Topological Aspects of the Dynamics of Fluids and Plasmas

1. Front Matter

   Pages i-xii

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2. Introductory Lectures

   1. Front Matter

      Pages 1-1

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   2. Relaxation Under Topological Constraints

      • H. K. Moffatt

      Pages 3-28

   3. Knot Theory, Jones’ Polynomials, Invariants of 3-Manifolds, and the Topological Theory of Fluid Dynamics

      • Kenneth C. Millett

      Pages 29-64

   4. Topology of Knots

      • Martin Scharlemann

      Pages 65-82

   5. Stretching and Alignment in General Flow Fields: Classical Trajectories from Reynolds Number Zero to Infinity

      • M. Tabor

      Pages 83-110

   6. Fast Dynamo Theory

      • Stephen Childress

      Pages 111-147

3. Relaxation and Minimum Energy States

   1. Front Matter

      Pages 149-149

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   2. Relaxation and Topology in Plasma Experiments

      • J. B. Taylor

      Pages 151-165

   3. Taylor’s Relaxation in an Unbounded Domain of Space

      • J. J. Aly

      Pages 167-175

   4. Force-Free Magnetic Fields with Constant Alpha

      • S. I. Vainshtein

      Pages 177-193

   5. Minimum Energy Magnetic Fields with Toroidal Topology

      • A. Y. K. Chui, H. K. Moffatt

      Pages 195-218

   6. Research Announcement on the “Energy” of Knots

      • Michael H. Freedman, Zheng-Xu He

      Pages 219-222

4. Helicity, Linkage, and Flow Topology

   1. Front Matter

      Pages 223-223

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   2. The Helicity of a Knotted Vortex Filament

      • R. L. Ricca, H. K. Moffatt

      Pages 225-236

   3. A Hierarchy of Linking Integrals

      • N. W. Evans, M. A. Berger

      Pages 237-248

   4. Borromeanism and Bordism

      • Peter Akhmet’ev, Alexander Ruzmaikin

      Pages 249-264

   5. Topology of Steady Fluid Flows

      • Viktor L. Ginzburg, Boris Khesin

      Pages 265-272

5. The Euler Equations: Extremal Properties and Finite Time Singularities

   1. Front Matter

      Pages 273-273

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   2. Extremal Properties and Hamiltonian Structure of the Euler Equations

      • T. G. Shepherd

      Pages 275-292

This volume contains papers arising out of the program of the Institute for Theoretical Physics (ITP) of the University of California at Santa Bar­ bara, August-December 1991, on the subject "Topological Fluid Dynamics". The first group of papers cover the lectures on Knot Theory, Relaxation un­ der Topological Constraints, Kinematics of Stretching, and Fast Dynamo Theory presented at the initial Pedagogical Workshop of the program. The remaining papers were presented at the subsequent NATO Advanced Re­ search Workshop or were written during the course of the program. We wish to acknowledge the support of the NATO Science Committee in making this workshop possible. The scope of "Topological Fluid Dynamics" was defined by an earlier Symposium of the International Union of Theoretical and Applied Mechan­ ics (IUTAM) held in Cambridge, England in August, 1989, the Proceedings of which were published (Eds. H.K. Moffatt and A. Tsinober) by Cambridge University Press in 1990. The proposal to hold an ITP program on this sub­ ject emerged from that Symposium, and we are grateful to John Greene and Charlie Kennel at whose encouragement the original proposal was formu­ lated. Topological fluid dynamics covers a range of problems, particularly those involving vortex tubes and/or magnetic flux tubes in nearly ideal fluids, for which topological structures can be identified and to some extent quantified.

• Fractal
• Plasma
• Volume
• fluid dynamics
• magnetohydrodynamics

• Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

  H. K. Moffatt

• Courant Institute of Mathematical Sciences, New York University, New York, USA

  G. M. Zaslavsky

• Institut de Mécanique de Grenoble, Institut National Polytechnique de Grenoble, Grenoble, France

  P. Comte

• Department of Mathematics, University of Arizona, Tucson, USA

  M. Tabor

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