Publication: Hamiltonian long wave expansions for free surfaces and interfaces

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Title Hamiltonian long wave expansions for free surfaces and interfaces
Authors/Editors* W. Craig, P. Guyenne, H. Kalisch
Where published* Commun. Pure Applied Math
How published* Journal
Year* 2005
Volume 58
Number 0
Publisher xxxxx
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 which 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 long wave scaling regimes. These include the Boussinesq and 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 \& Camassa \cite{ChoiCamassa96,ChoiCamassa99}, 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, and as well it holds promise as a basis for numerical simulations.
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