Serre-type equations in deep water

TL;DR

This paper reviews and derives deep water Serre-Green-Naghdi equations, highlighting their variational structures and multi-symplectic properties, with implications for developing structure-preserving numerical methods.

Contribution

It introduces the deep water analogue of Serre-Green-Naghdi equations and discusses their variational and multi-symplectic structures, expanding modeling tools for deep water waves.

Findings

Derivation of deep water Serre-Green-Naghdi equations

Analysis of variational and multi-symplectic structures

Potential for structure-preserving numerical methods

Abstract

This manuscript is devoted to the modelling of water waves in the deep water regime with some emphasis on the underlying variational structures. The present article should be considered as a review of some existing models and modelling approaches even if new results are presented as well. Namely, we derive the deep water analogue of the celebrated Serre-Green-Naghdi equations which have become the standard model in shallow water environments. The relation to existing models is discussed. Moreover, the multi-symplectic structure of these equations is reported as well. The results of this work can be used to develop various types of robust structure-preserving variational integrators in deep water. The methodology of constructing approximate models presented in this study can be naturally extrapolated to other physical flow regimes as well.