Dispersive Nonlinear Waves in Two‐Layer Flows with Free Surface. I. Model Derivation and General Properties

Abstract

In this paper we derive an approximate multi-dimensional model of dispersive waves propagating in a two-layer fluid with free surface. This model is a "two-layer" generalization of the Green-Naghdi model. Our derivation is based on Hamilton's principle. From the Lagrangian for the full-water problem we obtain an approximate Lagrangian with accuracy O(ε 2 ), where ε is the small parameter representing the ratio of a typical vertical scale to a typical horizontal scale. This approach allows us to derive governing equations in a compact and symmetric form. Important properties of the model are revealed. In particular, we introduce the notion of generalized vorticity and derive analogues of integrals of motion, such as Bernoulli integrals, which are well known in ideal Fluid Mechanics. Conservation laws for the total momentum and total energy are also obtained.

Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory

“…For every given n formula (8) yields a bound state λ n > 0; A ≈ λ 1 > λ 2 > · · · > λ N ≈ 0 with N the total number of bound states in the potential v 0 (x). In the KdV context the distribution (8), through a n = 2λ n , gives the amplitude of the n-th soliton in the soliton train as t → ∞. For the largest amplitude soliton we have the classical result…”

A Boussinesq system for two-way propagation of interfacial waves

“…Models valid for arbitrary depth ratio have been derived for example by Choi & Camassa [9]. 2 In a recent paper, Kataoka [21] showed that when H is near unity, the stability of solitary waves changes drastically for small density ratios r. Therefore one must be careful in evaluating the stability of air-water solitary waves. In other words, there may be differences between r = 0 and the true value r = 0.0013.…”

“…In the shallow water case, their set of equations is the two-layer version of the Green-Naghdi equations. The equations derived in [9] were recently extended to the free-surface configuration [2]. Solitary waves for two-layer flows have also been computed numerically as solutions to the full incompressible Euler equations in the presence of an interface by various authors -see for example [22].…”

Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part I

“…This study provides physical insight into the problem of the interaction of the shear flow in the wake of components of offshore structures with the ocean surface. The behavior of the linear instability of the flow is similar to the nonstratified flow, but the growth rates are smaller [1]. The reduction of the growth rates depends on the degree of stratification.…”