Topological Fluid Mechanics (MAGIC052)
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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)
Spring 2014 (Monday, January 20 to Friday, March 28)
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.
knowledge of magnetohydrodynamics: this will be developed where needed. knowledge of pure mathematics beyond basic group theory.
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.
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.
No bibliography has been specified for this course.
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
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