Hamiltonian long‐wave expansions for free surfaces and interfaces

Abstract

The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature.From the formulation, we develop a Hamiltonian perturbation theory for the long-wave limits, and we carry out a systematic analysis of the principal longwave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker [28], Benjamin [3,4], Ono [26], and many others. Our considerations include the Boussinesq and Korteweg-de Vries (KdV) regimes over finite-depth fluids, the Benjamin-Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long-wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface.Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa [6,7], we also derive novel systems of nonlinear dispersive long-wave equations in the large-amplitude, small-slope regime. Our formulation of the dynamical free-surface, free-interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations.

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

“…The limit cases (δ → 0, ∞ and γ → 0, 1) demand other scalings in the nondimensionalization than the ones presented in [16] (see Sect. A of [29] for example) and correspond to different regimes, such as the deep-water theory (from Benjamin [1] and Ono [35]), and lead to different models (see for example [7,14] in the rigid lid configuration, and [26,36,39] in the free surface case). In all of these cases, the calculations of our justification break: the dependence of the…”

“…The related Boussinesqtype systems have been justified (among many other asymptotic models) by Bona et al [7]. When the surface is not rigid and allowed to move as a free surface, it is known that there exist two different modes of wave motion, corresponding to different linear phase speeds (see Kakutani and Yamasaki [22], Leone et al [30], Michallet and Barthélemy [34] and Craig et al [14] for example). Accordingly, the KdV approximation states that any deformation of the surface and/or the interface will split up into four waves, each of them being lead by KdV equations.…”

Deep-water internal solitary waves near critical density ratio

“…A Hamiltonian formulation of the problem of a free interface between two ideal fluids, under rigid lid boundary conditions for the upper fluid, was also given by Benjamin & Bridges [21]. Craig & Groves [22] give a similar expression, by using the Dirichlet-Neumann operators for both the upper and lower fluid domains (see also [23]). The Hamiltonian can be expanded in series with respect to powers of the canonical variables.…”

A Boussinesq system for two-way propagation of interfacial waves

“…We will call it a system of extended Boussinesq equations (see also [11]). Linearizing (46) gives the same dispersion relation as before.…”