Two-layer Boussinesq models for coastal water waves

This paper presents three sets of two-layer Boussinesq models for highly dispersive and highly nonlinear water waves. These models are formulated in terms of depth-averaged velocities or velocities located at two arbitrary z locations within each layer and are fully nonlinear to the second order. Stokes-type expansions are used to theoretically analyze the linear and nonlinear properties of the models. The coefficients involved in the governing equations are determined from the minimization of the integral error between the linear wave celerity, shoaling gradient, second nonlinear harmonics of the equations and the related analytical solutions. The most promising model is applicable up to kh≈25.4 (where kh is the dimensionless water depth, k is the wave number, and h is the water depth) for the dispersive property, to kh≈6 for the second nonlinear property within 1% error, and to kh≤6 for the excellent shoaling property. The numerical implementation for one-dimensional governing equations on non-staggered grids is also presented by employing a fourth-order Adams-Bashforth-Moulton time integration and the high-accuracy finite difference method. Four demanding numerical experiments that require high accuracy of dispersion and nonlinearity are conducted to assess the performance of the models. The computational results are compared against the analytical solution and experimental data, good agreements are found.