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Structural Complexity and Dynamical Systems

  • Renzo L. RiccaEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, 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.

Keywords

Double Point Vortex Line Vortex Motion Magnetic Helicity Reidemeister Move 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Barenghi C.F., Ricca, R.L. & Samuels D.C. (2001) How tangled is a tangle? Physica D 157, 197–206.ADSzbMATHCrossRefMathSciNetGoogle Scholar
  2. 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. 3.
    Berger, M.A. & Field, G.B. (1984) The topological properties of magnetic helicity. J. Fluid Mech. 147, 133–148.ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bott, R. & Tu, L.W. (1982) Differential Forms in Algebraic Topology. Graduate texts in Mathematics 82, Springer, Berlin.zbMATHGoogle Scholar
  5. 5.
    Călugăreanu, G. (1961) Sur les classes d'isotopie des nœuds tridimensionnels et leurs invariants. Czechoslovak Math. J. 11, 588–625.MathSciNetGoogle Scholar
  6. 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.ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    Freedman, M.H. & He, Z.-X. (1991) Divergence-free fields: energy and asymptotic crossing number. Ann. Math. 134, 189–229.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hauser, H., Hagen, H. & Theisel, H. (2007) (Eds.) Topology-based Methods in Visualization. Springer, Berlin.zbMATHCrossRefGoogle Scholar
  9. 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.]zbMATHCrossRefGoogle Scholar
  10. 10.
    Lamb, H. (1879) Treatise on the Mathematical Theory of Motion of Fluids. Cambridge University Press, Cambridge.Google Scholar
  11. 11.
    Lord Kelvin (Thomson, W.) (1869) On vortex motion. Trans. Roy. S. Edinburgh 25, 217–260.Google Scholar
  12. 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. 13.
    Maxwell, J.C. (1873) A Treatise on Electricity and Magnetism. Clarendon Press, London.Google Scholar
  14. 14.
    Moffatt, H.K. (1969) The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.ADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Călugăreanu invariant. Proc. R. Soc. A 439, 411–429.ADSzbMATHCrossRefMathSciNetGoogle Scholar
  16. 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. 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.CrossRefGoogle Scholar
  18. 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.CrossRefGoogle Scholar
  19. 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. 20.
    Ricca, R.L. (2008a) Topology bounds energy of knots and links. Proc. R. Soc. A 464, 293–300.ADSzbMATHCrossRefMathSciNetGoogle Scholar
  21. 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. 22.
    Riemann, B. (1857) Lehrsätze aus der analysis situs für die theorie der integrale von zweiigliedrigren vollständingen differentialien. Crelle's J. 54, 105–110.zbMATHCrossRefGoogle Scholar
  23. 23.
    Saffmann, P.G. (1991) Vortex Dynamics. Cambridge University Press, Cambridge.Google Scholar
  24. 24.
    Weickert, J. & Hagen, H. (Eds.) (2006) Visualization and Processing of Tensor Fields. Springer, Berlin.zbMATHGoogle Scholar
  25. 25.
    Weintraub, S.H. (1997) Differential Forms. Academic Press Inc., San Diego.zbMATHGoogle Scholar
  26. 26.
    White, J.H. (1969) Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91, 693–728.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Woltjer, L. (1958) A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci. USA 44, 489–491.ADSzbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Mathematics and ApplicationsUniversity of Milano-BicoccaMilanoITALY

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