Mechanical balance laws for the Nwogu system

Abstract

Boussinesq systems, first introduced by J. Boussinesq in 1872, model small-amplitude shallow-water waves with weakly nonlinear and dispersive effects. Over time, various forms of these systems have been developed. A common feature is their solution variables: the time-dependent free-surface elevation and horizontal fluid velocity in the water column. In 1993, O. Nwogu introduced a modified Boussinesq system by taking the solution variable at a specific fluid depth. This innovation significantly improved the linear dispersion properties, extending the applicability of Boussinesq systems to greater water depths. Due to these enhanced properties, the Nwogu system has become a widely used tool for nearshore wave modelling. While Boussinesq systems are commonly applied to small-amplitude long waves, the mechanical properties of these systems—such as mass, momentum, and energy—have received comparatively little attention, even for the Nwogu system. By leveraging the conservation principle, which states that the rate of change of a quantity in a spatial region equals the net influx into that region, mechanical balance laws can be derived for water wave models. This study derives mass, momentum, and energy balance laws for the Nwogu system in a two-dimensional domain with realistic bottom topography, ensuring accuracy consistent with the system’s asymptotic precision.