Mechanical balance laws for fully nonlinear and weakly dispersive water waves

TL;DR

This paper derives and analyzes the balance laws for the Serre-Green-Naghdi system, a model for nonlinear dispersive water waves, demonstrating energy conservation and accurate wave shoaling predictions.

Contribution

It provides a detailed derivation of mass, momentum, and energy balance equations for the Serre-Green-Naghdi system, including variable bathymetry effects, and validates energy conservation numerically.

Findings

Energy is conserved nearly to machine precision in simulations.

The model accurately predicts wave shape and energy evolution during shoaling.

Energy loss in shallow-water theory is compensated by oscillations behind the bore front.

Abstract

The Serre-Green-Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre-Green-Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre-Green-Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling. One consequence of the present analysis is that the energy loss appearing in the shallow-water theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre-Green-Naghdi equations using a finite-element discretization coupled with an adaptive Runge-Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre-Green-Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling.