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General


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)
Semester

Spring 2012 (Monday, January 16 to Friday, March 23)

Timetable
  • Thu 10:05 - 10:55

Lecturer


Andrew Gilbert
Email a.d.gilbert@ex.ac.uk
Phone (01392) 725222
Interests Fluid mechanics, dynamo theory, stability, mixing.
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Photo of Andrew Gilbert
Profile: This course will be lectured by Andrew Gilbert (AG) and Mitchell Berger (MB) of the University of Exeter.

AG: Research Interests

* Fluid mechanics and magnetohydrodynamics

* Vortex dynamics and mixing

* Dynamo theory (modelling geophysical and astrophysical magnetic fields)

* Applications of dynamical systems to fluid flows

* Asymptotic methods for solving PDEs

MB: Research Interests

* Astrophysics, solar physics, and applications of geometry and topology

* Topology of magnetic fields

* Applications to fusion energy, solar activity, dynamo theory, and space physics

* Fluid mechanics and magnetohydrodynamics



Students


Photo of Laura Burgess
Laura Burgess
(Leeds)
Photo of Richard Clift
Richard Clift
(Loughborough)
Photo of Zhen Cui
Zhen Cui
(Exeter)
Photo of Sam Durston
Sam Durston
(Exeter)
Photo of Mark Esson
Mark Esson
(Birmingham)
Photo of Maxwell Fennelly
Maxwell Fennelly
(Southampton)
Photo of Sam Jones
Sam Jones
(Exeter)
Photo of John Schofield
John Schofield
(East Anglia)


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.

Bibliography


No bibliography has been specified for this course.

Assessment


No assessment information is available yet.

Assignments


Problem set 1: vorticity, magnetic fields and helicity

Files:Exam paper
Deadline: Tuesday 14 February 2012 (1015.0 days ago)
Instructions:The assessment for the course will be 3 equally weighted problem sets.
To pass the course, you will have to pass each problem set.
For this problem sheet you should attempt questions 1, 2, 3, 4, 6, 7. Good answers to 4 of these 6 questions will be sufficient to pass the problem set.
The deadline is midnight Tuesday 14th February and work should be submitted via the MAGIC web system.


Problem set 2: Vortex dynamics (part III of the course)

Files:Exam paper
Deadline: Thursday 8 March 2012 (992.0 days ago)
Instructions:Note that because we have reordered some of the lectures, ADG will be lecturing part III of the course on Vortex Dynamics, and this will correspond to problem set 2.
Mitch will then do part II of the course and give you problem set 3.


Problem Set 3: Topology, Helicity and Braids

Files:Exam paper
Deadline: Friday 6 April 2012 (963.0 days ago)
Instructions:For this problem sheet you should attempt 5 questions, and provide good answers to at least 4. The lecture materials can be found in files top-fl-mech-part2-A.pdf,top-fl-mech-part2-B.pdf,top-fl-mech-part2-C.pdf, and top-fl-mech-part2-D.pdf.
The deadline is midnight Friday 6 April and work should be submitted via the MAGIC web system.