Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom

The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. Having in mind a signalling problem, we search for optimal dispersive properties of the 1-D linear model over flat bottom and, using finite element and (pseudo-) spectral numerical codes, investigate its quality. For the optimization we restrict to the class of potentials with hyperbolic vertical profiles that are parametrized by the wavenumber. The optimal wavenumber is obtained by minimizing the kinetic energy for the given signal and produces good results for two realistic test cases. Besides this kinetic energy principle we also consider various ad-hoc least square type of minimization problems for the error of the phase or group velocity. The test cases are two examples of focussing wave groups with broad spectra for which accurate experimental data are available from MARIN hydrodynamic laboratory. To determine the quality of an 'optimized' wavenumber for the governing dynamics, we use accurate numerical simulations with the AB-equation to compare with VBM calculations for the whole range of possible wavenumbers. The comparison includes the errors in the signal at the focussing position, as well as the integrated errors of maximal and minimal wave heights along a spatial and temporal interval that is symmetric around the focussing event.