Analysis of the linear version of a highly dispersive potential water wave model using a spectral approach in the vertical
Résumé
h i g h l i g h t s • Highly dispersive potential flow model for water wave propagation. • Linear dispersion relation accurate in very deep water conditions (kh up to 100). • Accurate prediction of wave kinematics (orbital velocities) in deep water. • Validation of linear shoaling properties of the model. • Good prediction of reflected and transmitted waves on a Roseau-type bottom profile. a b s t r a c t The properties and accuracy of the linearized version of the fully dispersive and nonlinear wave model developed in Yates and Benoit (2015) and Raoult et al. (2016) are analyzed for both flat and variable bottom bathymetries. This model considers only a single layer of fluid and uses a basis of orthogonal Chebyshev polynomials to project the vertical structure of the potential. This approach results in an exponential convergence rate with the maximum degree of the Chebyshev polynomial, denoted N T , while only first-and second-order derivatives in space need to be evaluated. For the constant water depth case, the linear dispersion relation of the model is derived analytically, and expressions are established for N T ranging from 2 to 15. The analysis shows a rapid increase in accuracy in the deep water range with increasing N T. For instance, the relative error in the calculated wave celerity (in comparison with Stokes' analytical solution) remains smaller than 2.5% for deep water cases with kh up to 100 using N T ≥ 9 (k and h are the representative wavenumber and water depth, respectively). The wave kinematics, vertical profiles of the horizontal and vertical orbital velocities, converge to the Stokes profiles for kh up to 60 when using a sufficiently high value of N T. The vertically-averaged relative errors of the horizontal and vertical velocities remain below 6% and 3%, respectively, for kh up to 60 when using N T ≥ 11. The presented model shows better dispersive properties in deep water than several high-order Boussinesq-type models. For variable bottom bathymetries, the shoaling properties of the model are studied numerically, exhibiting good agreement with results from Stokes linear theory in the case of mild bottom slopes, using a sufficiently high value of N T with respect to the offshore relative water depth. For an offshore water depth of kh = 10 (i.e. more than 3 times the deep water limit), accurate wave heights in shallow water (kh = 0.25) are obtained with N T = 6 (or higher). Finally, the linear version of the model is validated with comparisons to analytical solutions of the reflection and transmission coefficients of regular waves over Roseau-type bathymetric profiles. Two bottom profiles are considered, including one with a steep slope, whose maximum value reaches about 1:0.7 (i.e. an angle of about 54.9 deg.). Using N T = 7, small differences (<0.4%) with the analytical solution are observed for the four considered cases, confirming the ability of the linear model to represent accurately the effects of steep bottom gradients on wave propagation dynamics.