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The Serre-Green-Naghdi system is a coupled, fully nonlinear system ofdispersive evolution equations which approximates the full water wave problem.The system is an extension of the well known shallow-water system to thesituation where the waves are long, but not so long that dispersive effects canbe neglected.  In the current work, the focus is on deriving mass, momentum and energydensities and fluxes associated with the Serre-Green-Naghdi system. Thesequantities arise from imposing balance equations of the same asymptotic orderas the evolution equations. In the case of an even bed, the conservationequations are satisfied exactly by the solutions of the Serre-Green-Naghdisystem. The case of variable bathymetry is more complicated, with mass andmomentum conservation satisfied exactly, and energy conservation satisfied onlyin a global sense. In all cases, the quantities found here reduce correctly tothe corresponding counterparts in both the Boussinesq and the shallow-waterscaling.  One consequence of the present analysis is that the energy loss appearing inthe shallow-water theory of undular bores is fully compensated by the emergenceof oscillations behind the bore front. The situation is analyzed numerically byapproximating solutions of the Serre-Green-Naghdi equations using afinite-element discretization coupled with an adaptive Runge-Kutta timeintegration scheme, and it is found that the energy is indeed conserved nearlyto machine precision. As a second application, the shoaling of solitary waveson a plane beach is analyzed. It appears that the Serre-Green-Naghdi equationsare capable of predicting both the shape of the free surface and the evolutionof kinetic and potential energy with good accuracy in the early stages ofshoaling.

Mechanical balance laws for fully nonlinear and weakly dispersive water waves