A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed

Abstract

A new multi-layer irrotational Boussinesq-type model is proposed for both linear and nonlinear surface water waves over mildly sloping seabeds. The model is formulated in terms of computational horizontal and vertical velocity components within each layer and satisfies exact kinematic and dynamic free-surface conditions as well as kinematic seabed conditions. Using a Stokes-type expansion, a theoretical analysis of the new multi-layer model is carried out to examine both linear and nonlinear properties, including wave celerity, velocity profiles, shoaling amplitude, second- and third-order transfer functions and amplitude dispersion. The dispersive coefficients in the governing equations are determined by optimizing the linear celerity or linear velocity profiles. For example, the four-layer model shows extremely high accuracy and is applicable up to $kh=667$–800 (where $k$ is the wavenumber and $h$ is a typical water depth) with a 1 % error in wave phase celerity, and up to $kh=352$–423 with a 1 % error in the linear velocity components. The super- and subharmonic transfer functions are extremely accurate up to $kh=300$ (1 % error), the third-order harmonics and amplitude dispersion are accurate up to $kh=477$ (1 % error), and the shoaling property is optimized to cover the range of $0<kh<300$, which presents a 0.06 % tolerance error in shoaling amplitude. The high-accuracy nature of the model increases its suitability for simulating random wave propagation from extremely deep to shallow waters over mildly sloping topographies. The model is implemented numerically on a non-staggered grid via a composite fourth-order Adams–Bashforth–Moulton time integration. The numerical results show good agreement with both the analytical solutions and experimental data.