Variational Boussinesq model for kinematics calculation of surface gravity waves over bathymetry

Abstract

Many of the widely used models for description of nonlinear surface gravity waves, in deep or shallow water, such as High Order Spectral Method (HOSM) and Boussinesq-type equations, rely on the elimination of the vertical coordinate from the basic three-dimensional Euler equations. From a numerical point of view such models are often computationally efficient, which is one of the main reasons that many such models are frequently used in studies on nonlinear surface waves. While surface-based models provide the time-evolution of surface quantities, typically the surface elevation η and velocity potential at the surface ϕ, they do not directly provide the water particle kinematics in the fluid interior. However, in many practical applications information about the water-particle kinematics is crucial. The present paper presents a new method for the calculation of water-particle kinematics, from information about surface quantities. The presented methodology is a non-perturbative approach based on the fully nonlinear Variational Boussinesq model, and can be applied to wave propagation over both constant and variable water depth. The proposed method is validated on several cases, including Stokes waves, a solitary wave, and irregular waves over flat bottom. We have carried out new laboratory experiments of regular waves over a shoal with measurements of the horizontal velocity specifically taken for validation of the method. We also employ recent laboratory experiments for validation of statistical properties of wave kinematics of long crested irregular waves propagating over a shoal.