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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 2014 (Monday, January 20 to Friday, March 28)

Timetable

  • Wed 10:05 - 10:55

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 (3 lectures)

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

Lecturers


Andrew Gilbert (main contact)
Email a.d.gilbert@ex.ac.uk
Phone (01392) 725222
Interests Fluid mechanics, dynamo theory, stability, mixing.
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

Mitchell Berger
Email m.berger@ex.ac.uk
Phone
Photo of Mitchell Berger


Students


Photo of Johar Ashfaque
Johar Ashfaque
(Liverpool)
Photo of Thomas Barker
Thomas Barker
(Manchester)
Photo of Adam CHILD
Adam CHILD
(Leeds)
Photo of Tom Clemo
Tom Clemo
(Exeter)
Photo of Vanessa Graber
Vanessa Graber
(Southampton)
Photo of Davide Maestrini
Davide Maestrini
(East Anglia)
Photo of David Nigro
David Nigro
(Manchester)
Photo of Alexander Ramage
Alexander Ramage
(Cardiff)
Photo of Yafet Sanchez Sanchez
Yafet Sanchez Sanchez
(Southampton)
Photo of Jay Unadkat
Jay Unadkat
(Manchester)
Photo of Alberto Villois
Alberto Villois
(East Anglia)
Photo of Stuart Wells
Stuart Wells
(Southampton)
Photo of Andrew Wheadon
Andrew Wheadon
(Exeter)


Bibliography


No bibliography has been specified for this course.

Assessment



Assessment for this course will be via a take-home examination which will be put online after the end of the course. You will have 2 weeks to complete the questions and to pass you will need to obtain at least 50%.

Topological Fluid Mechanics Examination

Files:Exam paper
Released: Sunday 13 April 2014 (1259.7 days ago)
Deadline: Sunday 27 April 2014 (1244.7 days ago)


Recorded Lectures


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