simulated by linear VBM (red solid) and linear AB-equation (green dashed). The test case shown is the ‘strongly focussing wave group’.

Fig. 2. Differences of the exact dispersive simulations with VBM-hyperbolic simulations, of the maximal crest height (left) and the minimal trough depth (right). Solid lines on the plots represent the optimal parameter as calculated by the kinetic optimization criterion