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Vortex- and magneto-dynamics—a topological perspective. (English) Zbl 0989.76093
Fokas, A. (ed.) et al., Mathematical physics 2000. International congress, London, GB, 2000. London: Imperial College Press. 170-182 (2000).
The vortex dynamics is reviewed. In particular, the author reconsiders the studies of Helmholtz and Thomson, and discusses the topological aspects of knot classification by Tait. The topological role of helicity is shown in a modern fashion, leading to the relation between knotted and linked vortex lines, and thus introducing results by H. K. Moffatt and R. L. Ricca [Proc. R. Soc. Lond., Ser. A 439, No. 1906, 411-429 (1992; Zbl 0771.57013)] and F. B. Fuller [Proc. Natl. Acad. Sci. USA 68, 815-819 (1971; Zbl 0212.26301)]. These aspects are studied in the case of magnetic fields, too. The meaning of magnetic knots and their role in the relaxation of chaotic fields are discussed, together with two-dimensional relaxation and analogous Euler flows. The outcome of the analysis is shown on a steady solution of ideal magnetohydrodynamics, leading to the author’s concluding remark: “the great, and enduring, difficulty of the Euler equations lies in their purity, within which the central intractable nonlinearity continues to defy progress at the heart of the still unsolved problem of turbulence – a problem that will continue to challenge and frustrate for many decades into the 21st century”.
For the entire collection see [Zbl 0949.00037].

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
76B47 Vortex flows for incompressible inviscid fluids
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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