2.
Topological equivalence classes and change of topology
2.1 Topological equivalence classes for frozen fields: ideal conditions, fluid flow map,
frozen field equations, integral Cauchy solutions, topological equivalence, physical
(magnetic) knot embedding, flux, zero-framing, Reidemeister moves, local flows.
2.2 Change of topology: pathological flow maps, multivaluedness, discontinuities,
reconnection, dissipation.
2.3 Measuring structural complexity: a test case: superfluid vortex tangle, isotropic
growth, measures of length, energy and complexity versus time, energy-complexity
correlation.
2.4 Articles included:
Ricca, R.L. & Berger, M.A. 1996 Topological ideas and fluid mechanics. Physics
Today 49 (12), 24-30.
Ricca, R.L. 1998 Applications of knot theory in fluid mechanics. In Knot Theory (ed.
V.F.R. Jones et al.), pp. 321-346. Banach Center Publs. 42, Polish Academy of
Sciences, Warsaw.
Further reading: a good introduction to classical fluid mechanics and magnetohydrodynamics is provided by:
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University
Press, Cambridge.
Davidson, P.A. 2001 An Introduction to Magnetohydrodynamics. Cambridge
University Press, Cambridge.
A knot theoretical approach to dynamical systems is discussed in:
Ghrist, R., Holmes, P. & Sullivan, M. 1997. Knots and Links in ThreeDimensional Flows. .Lecture Notes in Mathematics 1654, Springer, Berlin.
An up-dated introduction to the study of geometric and topological aspects of fluid
flows is given by:
Ricca, R.L. (Editor) 2001 An Introduction to the Geometry and Topology of Fluid
Flows. NATO ASI Series II 47, Kluwer, Dordrecht.
A formal mathematical survey of aspects of topological fluid mechanics is:
Arnold, V.I. & Khesin, B.A. 1998 Topological Methods in Hydrodynamics.
Applied Math. Sci. 125, Springer, Berlin.
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2.1 Topological equivalence classes for frozen fields
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2.2 Change of topology
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Two elliptic vortex rings undergo reconnection, forming a simple (Hopf) link obtained by
numerical integration of the Navier-Stokes equations done by vortex-in-cell methods
(from Aref & Zawadzki, Nature 354, 1991).
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2.3 Measuring structural complexity: a test case
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