An Introduction to the Geometry and Topology of Fluid Flows

345 Pages·2001·15.927 MB·English

by  H. Keith Moffatt (auth.), Renzo L. Ricca (eds.)

An Introduction to the Geometry and Topology
of Fluid Flows
NATO Science Series
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Series II: Mathematics, Physics and Chemistry - Vol. 47
An Introduction to the Geometry
and Topology of Fluid Flows
edited by
Renzo L. Ricca
Mathematics Department,
University College London,
London, United Kingdom
lIUII
"
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Study Institute on
Pedagogical Workshop on Geometry and Topology of Fluid Flows
Cambridge, United Kingdom
11 - 22 September 2000
A C.1. P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-0207-6 ISBN 978-94-010-0446-6 (eBook)
DOI 10.1007/978-94-010-0446-6
Printed on acid-free paper
AII Rights Reserved
©2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001
Softcover reprint of the hardcover 1s t edition 2001
No part of the material protected by this copyright notice may be reproduced or utilized in
any form or by any means, electronic or mechanical, including photocopying, recording
or by any information storage and retrieval system, without written permission from the
copyright owner.
Any ·d ea.r ;
'I,
CONTENTS
Preface IX
Photograph of H.K. Moffatt Xl
I. EIGHT PROBLEMS FOR THE XXI CENTURY 1
Some Remarks on Topological Fluid Mechanics 3
H.K. Moffatt
II. MATHEMATICS BACKGROUND 11
Differential Geometry of Curves and Surfaces 13
R. Langevin
Topology in Four Days 35
T. Tokieda
Elements of Classical Knot Theory 57
C. Weber
An Introduction to Knot Theory 77
L.H. Kauffman
Fluid Mechanics and Mathematical Structures 105
P. Boyland
vii
Vlll
III. GEOMETRY AND TOPOLOGY
OF FLUID FLOWS 135
Introduction to a Geometrical Theory of Fluid Flows
and Dynamical Systems 137
T. Kambe
Streamline Patterns and their Bifurcations Using Methods
from Dynamical Systems 167
M. Br¢ns
Topological Features of Inviscid Flows 183
R. Grhist (1 R. Komendarczyk
Geometric and Topological Aspects of Vortex Motion 203
R.L. Ricca
Topology Bounds the Energy 229
B.A. Khesin
Measures of Topological Structure in Magnetic Fields 239
M.A. Berger
Diffeomorphisms, Braids and Flows 253
A. Shnirelman
Variational Principles, Geometry and Topology
of Lagrangian-Averaged Fluid Dynamics 271
D.D. Holm
IV. RECONNECTIONS AND SINGULARITIES 293
The Geometry of Reconnection 295
G. Hornig
Euler Singularities from the Lagrangian Viewpoint 313
S. Childress
Analysis of a Candidate Flow for Hydrodynamic Blowup 329
R.B. Pelz
Subject Index 343
PREFACE
In editing this collection of lectures I kept in mind the spirit with which Hassan
Aref, Tsutomu Kambe, Richard Pelz and myselforganized the programme on
Geometry and Topology of Fluid Flows held at the Isaac Newton Institute for
Mathematical Sciences Cambridge, UK, September-December, 2000: to promote
and fosterinterchange of knowledge and information between mathematicians,
pure and applied, and physicists on modern applications of geometric and
topological techniques in fluid mechanics and magnetohydrodynamics. I therefore
asked the authors to contribute with papers that could reflect this spirit. I hope we
achieved our goal with a volume that will provide a valuable addition to your
library. If there is any merit in this our thanks must go to the contributors.
Mistakes, misprints and sloppy presentations of material, however, should be
entirely ascribed to this editor. I would like to thank my friends and co-organisers
of the GTFF programme, the Director of the Isaac Newton Institute, Professor
H.K. Moffatt, and the dedicated staff,for their continuing support and help
towards the organisation of that successful programme. I would also like to
acknowledge financial support from NATO through the Advanced Study Institute
held in Cambridge.
Renzo L. Ricca
ix
Henry Keith Moffatt
Photograph: R.L. Ricca - Cetraro (Italy), July 2001
I. EIGHT PROBLEMS FOR THE XXI CENTURY