Advanced

Rotational effects in water waves

Wahlén, Erik LU (2008)
Abstract (Swedish)
Popular Abstract in Swedish

Avhandlingen består av fyra artiklar som behandlar olika aspekter av vattenvågor med vorticitet.



Artikel I: Symmetry of steady periodic gravity water waves with vorticity.



Vi bevisar att en fortskridande, periodisk, två-dimensionell, rotationell gravitationsvåg, med en ytprofil som är monoton mellan vågdalar och vågtoppar, måste vara symmetrisk kring vågtoppen, oavsett hur fördelningen av vorticitet ser ut inuti vätskan.



Artikel II: Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity.



Vi presenterar existensteorier för flera olika familjer av solitära vågor... (More)
Popular Abstract in Swedish

Avhandlingen består av fyra artiklar som behandlar olika aspekter av vattenvågor med vorticitet.



Artikel I: Symmetry of steady periodic gravity water waves with vorticity.



Vi bevisar att en fortskridande, periodisk, två-dimensionell, rotationell gravitationsvåg, med en ytprofil som är monoton mellan vågdalar och vågtoppar, måste vara symmetrisk kring vågtoppen, oavsett hur fördelningen av vorticitet ser ut inuti vätskan.



Artikel II: Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity.



Vi presenterar existensteorier för flera olika familjer av solitära vågor med liten amplitud som löser det klassiska två-dimensionella vattenvågsproblemet med ytspänning och vorticitet. Det lokala bifurkationsdiagrammet visar sig vara detsamma som för irrotationella solitära vågor, oavsett hur vorticiteten är fördelad. Det hydrodynamiska problet formuleras som ett oändligtdimensionellt Hamiltonskt system i vilket den horisontella rumsvariabeln har rollen som tidsvariabel. Sedan används ändligtdimensionell reduktion för att ersätta systemet med ett lokalt ekvivalent Hamiltonskt system med ändligt många frihetsgrader. Homokliniska lösningar till det reducerade systemet, som motsvarar solitära vågor, hittas genom olika metoder för dynamiska system.





Artikel III: A Hamiltonian formulation of water waves with constant vorticity.



Vi visar att de styrande ekvationerna för två-dimensionella vattenvågor med konstant vorticitet kan formuleras som ett kanoniskt Hamiltonskt system, i vilket en av de kanoniska variablerna är ythöjden.



Artikel IV: Hamiltonian long-wave approximations of water waves with constant vorticity



Med utgångspunkt i den Hamiltonska formuleringen i Artikel III härleder vi flera olika approximativa modeller för långa vattenvågor. Dessa approximativa modeller är också Hamiltonska och sambandet mellan de olika symplektiska strukturerna beskrivs med en enkel transformationsteori. (Less)
Abstract
Abstract: This thesis consists of four papers related to various aspects of water waves with vorticity.



Paper I: Symmetry of steady periodic gravity water waves with vorticity.



We prove that steady periodic two-dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid.



Paper II: Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity.



We present existence theories for several families of small-amplitude solitary-wave solutions to the classical two-dimensional... (More)
Abstract: This thesis consists of four papers related to various aspects of water waves with vorticity.



Paper I: Symmetry of steady periodic gravity water waves with vorticity.



We prove that steady periodic two-dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid.



Paper II: Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity.



We present existence theories for several families of small-amplitude solitary-wave solutions to the classical two-dimensional water-wave problem in the presence of surface tension and vorticity. The established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods.



Paper III: A Hamiltonian formulation of water waves with constant vorticity.



We show that the governing equations for two-dimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the surface elevation.



Paper IV: Hamiltonian long-wave approximations of water waves with constant vorticity



Starting with the Hamiltonian formulation in Paper III 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. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor Escher, Joachim, Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1 30167 Hannover, Germany
organization
publishing date
type
Thesis
publication status
published
subject
pages
105 pages
defense location
Aula C, Centre for Mathematical Sciences, Sölvegatan 18, Lund
defense date
2008-03-19 13:15
ISSN
1404-0034
ISBN
978-91-628-7397-4
language
English
LU publication?
yes
id
09d88fad-9ec5-4f04-a0da-5a9812380adc (old id 1031485)
date added to LUP
2008-02-18 10:34:19
date last changed
2016-09-19 08:45:00
@misc{09d88fad-9ec5-4f04-a0da-5a9812380adc,
  abstract     = {Abstract: This thesis consists of four papers related to various aspects of water waves with vorticity.<br/><br>
<br/><br>
Paper I: Symmetry of steady periodic gravity water waves with vorticity. <br/><br>
<br/><br>
 We prove that steady periodic two-dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid.<br/><br>
<br/><br>
Paper II: Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity.<br/><br>
<br/><br>
 We present existence theories for several families of small-amplitude solitary-wave solutions to the classical two-dimensional water-wave problem in the presence of surface tension and vorticity. The established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods.<br/><br>
<br/><br>
Paper III: A Hamiltonian formulation of water waves with constant vorticity.<br/><br>
<br/><br>
 We show that the governing equations for two-dimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the surface elevation.<br/><br>
 <br/><br>
Paper IV: Hamiltonian long-wave approximations of water waves with constant vorticity<br/><br>
 <br/><br>
 Starting with the Hamiltonian formulation in Paper III 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.},
  author       = {Wahlén, Erik},
  isbn         = {978-91-628-7397-4},
  issn         = {1404-0034},
  language     = {eng},
  pages        = {105},
  title        = {Rotational effects in water waves},
  year         = {2008},
}