Hamiltonian long-wave approximations of water waves with constant vorticity

doi:10.1016/j.physleta.2007.12.018

Physics Letters A

Volume 372, Issue 15, 7 April 2008, Pages 2597-2602

https://doi.org/10.1016/j.physleta.2007.12.018 Get rights and content

Abstract

Starting from a Hamiltonian formulation of water waves with constant vorticity we derive several long-wave approximations. These approximate models are also Hamiltonian and the connection between the symplectic structures is described by a simple transformation theory.

Introduction

An elegant way of deriving the classical model equations for long irrotational water waves is to expand the Hamiltonian in a power series in small parameters and truncate at the desired order. By keeping track of how the symplectic structure is transformed when the small parameters are introduced, the connection between the Hamiltonian formulation of the original problem and the model equations becomes apparent (see [1], [2]). The Hamiltonian for irrotational water waves involves the Dirichlet–Neumann operator, which maps Dirichlet boundary data for a harmonic function to Neumann data. Although the operator is non-local, it can be approximated by differential operators in the long-wave regime.

The aim of this Letter is to carry out the same procedure in the case of a underlying shear flow with constant vorticity. A Hamiltonian formulation for water waves with constant vorticity, which also includes the Dirichlet–Neumann operator, was recently derived in [3] (see also [4]). While approximate models for water waves on shear flows, both with constant and more general vorticities, have been derived previously [5], [6], [7], [8], [32], we are not aware of any derivations within the Hamiltonian framework. We will work formally, leaving aside questions of function spaces etc. Lately several results concerning the rigorous justification of model equations for irrotational water waves have appeared—see, e.g., [9], [10], [11], [12], [13], [33], [35]. It would be interesting to see if the same methods can be applied in the case of constant vorticity. Note however that there is to this date no local well-posedness result for the constant vorticity water wave problem with decay at spatial infinity. While the justification of the full dynamics of the model equations is an open problem, there are several papers treating solitary travelling waves with constant vorticity. Long-wave approximations as well as numerical results can be found in [14], [15], [16], [17], [18], [34], while existence results for the full water wave problem are given in [19], [20], [21] (see also [22], [23] for particle trajectories and [24] for solitary capillary-gravity waves). To the lowest order these waves are described by a stationary KdV equation.

Section snippets

Preliminaries

Let us briefly recall the governing equations for two-dimensional water waves with constant vorticity. The fluid domain $Ω_{η} = {(x, y) \in R^{2} : - h < y < η (t, x)}$ is bounded from below by a flat rigid bottom $B = {(x, y) \in R^{2} : y = - h}$ and above by a free surface $S_{η} = {(x, y) \in R^{2} : y = η (t, x})$ , which we assume to be the graph of a function. We will concentrate on waves with decay and so we assume that $η (t, x) \to 0$ as $| x | \to \infty$ . In the case of periodic waves η is assumed to be periodic and the fluid domain is restricted to a periodic

Hamiltonian transformation and perturbation theory

We will derive approximations of (2.1), (2.2) in various parameter regimes. In order to describe these regimes, we introduce two dimensionless scaling parameters $α = a / h$ and $β = {(h / l)}^{2}$ , where a is a typical amplitude and l is a measure of the width of the wave. Different parameter regimes are introduced by scaling both independent and dependent variables in the governing equations. Although this will not change the Hamiltonian nature of the equations, the Hamiltonian function and the structure map

The linearised equations

Before proceeding to the long-wave expansions, it is instructive to look at the equations of motion linearised around a steady state. The steady state is in our case the underlying shear flow, represented by $η = 0$ and $ξ = 0$ . In terms of the parameters described in the previous section, the small-amplitude regime occurs when $α = μ$ is small while $β = O (1)$ . It is introduced by scaling the dependent variables $η = μ η_{1}, ξ = μ ξ_{1} .$ We then obtain $H (η, ξ) = \sum_{j = 2}^{\infty} μ^{j} H^{(j)} (η_{1}, ξ_{1}),$ where $H^{(2)} (η_{1}, ξ_{1}) = \int_{R} (\frac{ξ_{1} G_{0} (η_{1}) ξ_{1}}{2} + \frac{g η_{1}^{2}}{2}) d x,$ $H^{(3)} ($

Shallow water approximation

The shallow water regime, $α = O (1)$ and $β = μ^{2}$ , where μ is a small parameter, is introduced by the transformation to stretched variables $x_{1} = μ x, t_{1} = μ t$ and scaling of ξ $ξ_{1} = μ ξ .$ This induces the transformation $D = μ D_{1} .$ The terms in the expansion of the Dirichlet–Neumann operator become $G_{0} = μ D_{1} \tanh (μ h D_{1}),$ $G_{1} (η) = μ^{2} (D_{1} η D_{1} - D_{1} \tanh (μ h D_{1}) η D_{1} \tanh (μ h D_{1})),$ etc. These expression can then be expanded in the small parameter μ, using $\tanh (μ h D_{1}) = μ h D_{1} - \frac{1}{3} {(μ h)}^{3} D_{1}^{3} + \frac{2}{15} {(μ h)}^{5} D_{1}^{5} + \dots .$ We thus obtain $H (η, ξ) = \int_{R} (\frac{g η^{2}}{2} - \frac{ω ξ_{1 x_{1}} η^{2}}{2} + \frac{ω^{2} η^{3}}{6} + (η + h) ξ_{1 x_{1}}$

The Boussinesq regime

In this regime one studies long, small-amplitude waves. In other words, $α = β = μ^{2}$ is a small parameter. The regime is induced by introducing stretched variables $x_{1} = μ x, t_{1} = μ t,$ and scaled dependent variables $η_{1} = \frac{1}{μ^{2}} η, ξ_{1} = \frac{1}{μ} ξ .$ The corresponding structure map is $μ^{−3} J$ (if we incorporate the stretching of t).

In the first approximation of the Hamiltonian, terms up to order $O (μ^{4})$ are retained. All but the first term in the Taylor series for $G (η)$ may therefore be neglected. Expanding $G_{0}$ in μ and keeping terms

Unidirectional waves of KdV type

We now concentrate on unidirectional waves. Let us first nondimensionalise the equations by setting $(x, y) = \frac{1}{h} (x^{'}, y^{'}), t = t^{'} {(\frac{g}{h})}^{1 / 2}, u = \frac{u^{'}}{\sqrt{g h}},$ where a prime denotes a dimensional variable. This is equivalent to setting $g = h = 1$ and introducing the nondimensional vorticity $\tilde{ω} = \frac{ω h}{\sqrt{g h}}$ .

In the lowest order the Boussinesq equation is the linear system ${\begin{matrix} η_{t} = - u_{x}, \\ u_{t} = - η_{x} - \hat{ω} u_{x}, \end{matrix}$ with Hamiltonian $H = \frac{1}{2} \int_{R} (η^{2} + u^{2}) d x$ and structure map $J = - \partial_{x} (\begin{matrix} 0 & 1 \\ 1 & \hat{ω} \end{matrix}) .$ The system can be diagonalised by introducing characteristic coordinates.

Acknowledgements

The author is grateful for constructive suggestions made by the referees.

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Hamiltonian long-wave approximations of water waves with constant vorticity

Abstract

Introduction

Section snippets

Preliminaries

Hamiltonian transformation and perturbation theory

The linearised equations

Shallow water approximation

The Boussinesq regime

Unidirectional waves of KdV type

Acknowledgements

Wave Motion

J. Comput. Phys.

Wave Motion

Phys. Lett. A

Commun. Pure Appl. Math.

Lett. Math. Phys.

Phys. Fluids A

J. Math. Phys.

J. Fluid Mech.

A Modern Introduction to the Mathematical Theory of Water Waves

Arch. Ration. Mech. Anal.

Osaka J. Math.

Commun. Pure Appl. Math.

Arch. Ration. Mech. Anal.

GAMM Mitt. Ges. Angew. Math. Mech.

Phys. Fluids