Variational Principles for Water Waves

We describe the Hamiltonian structures, including the Poisson brackets and Hamiltonians, for free boundary problems for incompressible fluid flows with vorticity. The Hamiltonian structure is used to obtain variational principles for stationary gravity waves both for irrotational flows as well as flows with vorticity.Comment: 20 page

Practical use of variational principles for modeling water waves

This paper describes a method for deriving approximate equations for irrotational water waves. The method is based on a 'relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible. This formulation is particularly suitable for the construction of approximate water wave models, since it allows more freedom while preserving the variational structure. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters, as well as arbitrary depths. Using subordinate constraints (e.g., irrotationality or free surface impermeability) in various combinations, several model equations are derived, some being well-known, other being new. The models obtained are studied analytically and exact travelling wave solutions are constructed when possible.Comment: 30 pages, 1 figure, 62 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Modeling water waves beyond perturbations

In this chapter, we illustrate the advantage of variational principles for modeling water waves from an elementary practical viewpoint. The method is based on a `relaxed' variational principle, i.e., on a Lagrangian involving as many variables as possible, and imposing some suitable subordinate constraints. This approach allows the construction of approximations without necessarily relying on a small parameter. This is illustrated via simple examples, namely the Serre equations in shallow water, a generalization of the Klein-Gordon equation in deep water and how to unify these equations in arbitrary depth. The chapter ends with a discussion and caution on how this approach should be used in practice.Comment: 15 pages, 1 figure, 39 references. This document is a contributed chapter to an upcoming volume to be published by Springer in Lecture Notes in Physics Series. Other author's papers can be downloaded at http://www.denys-dutykh.com

Modelling of nonlinear wave-buoy dynamics using constrained variational methods

We consider a comprehensive mathematical and numerical strategy to couple water-wave motion with rigid ship dynamics using variational principles. We present a methodology that applies to three-dimensional potential flow water waves and ship dynamics. For simplicity, in this paper we demonstrate the method for shallow-water waves coupled to buoy motion in two dimensions, the latter being the symmetric motion of a crosssection of a ship. The novelty in the presented model is that it employs a Lagrange multiplier to impose a physical restriction on the water height under the buoy in the form of an inequality constraint. A system of evolution equations can be obtained from the model and consists of the classical shallow-water equations for shallow, incompressible and irrotational waves, and relevant equations for the dynamics of the wave-energy buoy. One of the advantages of the variational approach followed is that, when combined with symplectic integrators, it eliminates any numerical damping and preserves the discrete energy; this is confirmed in our numerical results

An alternative view on the Bateman-Luke variational principle

A new derivation of the Bernoulli equation for water waves in three-dimensional rotating and translating coordinate systems is given. An alternative view on the Bateman-Luke variational principle is presented. The variational principle recovers the boundary value problem governing the motion of potential water waves in a container undergoing prescribed rigid-body motion in three dimensions. A mathematical theory is presented for the problem of three-dimensional interactions between potential surface waves and a floating structure with interior potential fluid sloshing. The complete set of equations of motion for the exterior gravity-driven water waves, and the exact nonlinear hydrodynamic equations of motion for the linear momentum and angular momentum of the floating structure containing fluid, are derived from a second variational principle

A water wave model with horizontal circulation and accurate dispersion

We describe a new water wave model which is variational, and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite element profile with a small number of elements (say), leading to a framework for efficient modelling of the interaction of steepening and breaking waves near the shore with a large-scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity (or circulation). We show that the potential flow water wave equations and the shallow-water equations are recovered in the relevant limits, and provide approximate shock relations for the model which can be used in numerical schemes to model breaking waves

Variational water-wave model with accurate dispersion and vertical vorticity

A new water-wave model has been derived which is based on variational techniques and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite-element profile with a small number of elements (say), leading to a framework for efficient modeling of the interaction of steepening and breaking waves near the shore with a large-scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity. It is shown that the potential-flow water-wave equations and the shallow-water equations are recovered in the relevant limits. Approximate shock relations are provided, which can be used in numerical schemes to model breaking waves

Effective coastal boundary conditions for dispersive tsunami propagation

We aim to improve the techniques to predict tsunami wave heights along the coast. The modeling of tsunamis with the shallow water equations has been very successful, but is somewhat simplistic because wave dispersion is neglected. To bypass this shortcoming, we use the (linearized) variational Boussinesq model derived by Klopman et al. [J. Fluid Mech. 657, 36--63, 2010]. Another shortcoming is that the complicated interactions between incoming and reflected waves near the shore are usually simplified by a fixed wall boundary condition at a certain shallow depth contour. To alleviate this shortcoming, we explore and present in one spatial dimension a so-called effective boundary condition (EBC). From the deep ocean to the seaward boundary, i.e., the simulation area, we model wave propagation numerically. Given the measurements of the incoming wave at the seaward boundary, we model the wave dynamics towards the shoreline analytically, based on shallow water theory and the Wentzel-Kramer-Brillouin (WKB) approximation, as well as extensions to the dispersive, Boussinesq model. The reflected wave is then influxed back into the simulation area using the EBC. The coupling between the two areas, one done numerically and one analytically, via the EBC is handled using variational principles, to preserve the overall energy in both areas. We verify and validate our approach in a series of numerical test cases of increasing complexity, including a case akin to tsunami propagation to the coastline at Aceh, Sumatra, Indonesia

New variational and multisymplectic formulations of the Euler-Poincar\'e equation on the Virasoro-Bott group using the inverse map

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincar\'e equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with $2$-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.Comment: 19 page

Two-timing, variational principles and waves

In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples