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Discrete & Continuous Dynamical Systems - A

2014 , Volume 34 , Issue 8

Special Issue on Water Waves

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Introduction
Adrian Constantin
2014, 34(8): i-iii doi: 10.3934/dcds.2014.34.8i +[Abstract](60) +[PDF](160.7KB)
Abstract:
We survey the content of the present special issue devoted to nonlinear water waves.
One-dimensional weakly nonlinear model equations for Rossby waves
David Henry and Rossen Ivanov
2014, 34(8): 3025-3034 doi: 10.3934/dcds.2014.34.3025 +[Abstract](59) +[PDF](342.5KB)
Abstract:
In this study we explore several possibilities for modelling weakly nonlinear Rossby waves in fluid of constant depth, which propagate predominantly in one direction. The model equations obtained include the BBM equation, as well as the integrable KdV and Degasperis-Procesi equations.
Recovering surface profiles of solitary waves on a uniform stream from pressure measurements
Hung-Chu Hsu
2014, 34(8): 3035-3043 doi: 10.3934/dcds.2014.34.3035 +[Abstract](66) +[PDF](290.4KB)
Abstract:
In this paper, we derive an explicit formula that permits to recover the free surface wave profile of an irrotational solitary wave with a uniform underlying current from pressure data measured at the flat bed of the fluid. The formula is valid for the governing equations and applies to waves of small and large amplitude.
Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories
Delia Ionescu-Kruse and Anca-Voichita Matioc
2014, 34(8): 3045-3060 doi: 10.3934/dcds.2014.34.3045 +[Abstract](140) +[PDF](452.2KB)
Abstract:
We consider the two-dimensional equatorial water-wave problem with constant vorticity in the $f$-plane approximation. Within the framework of small-amplitude waves, we derive the dispersion relations and we find the analytic solutions of the nonlinear differential equation system describing the particle paths below such waves. We show that the solutions obtained are not closed curves. Some remarks on the stagnation points are also provided.
On the nature of large and rogue waves
Mikhail Kovalyov
2014, 34(8): 3061-3093 doi: 10.3934/dcds.2014.34.3061 +[Abstract](56) +[PDF](7638.6KB)
Abstract:
In this paper we discuss a model of large and rogue waves in non-necessarily shallow water. We assume that the relevant portion of the flow is restricted to a near-surface layer, assumption which enables us to use the Kadomtsev-Petviashvili equation. The shape and behavior of several types of waves predicted by some singular solutions of the Kadomtsev-Petviashvili equation is compared to the physical waves observed in the ocean.
Particle trajectories in extreme Stokes waves over infinite depth
Tony Lyons
2014, 34(8): 3095-3107 doi: 10.3934/dcds.2014.34.3095 +[Abstract](59) +[PDF](347.8KB)
Abstract:
We investigate the velocity field of fluid particles in an extreme water wave over infinite depth. It is shown that the trajectories of particles within the fluid and along the free surface do not form closed paths over the course of one period, but rather undergo a positive drift in the direction of wave propagation. In addition it is shown that the wave crest cannot form a stagnation point despite the velocity of the fluid particles being zero there.
Dispersion relations for periodic water waves with surface tension and discontinuous vorticity
Calin Iulian Martin
2014, 34(8): 3109-3123 doi: 10.3934/dcds.2014.34.3109 +[Abstract](87) +[PDF](376.9KB)
Abstract:
We derive the dispersion relation for water waves with surface tension and having a piecewise constant vorticity distribution. More precisely, we consider here two scenarios; the first one is that of a flow with constant non-zero vorticity adjacent to the flat bed while above this layer of vorticity we assume the flow to be irrotational. The second type of flow has a layer of non-vanishing vorticity adjacent to the free surface and is irrotational below.
A characterization of the symmetric steady water waves in terms of the underlying flow
Bogdan-Vasile Matioc
2014, 34(8): 3125-3133 doi: 10.3934/dcds.2014.34.3125 +[Abstract](52) +[PDF](331.2KB)
Abstract:
In this paper we present a characterization of the symmetric rotational periodic gravity water waves of finite depth and without stagnation points in terms of the underlying flow. Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only if there exists a vertical line within the fluid domain such that all the fluid particles located on that line minimize there simultaneously their distance to the fluid bed as they move about. Our analysis uses the moving plane method, sharp elliptic maximum principles, and the principle of analytic continuation.
A boundary integral formulation for particle trajectories in Stokes waves
André Nachbin and Roberto Ribeiro-Junior
2014, 34(8): 3135-3153 doi: 10.3934/dcds.2014.34.3135 +[Abstract](80) +[PDF](979.2KB)
Abstract:
Recently important theorems have been established presenting qualitative results for particle trajectories below a Stokes wave. A diversity of orbit patterns were described, including the case of a closed orbit when a Stokes wave propagates in the presence of an adverse current. In this work these results are revisited in a quantitative fashion through a boundary integral formulation which leads to very accurate numerical simulations of particle trajectories. The boundary integral formulation allows the accurate evaluation of the vector field of the (particle's) dynamical system, without resorting to a series expansion and a small parameter. Accurate trajectories are benchmarked against well known expansions for weakly nonlinear waves. Simulations are then performed beyond this regime. Closed orbits are found in the presence of an adverse current, as well as non-smooth trajectories that have not been reported. These occur for both adverse and favorable currents.
Steady-states and traveling-wave solutions of the generalized Constantin--Lax--Majda equation
Hisashi Okamoto, Takashi Sakajo and Marcus Wunsch
2014, 34(8): 3155-3170 doi: 10.3934/dcds.2014.34.3155 +[Abstract](94) +[PDF](730.6KB)
Abstract:
Steady-states and traveling-waves of the generalized Constantin--Lax--Majda equation are computed and their asymptotic behavior is described. Their relation with possible blow-up and the Benjamin--Ono equation is discussed.
Progressive waves on a blunt interface
Michael Stiassnie and Raphael Stuhlmeier
2014, 34(8): 3171-3182 doi: 10.3934/dcds.2014.34.3171 +[Abstract](56) +[PDF](861.3KB)
Abstract:
We present a new exact solution describing progressive waves on a blunt interface based on Gerstner's trochoidal wave. The second-order irrotational theory is developed for a sharp interface, and subsequently for three fluid layers, the upper and lower of which may approach one another to form the so-called blunt interface. This situation is captured analogously by our exact rotational solution. We establish remarkable agreement between the exact and second-order theories, and present applications to surface water waves.
Internal Gerstner waves on a sloping bed
Raphael Stuhlmeier
2014, 34(8): 3183-3192 doi: 10.3934/dcds.2014.34.3183 +[Abstract](45) +[PDF](313.9KB)
Abstract:
We provide an explicit solution to the full, nonlinear governing equations for gravity water waves describing internal edge waves along a sloping bed. This solution is based on the Gerstner edge wave. We discuss the relation of this internal, trochoidal edge wave to the analogous wave found in the linear theory, compare it with the classical Gerstner wave, as well as discuss the inclusion of Coriolis forces in the f-plane approximation.
Energy-minimising parallel flows with prescribed vorticity distribution
J. F. Toland
2014, 34(8): 3193-3210 doi: 10.3934/dcds.2014.34.3193 +[Abstract](43) +[PDF](440.0KB)
Abstract:
This note concerns a nonlinear differential equation problem in which both the nonlinearity in the equation and its solution are determined by other constraints. The question under consideration arises from a study of two-dimensional steady parallel-flows of a perfect fluid governed by Euler's equations and a free-boundary condition, when the distribution of vorticity is arbitrary but prescribed.
Non-existence of global energy minimisers in Stokes waves problems
J. F. Toland
2014, 34(8): 3211-3217 doi: 10.3934/dcds.2014.34.3211 +[Abstract](64) +[PDF](143.5KB)
Abstract:
Recently it was shown that a wave profile which minimises total energy, elastic plus hydrodynamic, subject to the vorticity distribution being prescribed, gives rise to a steady hydroelastic wave. Using this formulation, the existence of non-trivial minimisers leading to such waves was established for certain non-zero values of the elastic constants used to model the surface. Here we show that when these constants are zero, global minimisers do not exist except in a unique set of circumstances.
Pressure beneath a traveling wave with constant vorticity
Vishal Vasan and Katie Oliveras
2014, 34(8): 3219-3239 doi: 10.3934/dcds.2014.34.3219 +[Abstract](88) +[PDF](1988.6KB)
Abstract:
The main focus of this paper is to derive a direct relationship between the surface of an inviscid traveling gravity wave in two dimensions, and the pressure at the bottom of the fluid without approximation, including the effects of constant vorticity. Using this relationship, we reconstruct both the pressure and streamlines throughout the fluid domain. We compare our numerical results with various analytical results (such as the bounds presented in [7-10])as well as known numerical results (see [16]).
Steady stratified periodic gravity waves with surface tension I: Local bifurcation
Samuel Walsh
2014, 34(8): 3241-3285 doi: 10.3934/dcds.2014.34.3241 +[Abstract](71) +[PDF](650.2KB)
Abstract:
In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of small-amplitude solutions. This is accomplished by first constructing a 1-parameter family of laminar flow solutions, $\mathcal{T}$, then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from $\mathcal{T}$ at an eigenvalue of the linearized problem.
Steady stratified periodic gravity waves with surface tension II: Global bifurcation
Samuel Walsh
2014, 34(8): 3287-3315 doi: 10.3934/dcds.2014.34.3287 +[Abstract](81) +[PDF](522.4KB)
Abstract:
In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of global continua of classical solutions that are periodic and traveling. This is accomplished by globally continuing the curves of small-amplitude solutions obtained by the author in [25]. We do this in two ways: first, by means of a degree theoretic theorem in the spirit of Rabinowitz, and second via the analytic continuation method of Dancer.

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