Hamiltonian structure for two-dimensional extended Green-Naghdi equations

TL;DR

This paper extends the two-dimensional Green-Naghdi model for shallow-water waves to include higher-order dispersive effects, analyzes its Hamiltonian structure, and compares it with Zakharov's formulation, providing new insights into wave modeling.

Contribution

The paper introduces a higher-order extended Green-Naghdi model with Hamiltonian structure and establishes its equivalence to Zakharov's formulation for surface gravity waves.

Findings

Extended GN system includes higher-order dispersive effects.

The extended model maintains the Hamiltonian structure of the original GN system.

Demonstrated equivalence between Zakharov's formulation and the extended GN system.

Abstract

The two-dimensional Green-Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original GN model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.