Two-dimensional parabolic modeling of extended Boussinesq equations
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abstract
A frequency domain wave transformation model is derived from a set of time-dependent extended Boussinesq equations. In contrast to an earlier study, this model is derived directly from the (, u) form of the equations. The resulting model maintains the excellent dispersive properties of the original equations and also accurately mimics the shoaling behavior of dispersive linear theory for a wide range of water depths. The model is formulated in terms of a free parameter that can be tuned for optimum shoaling behavior. We tune the parameter in two ways; the first seeks the minimum error for the shoaling parameter while using the optimum value for the dispersion parameter, while the second seeks the minimum combined error for both dispersion and shoaling. These lead to two different sets of free parameters, with the second method leading to more favorable linear shoaling behavior than the first. The effectiveness of the linear terms of the model is demonstrated by using them, with both sets of free parameters, to propagate waves over a shoal; comparisons to both experimental data and a linear mild-slope parabolic model are performed, and agreement is favorable for both sets of parameters. The full model, with nonlinear terms and both sets of parameters, is then used to simulate a laboratory experiment involving harmonic generation and nonlinear wave focusing. Results indicate that the model can reproduce the characteristics of a lowest-order Kadomtsev-Petviashvili (KP) model in shallow water. The model also shows improved agreement with data relative to the KP model and another dispersive frequency-domain model in intermediate water depth. Additionally, there is little difference between the results from either set of parameters used in the model, indicating that for these cases it is not clear which optimization is most ideal.
