MAGIC052: Topological Fluid Mechanics

Course details

Semester

Spring 2010
Monday, January 11th to Friday, March 19th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
0

Timetable

Wednesdays
11:05 - 11:55

Announcements

Part I (AG): basics, helicity and relaxation (4 lectures)
Part III (AG): dynamics of vortex filaments and singularities (2 lectures)
  • There will be a set of Latex notes available in the files top-fl-mech-latex-I.pdf, top-fl-mech-latex-III.pdf, which you can download from the web page. These notes are detailed but do not include pictures.

  • The Latex notes include bibliography and exercises. There is no single source that is recommended, but the notes of Steve Childress are good, as are original papers by Keith Moffatt for I, and papers of Renzo Ricca and Keith Moffatt for III.

  • The actual lectures will be slides which will go on as top-fl-mech-I-a.pdf, -b.pdf, etc. These will be in lecture form, and so more notes with pictures. I am breaking the sequence of slides into sets of 10 slides: each file does not necessarily correspond to a lecture.

Description

The title Topological Fluid Mechanics covers a range of methods for understanding fluid mechanics (and related areas) in terms of the geometry and topology of continuous fields. For example in ideal fluid mechanics the vorticity field can be considered: by Kelvin's theorem the field is frozen, moving in the fluid flow and its topology is conserved. Topological invariants can thus be used to describe aspects of the fluid flow. There are similar applications in magnetohydrodynamics, relevant to the Solar magnetic field.


This course will be lectured by Andrew Gilbert and Mitchell Berger (University of Exeter)

Prerequisites

PREREQUISITES:

knowledge of vector calculus and fluid mechanics up to 3rd year undergraduate level.
basic knowledge of pure mathematics, in particular group theory up to 2nd year undergraduate level.

NOT REQUIRED:

knowledge of magnetohydrodynamics: this will be developed where needed.
knowledge of pure mathematics beyond basic group theory.

NOTE:

ideas will be developed in concert with, and motivated by, applications and strongly based on examples. The course will have an applied mathematics feel to it, rather than a very formal development.

Syllabus

Outline Syllabus

This course will be lectured by Andrew Gilbert (AG) and Mitchell Berger (MB) of the University of Exeter.

Part I (AG): basics, helicity and relaxation (3 lectures)

Background and motivation, hydrodynamics and magnetohydrodynamics. Revision of Kelvins theorem and magnetic analogies.
Fluid, magnetic and cross helicity, geometrical interpretation.
Magnetic relaxation.

Part II (MB): knots, tangles, braids and applications (4 lectures)

Link, twist and writhe of flux and vortex tubes.
Braiding of flux and vortex tubes.
Vortex tangles in quantum fluids and vortex tubes in turbulence, crossing numbers.
Chaotic mixing, stirrer protocols, pA maps and topological entropy.

Part III (AG): dynamics of vortex filaments and singularities (2 lectures)

Vortex tube dynamics, local induction approximation, invariants, solitons. The singularity problem and approaches.

Lecturer

  • AG

    Prof Andrew Gilbert

    University
    University of Exeter

Bibliography

No bibliography has been specified for this course.

Assessment

Attention needed

Assessment information will be available nearer the time.

Files

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Lectures

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