A new two-layer Boussinesq model for coastal waves from deep to shallow water: Derivation and analysis

Abstract

We derive a new two-layer Boussinesq model with high accuracy in linear and nonlinear properties and in interior kinematic property from deep to shallow water. This model is formulated in terms of computational horizontal and vertical velocities defined in each layer. The highest derivative in the equations is limited to three, which is convenient for numerical discretization. Stokes-type expansions are used to theoretically analyze the linear and nonlinear properties of the new two-layer Boussinesq model. The dispersive coefficients involved in the governing equations are determined by minimizing the integral error between the linear wave celerity of the Boussinesq model and the analytical solution. Shoaling coefficients are also optimized to expand the application range of the model to the mildly varying bathymetries. The most attractive aspect of this work is that the newly developed two-layer model exhibits high accuracy in linear, nonlinear, shoaling, and kinematic properties from extremely deep to shallow water. The analyses show that the resultant model is applicable to up to kh≈ 53 in linear dispersion, up to kh≈ 30 in the second nonlinear property within 1% error, and up to 0<kh< 60 in linear shoaling property with 0.13% error. Compared with that of most existing Boussinesq-type models, the accuracy of the horizontal and vertical velocities of the new model along the water column is improved significantly, and the model can be applicable to up to kh≈ 23.2 within 1% error.