Hamiltonian structure for two-dimensional extended Green–Naghdi equations

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.

Isobe–Kakinuma model for water waves as a higher order shallow water approximation

“…on some time interval independent of δ, where η IK is a solution to the Isobe-Kakinuma model (1.4). Here we remark that Y. Matsuno [15,16] derived extended Green-Naghdi equations as higher order shallow water approximations in the strongly nonlinear regime. His δ 2N model is an approximation of the full water wave equations with an error of order δ 2N +2 .…”

Well-posedness of the Green–Naghdi and Boussinesq–Peregrine systems

“…For a clear and modern exposition, it is shown in [32] that the Green-Naghdi system can be derived as an asymptotic model from the water waves system (namely the "exact" equations for the propagation of surface gravity waves), by assuming that the typical horizontal length of the flow is much larger than the depth of the fluid layer -that is in the shallow-water regime-and that the flow is irrotational. Roughly speaking, a Taylor expansion with respect to the small "shallow-water parameter" yields at first order the Saint-Venant system, and at second order the Green-Naghdi system (see for instance [38] for higher order systems). As a relatively simple fully nonlinear model (that is without restriction on the amplitude of the waves) formally improving the precision of the Saint-Venant system, the Green-Naghdi system is widely used to model and numerically simulate the propagation of surface waves, in particular in coastal oceanography.…”

On wave breaking for Boussinesq-type models

“…We also stress that the discretizations employed has no exact energy/entropy stability properties. For these issues, we refer to [47,2,57] and references therein, concerning the PDE continuous modelling, and to [76,26,91] and references therein, for aspects related to the energy stable approximation of the shallow water equations. The construction of exactly energy preserving schemes for dispersive equations is still a subject of research, the interested reader may refer to [84,35,93,92] for some recent results concerning dispersive equations.…”