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1998 | 42 | 1 | 321-346
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Applications of knot theory in fluid mechanics

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Języki publikacji
EN
Abstrakty
EN
In this paper we present an overview of some recent results on applications of knot theory in fluid mechanics, as part of a new discipline called `topological fluid mechanics' (TFM). The choice of the topics covered here is deliberately restricted to those areas that involve mainly a combination of ideal fluid mechanics techniques and knot theory concepts, complemented with a brief description of some other concepts that have important applications in fluid systems. We begin with the concept of topological equivalence of fluid flow maps, giving a definition of knotted and linked flux-tubes. In the fluid mechanics context Reidemeister's moves are interpreted in terms of local actions of fluid flows performed on fluid structures. An old theorem of Lichtenstein (1925) concerning the isotopic evolution of vortex structures in the context of the Euler equations is re-proposed and discussed in the TFM context for the first time. Then, we review the relationship between helicity and linking numbers and we present some recent results on magnetic relaxation of linked, knotted and braided structures in magnetohydrodynamics. In the context of the Euler equations (and under certain approximations given by the so-called `localized induction' for vortex structures) we briefly examine some interesting relationships between integrability and existence and stability of vortex filaments in the shape of torus knots. We conclude with an overview of some new results concerning electrically charged knots embedded in a viscous fluid, elastic relaxation of strings and braids and relationships between energy levels and topological information. Some simple bounds on elastic energy levels given by global geometric quantities and topological quantities are presented and discussed for the first time.
Słowa kluczowe
Rocznik
Tom
42
Numer
1
Strony
321-346
Opis fizyczny
Daty
wydano
1998
Twórcy
  • Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K.
Bibliografia
  • [1] H. Alfvén, Cosmic Electrodynamics, Oxford University Press, 1950.
  • [2] H. Aref and I. Zawadzki, Linking of vortex rings, Nature 354 (1991), 50-53.
  • [3] V.I. Arnold, The asymptotic Hopf invariant and its applications, in: Summer School in Diff. Eqs., Proc. Acad. Sci. Armenian S.S.R., Erevan, 1974, 229-256 (in Russian).
  • [4] V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics, Ann. Rev. Fluid Mech. 24 (1992), 145-166.
  • [5] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
  • [6] T. Bedford and J. Swift (ed.), New Directions in Dynamical Systems, Cambridge University Press, 1988.
  • [7] M.A. Berger, Third order link invariants, J. Phys. A: Math. & Gen. 23 (1990), 2787-2793.
  • [8] M.A. Berger, Third order braid invariants, J. Phys. A: Math. & Gen. 24 (1991), 4027-4036.
  • [9] M.A. Berger, Energy-crossing number relations for braided magnetic fields, Phys. Rev. Lett. 70 (1993), 705-708.
  • [10] M.A. Berger and G.B. Field, The topological properties of magnetic helicity, J. Fluid Mech. 147 (1984), 133-148.
  • [11] J.S. Birman, Braids, links and mapping class group, Annals of Math. Studies 82, Princeton University Press, 1976.
  • [12] J.S. Birman and R.F. Williams, Knotted periodic orbits in dynamical systems I: Lorenz's equations, Topology 22 (1983), 47-82.
  • [13] O.N. Boratav, R.B. Pelz and N.J. Zabusky, Reconnection in orthogonally interacting vortex tubes: direct numerical simulations and quantifications, Phys. Fluids A 4 (1992), 581-605.
  • [14] R.J. Bray, L.E. Cram, C.J. Durrant and R.E. Loughhead, Plasma Loops in the Solar Corona, Cambridge University Press, 1991.
  • [15] G. Călugăreanu, On isotopy classes of three-dimensional knots and their invariants, Czechoslovak Math. J. T11 (1961), 588-625 (in French).
  • [16] A.Y.K. Chui and H.K. Moffatt, The energy and helicity of knotted magnetic flux-tubes, Proc. Roy. Soc. Lond. A 451 (1995), 609-629.
  • [17] N.W. Evans and M.A. Berger, A hierarchy of linking integrals, in: Topological Aspects of the Dynamics of Fluids and Plasmas, H.K. Moffatt et al. (ed.), Kluwer, Dordrecht, The Netherlands, 1992, 237-248.
  • [18] M.H. Freedman, A note on topology and magnetic energy in incompressible perfectly conducting fluids, J. Fluid Mech. 194 (1988), 549-551.
  • [19] M.H. Freedman and M.A. Berger, Combinatorial relaxation of magnetic fields, Geophys. Astrophys. Fluid Dynamics 73 (1993), 91-96.
  • [20] M.H. Freedman and Z.-X. He, Divergence-free fields: energy and asymptotic crossing number, Ann. Math. 134 (1991), 189-229.
  • [21] M.H. Freedman and Z.-X. He, Research announcement on the ``energy'' of knots, in: Topological Aspects of the Dynamics of Fluids and Plasmas, H.K. Moffatt et al. (ed.), Kluwer, Dordrecht, The Netherlands, 1992, 219-222.
  • [22] M.H. Freedman, Z.-X. He and Z. Wang, Möbius energy of knots and unknots, Ann. Math. 139 (1994), 1-50.
  • [23] S. Fukuhara, Energy of a knot, in: A Fete of Topology: Papers Dedicated to Itiro Tamura, Y.T. Matsumoto and S. Morita (ed.), Academic Press, New York, 1988, 443-451.
  • [24] F.B. Fuller, The writhing number of a space curve, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 815-819.
  • [25] C.F. Gauss, Works, Königliche Gesellschaft der Wissenschaften zu Göttingen, 1877 (in German).
  • [26] V.L. Hansen, Braids and Coverings, London Math. Soc. 18, Cambridge University Press, 1989.
  • [27] H. Helmholtz, On integrals of the hydrodynamical equations, which express vortex motion, J. Reine Angew. Math. 55 (1858), 25-55 (in German).
  • [28] P.J. Holmes and R.F. Williams, Knotted periodic orbits in suspensions of Smale's horseshoe: torus knots and bifurcation sequences, Arch. Rat. Mech. Anal. 90 (1985), 115-194.
  • [29] J. Jiménez (ed.), The Global Geometry of Turbulence, NATO ASI B 268, Plenum Press, New York, 1991.
  • [30] B.B. Kadomtsev, Magnetic field line reconnection, Rep. Prog. Phys. 50 (1987), 115-143.
  • [31] L.H. Kauffman, On Knots, Annals Study 115, Princeton University Press, 1987.
  • [32] L.H. Kauffman, [a] Knots and Applications, World Scientific, Singapore, 1995.
  • [33] L.H. Kauffman, [b] The interface of Knots and Physics, Proc. Symp. Appl. Maths. 51, Am. Math. Soc., 1995.
  • [34] J.P. Keener, Knotted vortex filaments in an ideal fluid, J. Fluid Mech. 211 (1990), 629-651.
  • [35] Lord Kelvin (W.T. Thomson), On vortex motion, Trans. Roy. Soc. Edinb. 25 (1869), 217-260.
  • [36] Lord Kelvin (W.T. Thomson), Vortex statics, Proc. Roy. Soc. Edinb. ss. 1875-76 (1875), 115-128.
  • [37] B.A. Khesin and Yu.V. Chekanov, Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity in D dimensions, Physica D 40 (1989), 119-131.
  • [38] S. Kida, A vortex filament moving without change of form, J. Fluid Mech. 112 (1981), 397-409.
  • [39] D. Kim and R. Kusner, Torus knots extremizing the conformal energy, Experimental Math. 2 (1993), 1-9.
  • [40] E.A. Kuznetsov and A.V. Mikhailov, On the topological meaning of canonical Clebsch variables, Phys. Lett. 77 A (1980), 37-38.
  • [41] J. Langer and R. Perline, Poisson geometry of the filament equation, J. Nonlinear Sci. 1 (1991), 71-93.
  • [42] J. Langer and D.A. Singer, Knotted elastic curves in $ℝ^3$, J. London Math. Soc. 30 (1984), 512-520.
  • [43] J. Langer and D.A. Singer, Curve straightening and a minimax argument for closed elastic curves, Topology 24 (1985), 75-88.
  • [44] T. Levi-Civita, On mechanical actions due to a filiform flux of electricity, Rend. R. Acc. Lincei 18 (1909), 41-50 (in Italian).
  • [45] L. Lichtenstein, On some existence problems of the hydrodynamics of homogeneous and incompressible, inviscid fluid and the vortex laws of Helmholtz, Math. Zeit. 23 (1925), 89-154 (in German).
  • [46] S.J. Lomonaco, The modern legacies of Thomson's atomic vortex theory in classical electrodynamics, in: The interface of Knots and Physics, Proc. Symp. Appl. Maths. 51, Am. Math. Soc., 1995.
  • [47] H.J. Lugt, Vortex Flow in Nature and Technology, J. Wiley & Sons, New York, 1983.
  • [48] J. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D 7 (1983), 305-323.
  • [49] J.C. Maxwell, A Treatise on Electricity and Magnetism, MacMillan & Co., Oxford, 1873.
  • [50] K.R. Meyer and D.G. Saari, Hamiltonian Dynamical Systems, Contemp. Math. 81, New York, 1988.
  • [51] J. Milnor, On the total curvature of knots, Ann. Math. 52 (1950), 248-257.
  • [52] H.K. Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 35 (1969), 117-129.
  • [53] H.K. Moffatt, The energy spectrum of knots and links, Nature 347 (1990), 367-369.
  • [54] H.K. Moffatt, Relaxation under topological constraints, in: Topological Aspects of the Dynamics of Fluids and Plasmas, H.K. Moffatt et al. (ed.), Kluwer, Dordrecht, The Netherlands, 1992, 3-28.
  • [55] H.K. Moffatt and R.L. Ricca, Interpretation of invariants of the Betchov-Da Rios equations and of the Euler equations, in: The Global Geometry of Turbulence, J. Jiménez (ed.), NATO ASI B 268, Plenum Press, New York, 1991, 257-264.
  • [56] H.K. Moffatt and R.L. Ricca, Helicity and the Călugăreanu invariant, Proc. R. Soc. Lond. A 439 (1992), 411-429.
  • [57] H.K. Moffatt and A. Tsinober (ed.), Topological Fluid Mechanics, Cambridge University Press, 1990.
  • [58] H.K. Moffatt, G.M. Zaslavsky, P. Comte and M. Tabor (ed.), Topological Aspects of the Dynamics of Fluids and Plasmas, Kluwer, Dordrecht, The Netherlands, 1992.
  • [59] J.-J. Moreau, Constants of a vortex filament in a barotropic perfect fluid, C. R. Acad. Sci. Paris 2 (1961), 2810-2812 (in French).
  • [60] J. O'Hara, Energy of a knot, Topology 30 (1991), 241-247.
  • [61] R.L. Ricca, Global geometric invariants of the Da Rios-Betchov equations, Internatl. Symp. on Cont. Mech., USSR Acad. Sci., Perm-Moscow, 1990.
  • [62] R.L. Ricca, Torus knots and polynomial invariants for a class of soliton equations, Chaos 3 (1993), 83-91 [Erratum: Chaos 5 (1995), 346].
  • [63] R.L. Ricca, [a] Geometric and topological aspects of vortex filament dynamics under LIA, in: Small-Scale Structures in Three-Dimensional Hydro and Magnetohydrodynamics Turbulence, M. Meneguzzi et al. (ed.), Lecture notes in Physics 462, Springer-Verlag, 1995, 99-104.
  • [64] R.L. Ricca, [b] The energy spectrum of a twisted flexible string under elastic relaxation, J. Physics A: Math. & Gen. 28 (1995), 2335-2352.
  • [65] R.L. Ricca, [a] Dynamics of knotted magnetic flux-tubes, in: XIX Int. Cong. Theor. Appl. Mech. - Abstracts, IUTAM, Kyoto, 1996, 94.
  • [66] R.L. Ricca, [b] The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dyn. Res. 18 (1996), 245-268.
  • [67] R.L. Ricca and M.A. Berger, Topological ideas and fluid mechanics, Phys. Today 49, 12 (1996), 24-30.
  • [68] R.L. Ricca and H.K. Moffatt, The helicity of a knotted vortex filament, in: Topological Aspects of the Dynamics of Fluids and Plasmas, H.K. Moffatt et al. (ed.), Kluwer, Dordrecht, The Netherlands, 1992, 225-236.
  • D. Rolfsen, Knots and Links, Publish or Perish Inc., Berkeley (CA), 1976.
  • [70] A. Schilham, Work done at the Department of Mathematics, University College London, 1995.
  • [71] T. Schlick and W. Olson, Trefoil knotting revealed by molecular dynamics simulations of supercoiled DNA, Science 257 (1992), 1110-1115.
  • [72] J.K. Simon, Energy functions for polygonal knots, in: Random Knotting and Linking, K.C. Millett & D.W. Sumners (ed.), World Scientific, Singapore, 1994.
  • [73] De W. Sumners (ed.), New Scientific Applications of Geometry and Topology, PSAM 45, Am. Math. Soc., Providence, 1992.
  • [74] P.G. Tait, On Knots I, II, III, Scientific Papers 1, Cambridge University Press, 1898, 273-347.
  • [75] F. Tanaka and H. Takahashi, Elastic theory of supercoiled DNA, J. Chem. Phys. 83 (1985), 6017-6026.
  • [76] P. Traczyk, A new proof of Markov's braid theorem, this volume.
  • [77] A.V. Tur and V.V. Yanovsky, Invariants in dissipationless hydrodynamic media, J. Fluid Mech. (1993), 67-106.
  • [78] T. Uezu, Topological structure in flow systems, Prog. Theor. Phys. 83 (1990), 850-874.
  • [79] M. Van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford (CA), 1988.
  • [80] H. Villat, Lectures on the Theory of Vortices, Gauthier-Villars, Paris, 1930 (in French).
  • [81] P. Vogel, Representation of links by braids: a new algorithm, Comment. Math. Helvetici 65 (1990), 104-113.
  • [82] M. Wadati and H. Tsuru, Elastic model of looped DNA, Physica D 21 (1986), 213-226.
  • [83] J.H. White, Self-linking and the Gauss integral in higher dimensions, Am. J. Math. 91 (1969), 693-728.
  • [84] J.H.C. Whitehead, An expression of Hopf's invariant as an integral, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 117-123.
  • [85] L. Woltjer, A theorem on force-free magnetic fields, Proc. Natl. Acad. Sci. USA 44 (1958), 489-491.
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