Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation

TL;DR

This paper introduces a general asymptotic method to derive two-dimensional time-dependent models for ocean wave propagation, including a new dispersive model that bridges deep and shallow water regimes.

Contribution

The paper presents a unified asymptotic framework for deriving water wave models and introduces a new dispersive model valid for small wave steepness and uneven bottoms.

Findings

Derived a system of surface equations involving Dirichlet-Neumann operator

Developed a new fully dispersive wave model for small steepness

Proposed coupling of dispersive and nonlinear shallow water models

Abstract

A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet-Neumann operator and we show that all the asymptotic models can be recovered by a simple asymptotic expansion of this operator, in function of the shallowness parameter (shallow water limit) or the steepness parameter (deep water limit). Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to uneven bottom the approach developed by Matsuno \cite{matsuno3} and Choi \cite{choi}. This model is still valid in shallow water but with less precision than what can be achieved with Green-Naghdi model, when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equations with the fully nonlinear shallow water Green-Naghdi equations represents a relevant model for describing ocean wave propagation from deep to shallow waters.