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2019 | Buch

Nonlinear Water Waves

An Interdisciplinary Interface

herausgegeben von: Dr. David Henry, Dr. Konstantinos Kalimeris, Emilian I. Părău, Prof. Jean-Marc Vanden-Broeck, Prof. Erik Wahlén

Verlag: Springer International Publishing

Buchreihe : Tutorials, Schools, and Workshops in the Mathematical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The motion of water is governed by a set of mathematical equations which are extremely complicated and intractable. This is not surprising when one considers the highly diverse and intricate physical phenomena which may be exhibited by a given body of water. Recent mathematical advances have enabled researchers to make major progress in this field, reflected in the topics featured in this volume.

Cutting-edge techniques and tools from mathematical analysis have generated strong rigorous results concerning the qualitative and quantitative physical properties of solutions of the governing equations. Furthermore, accurate numerical computations of fully-nonlinear steady and unsteady water waves in two and three dimensions have contributed to the discovery of new types of waves. Model equations have been derived in the long-wave and modulational regime using Hamiltonian formulations and solved numerically.

This book brings together interdisciplinary researchers working in the field of nonlinear water waves, whose contributions range from survey articles to new research results which address a variety of aspects in nonlinear water waves. It is motivated by a workshop which was organised at the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna, November 27-December 7, 2017. The key aim of the workshop was to describe, and foster, new approaches to research in this field. This is reflected in the contents of this book, which is aimed to stimulate both experienced researchers and students alike.

Inhaltsverzeichnis

Frontmatter

Modeling Surface Waves Over Highly Variable Topographies

Abstract

This article introduces some applied mathematics research problems on surface water waves propagating in the presence of highly variable bottom topographies. Asymptotic problems arise from variable coefficient partial differential equations regarding both its solutions, as well as the differential operators’ reduced modeling. Two simple problems are first introduced, setting the main ideas.

André Nachbin

Global Diffeomorphism of the Lagrangian Flow-map for a Pollard-like Internal Water Wave

Abstract

In this article we provide an overview of a rigorous justification of the global validity of the fluid motion described by a new exact and explicit solution prescribed in terms of Lagrangian variables of the nonlinear geophysical equations. More precisely, the three-dimensional Lagrangian flow-map describing this exact and explicit solution is proven to be a global diffeomorphism from the labelling domain into the fluid domain. Then, the flow motion is shown to be dynamically possible.

Mateusz Kluczek, Adrián Rodríguez-Sanjurjo

The Unified Transform and the Water Wave Problem

Abstract

The unified transform, also known as the Fokas method, was introduced in 1997 by one of the authors Fokas (Proc R Soc Lond A: Math Phys Eng Sci 453(1962):1411–1443, 1997 ) for the analysis of nonlinear initial-boundary value problems. Later, it was realised that this method also yields novel results for linear problems. In 2006, the classical water wave problem was studied via the Fokas method (Ablowitz et al., J Fluid Mech 562:313–343, 2006), yielding a novel non-local formulation. In this paper we review the unified transform, with particular emphasis on its application in water wave in two spacial dimensions with moving boundaries.

A. S. Fokas, K. Kalimeris

HOS Simulations of Nonlinear Water Waves in Complex Media

Abstract

We present an overview of recent extensions of the high-order spectral method of Craig and Sulem (J Comput Phys 108:73–83, 1993) to simulating nonlinear water waves in a complex environment. Under consideration are cases of wave propagation in the presence of fragmented sea ice, variable bathymetry and a vertically sheared current. Key components of this method, which apply to all three cases, include reduction of the full problem to a lower-dimensional system involving boundary variables alone, and a Taylor series representation of the Dirichlet–Neumann operator. This results in a very efficient and accurate numerical solver by using the fast Fourier transform. Two-dimensional simulations of unsteady wave phenomena are shown to illustrate the performance and versatility of this approach.

Philippe Guyenne

Stokes Waves in a Constant Vorticity Flow

Abstract

The Stokes wave problem in a constant vorticity flow is formulated via conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a fold develops in the wave speed versus amplitude plane, and a gap as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself, enclosing a bubble of air. More folds and gaps follow as the vorticity strength increases further. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while a fluid disk in rigid body rotation at the ends of the gaps. Touching waves at the boundaries of higher gaps contain more fluid disks.

Sergey A. Dyachenko, Vera Mikyoung Hur

Integrable Models of Internal Gravity Water Waves Beneath a Flat Surface

Abstract

A two-layer fluid system separated by a pycnocline in the form of an internal wave is considered. The lower layer is bounded below by a flat bottom and the upper layer is bounded above by a flat surface. The fluids are incompressible and inviscid and Coriolis forces as well as currents are taken into consideration. A Hamiltonian formulation is presented and appropriate scaling leads to a KdV approximation. Additionally, considering the lower layer to be infinitely deep leads to a Benjamin–Ono approximation.

Alan C. Compelli, Rossen I. Ivanov, Tony Lyons

Numerical Simulations of Overturned Traveling Waves

Abstract

Dimension-breaking continuation as a numerical technique for computing large amplitude, overturned traveling waves is presented. Dimension-breaking bifurcations from branches of planar waves are presented in two weakly-nonlinear model equations as well as in the vortex sheet formulation of the water wave problem, with the small scale approximation (Ambrose et al., J Comput Phys 247:168–191, 2013; Akers and Reeger, Wave Motion 68:210–217, 2017). The challenges and potential of this method toward computing overturned traveling waves at the interface between three-dimensional fluids is reviewed. Numerical simulations of dimension-breaking continuation are presented in each model. Overturned traveling three-dimensional waves are presented in the vortex sheet system.

Benjamin F. Akers, Matthew Seiders

A Model for the Periodic Water Wave Problem and Its Long Wave Amplitude Equations

Abstract

We are interested in the validity of the KdV and of the long wave NLS approximation for the water wave problem over a periodic bottom. Approximation estimates are non-trivial, since solutions of order \( \mathcal {O}(\varepsilon ^2) \), resp. \( \mathcal {O}(\varepsilon ) \), have to be controlled on an \( \mathcal {O}(1/\varepsilon ^{3}) \), resp. \( \mathcal {O}(1/\varepsilon ^{2}) \), time scale. In contrast to the spatially homogeneous case, in the periodic case new quadratic resonances occur and make a more involved analysis necessary. For a phenomenological model we present some results and explain the underlying ideas. The focus is on results which are robust in the sense that they hold under very weak non-resonance conditions without a detailed discussion of the resonances. This robustness is achieved by working in spaces of analytic functions. We explain that, if analyticity is dropped, the KdV and the long wave NLS approximation make wrong predictions in case of unstable resonances and suitably chosen periodic boundary conditions. Finally we outline, how, we think, the presented ideas can be transferred to the water wave problem.

Roman Bauer, Patrick Cummings, Guido Schneider

On Recent Numerical Methods for Steady Periodic Water Waves

Abstract

The study of steady periodic water waves, analytically as well as numerically, is a very active field of research. We describe some of the more recent numerical approaches to computing these waves numerically as well as the corresponding results. The focus of this work is on the different formulations as well as their limitations and similarities.

Dominic Amann

Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory

Abstract

We review the theory of wave interaction in finite and infinite depth. Both of these strands of water-wave research begin with the deterministic governing equations for water waves, from which simplified equations can be derived to model situations of interest, such as the mild slope and modified mild slope equations, the Zakharov equation, or the nonlinear Schrödinger equation. These deterministic equations yield accompanying stochastic equations for averaged quantities of the sea-state, like the spectrum or bispectrum. We discuss several of these in depth, touching on recent results about the stability of open ocean spectra to inhomogeneous disturbances, as well as new stochastic equations for the nearshore.

Raphael Stuhlmeier, Teodor Vrecica, Yaron Toledo

Gravity-Capillary and Flexural-Gravity Solitary Waves

Abstract

Solitary gravity-capillary and flexural-gravity waves in two and three dimensions of space are reviewed in this paper. Numerical methods used to compute the solitary waves are described in detail and typical solutions found over the years are presented. Similarities and differences between the solutions for the two physical problems are discussed.

Emilian I. Părău, Jean-Marc Vanden-Broeck

A Method for Identifying Stability Regimes Using Roots of a Reduced-Order Polynomial

Abstract

For dispersive Hamiltonian partial differential equations of order 2N + 1, \(N\in \mathbb {Z}\), there are two criteria to analyse to examine the stability of small-amplitude, periodic travelling wave solutions to high-frequency perturbations. The first necessary condition for instability is given via the dispersion relation. The second criterion for instability is the signature of the eigenvalues of the spectral stability problem given by the sign of the Hamiltonian. In this work, we show how to combine these two conditions for instability into a polynomial of degree N. If the polynomial contains no real roots, then the travelling wave solutions are stable. We present the method for deriving the polynomial and analyse its roots using Sturm’s theory via an example.

Olga Trichtchenko

Titel: Nonlinear Water Waves
herausgegeben von: Dr. David Henry
Dr. Konstantinos Kalimeris
Emilian I. Părău
Prof. Jean-Marc Vanden-Broeck
Prof. Erik Wahlén
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-33536-6
Print ISBN: 978-3-030-33535-9
DOI: https://doi.org/10.1007/978-3-030-33536-6