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Structural Complexity and Dynamical Systems

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Lectures on Topological Fluid Mechanics

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1973))

Abstract

With this paper we want to pay tribute to 150 years of work on topological fluid mechanics. For this, we review Helmholtz's (1858) original contribution on topological issues related to vortex motion. Some recent results on aspects of structural complexity analysis of fluid flows are presented and discussed, as well as new results on topological bounds on the energy of magnetic knots and links in ideal magnetohydrodynamics, and on helicity-crossing number relations in dissipative fluids.

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References

  1. Barenghi C.F., Ricca, R.L. & Samuels D.C. (2001) How tangled is a tangle? Physica D 157, 197–206.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Barenghi C.F., Ricca, R.L. & Samuels D.C. (2002) Complexity measures of tangled vortex filaments. In Tubes, Sheets and Singularities in Fluid Dynamics (ed. K. Bajer & H.K. Moffatt), pp. 69–74. NATO ASI Series, Kluwer.

    Google Scholar 

  3. Berger, M.A. & Field, G.B. (1984) The topological properties of magnetic helicity. J. Fluid Mech. 147, 133–148.

    Article  ADS  MathSciNet  Google Scholar 

  4. Bott, R. & Tu, L.W. (1982) Differential Forms in Algebraic Topology. Graduate texts in Mathematics 82, Springer, Berlin.

    MATH  Google Scholar 

  5. Călugăreanu, G. (1961) Sur les classes d'isotopie des nœuds tridimensionnels et leurs invariants. Czechoslovak Math. J. 11, 588–625.

    MathSciNet  Google Scholar 

  6. Chong, M.S., Perry, A.E. & Cantwell, B.J. (1990) A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777.

    Article  ADS  MathSciNet  Google Scholar 

  7. Freedman, M.H. & He, Z.-X. (1991) Divergence-free fields: energy and asymptotic crossing number. Ann. Math. 134, 189–229.

    Article  MATH  MathSciNet  Google Scholar 

  8. Hauser, H., Hagen, H. & Theisel, H. (2007) (Eds.) Topology-based Methods in Visualization. Springer, Berlin.

    Book  MATH  Google Scholar 

  9. Helmholtz, H. (1858) Über integrale der hydrodynamischen gleichungen welche den wirbelbewegungen entsprechen. Crelle's J. 55, 25–55. [Transalted by P.G. Tait: (1867) On integrals of the hydrodynamical equations, which express vortex motion. Phil. Mag. 33, 485–512.]

    Article  MATH  Google Scholar 

  10. Lamb, H. (1879) Treatise on the Mathematical Theory of Motion of Fluids. Cambridge University Press, Cambridge.

    Google Scholar 

  11. Lord Kelvin (Thomson, W.) (1869) On vortex motion. Trans. Roy. S. Edinburgh 25, 217–260.

    Google Scholar 

  12. Ma, T. & Wang, S. (2005) Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics. Mathematical Surveys and Monographs 119, American Mathematical Society.

    Google Scholar 

  13. Maxwell, J.C. (1873) A Treatise on Electricity and Magnetism. Clarendon Press, London.

    Google Scholar 

  14. Moffatt, H.K. (1969) The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.

    Article  ADS  MATH  Google Scholar 

  15. Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Călugăreanu invariant. Proc. R. Soc. A 439, 411–429.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. 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.

    Google Scholar 

  17. Ricca, R.L. (2000) Towards a complexity measure theory for vortex tangles. In Knots in Hellas '98 (ed. C. McA. Gordon et al.), pp. 361–379. Series on Knots & Everything 24, World Scientific, Singapore.

    Chapter  Google Scholar 

  18. Ricca, R.L. (2001) Tropicity and complexity measures for vortex tangles. In Quantized Vortex Dynamics and Superfluid Turbulence (ed. C.F. Barenghi et al.), pp. 366–372. Springer Lecture Notes in Physics 571, Springer, Berlin.

    Chapter  Google Scholar 

  19. Ricca, R.L. (2005) Structural complexity. In Encyclopedia of Nonlinear Science (ed. A. Scott), pp. 885–887. Routledge, New York and London.

    Google Scholar 

  20. Ricca, R.L. (2008a) Topology bounds energy of knots and links. Proc. R. Soc. A 464, 293–300.

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Ricca, R.L. (2008b) Momenta of a vortex tangle by structural complexity analysis. Physica D, in press. (doi.10.1016/j.physd.2008.01.002).

    Google Scholar 

  22. Riemann, B. (1857) Lehrsätze aus der analysis situs für die theorie der integrale von zweiigliedrigren vollständingen differentialien. Crelle's J. 54, 105–110.

    Article  MATH  Google Scholar 

  23. Saffmann, P.G. (1991) Vortex Dynamics. Cambridge University Press, Cambridge.

    Google Scholar 

  24. Weickert, J. & Hagen, H. (Eds.) (2006) Visualization and Processing of Tensor Fields. Springer, Berlin.

    MATH  Google Scholar 

  25. Weintraub, S.H. (1997) Differential Forms. Academic Press Inc., San Diego.

    MATH  Google Scholar 

  26. White, J.H. (1969) Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91, 693–728.

    Article  MATH  MathSciNet  Google Scholar 

  27. Woltjer, L. (1958) A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci. USA 44, 489–491.

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Applications, University of Milano-Bicocca, Via Cozzi 53, 20125, Milano, ITALY

    Renzo L. Ricca

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  1. Renzo L. Ricca
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Correspondence to Renzo L. Ricca .

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Editors and Affiliations

  1. Dipto. Matematica e Applicazioni, Università Milano-Bicocca, Via R. Cozzi 53, Milano, 20125, Italy

    Renzo L. Ricca

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Ricca, R.L. (2009). Structural Complexity and Dynamical Systems. In: Ricca, R. (eds) Lectures on Topological Fluid Mechanics. Lecture Notes in Mathematics(), vol 1973. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00837-5_6

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