On the Relation between Surface Profiles and Internal Flow Properties in Long-Wave Models

Abstract

In this work, we investigate the internal velocity field in a number of Boussinesq models in non-uniform situations. A coupled BBM-BBM type system of equations is derived in the assumption of water wave propagating over an uneven bottom. The focus is on formulating mass, momentum and energy densities and fluxes associated with the BBM-BBM system over an uneven bottom. These densities and the associated fluxes arise from establishing mechanical balance equations of the same asymptotic order as the evolution equations. The BBM-BBM type system derived here is solved numerically by applying a Fourier collocation method coupled with a four stage Runge-Kutta time integration scheme. We look at the propagation of waves over a slope, and how the reconstruction of the flow under the surface is connected with shoaling and wave breaking. The mass conservation equations are used to quantify the role of reflection in the shoaling of solitary waves. Moreover, the principle of conservation of energy is used to develop an equation relating the waveheight and undisturbed depth to the initial undisturbed depth and the incident waveheight. Boussinesq’s shoaling law is approximately recovered for waves of very small waveheight. Shoaling and breaking results for the different Boussinesq systems are plotted. Internal properties of the flow are also in focus in the case of a background shear flow. The Boussinesq -type equations for water waves with constant vorticity are derived in the Boussinesq regime. We reduced the Boussinesq -type equations to the Korteweg-de Vries (KdV) equation in the unidirectional case. We found the approximate velocity field associated with exact solutions of KdV equation including shear flow. The influence of the shear flow on particle trajectories and breaking of surface waves are studied using the approximate velocity field.