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Creation and dynamics of knotted vortices

Nature Physics volume 9, pages 253–258 (2013)Cite this article

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Abstract

Knots and links have been conjectured to play a fundamental role in a wide range of physical fields, including plasmas and fluids, both quantum and classical. In fluids, the fundamental knottedness-carrying excitations occur in the form of linked and knotted vortex loops, which have been conjectured to exist for over a century. Although they have been the subject of considerable theoretical study, their creation in the laboratory has remained an outstanding experimental goal. Here we report the creation of isolated trefoil vortex knots and pairs of linked vortex rings in water using a new method of accelerating specially shaped hydrofoils. Using a high-speed scanning tomography apparatus, we measure their three-dimensional topological and geometrical evolution in detail. In both cases we observe that the linked vortices stretch themselves and then deform—as dictated by their geometrically determined energy—towards a series of local vortex reconnections. This work establishes the existence and dynamics of knotted vortices in real fluids.

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Figure 1: The creation of vortices with designed shape and topology.
Figure 2: Scaling of trefoil knot vortex loops.
Figure 3: Topological evolution of linked and knotted vortex loops in three dimensions.
Figure 4: Topological and energetic evolution of several types of vortex loop.
Figure 5: Reconstruction of the vortex core and flow field from raw 3D data.

References

  1. Witten, E. Quantum-field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  2. Faddeev, L. & Niemi, A. Stable knot-like structures in classical field theory. Nature 387, 58–61 (1997).

    Article  ADS  Google Scholar 

  3. Manton, N. & Sutcliffe, P. M. Topological Solitons (Cambridge Univ. Press, 2004).

    Book  Google Scholar 

  4. Smalyukh, I. I., Lansac, Y., Clark, N. A. & Trivedi, R. P. Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids. Nature Mater. 9, 139–145 (2010).

    Article  ADS  Google Scholar 

  5. Tkalec, U. et al. Reconfigurable knots and links in chiral nematic colloids. Science 333, 62–65 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  6. Irvine, W. T. M. & Bouwmeester, D. Linked and knotted beams of light. Nature Phys. 4, 716–720 (2008).

    Article  ADS  Google Scholar 

  7. Irvine, W. T. M. Linked and knotted beams of light, conservation of helicity and the flow of null electromagnetic fields. J. Phys. A 43, 385203 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  8. Dennis, M. R., King, R. P., Jack, B., O’holleran, K. & Padgett, M. Isolated optical vortex knots. Nature Phys. 6, 118–121 (2010).

    Article  ADS  Google Scholar 

  9. Chandrasekhar, S. & Kendall, P. C. On force-free magnetic fields. Proc. Natl Acad. Sci. USA 42, 1–5 (1956).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  11. Berger, M. A. Introduction to magnetic helicity. Plasma Phys. Control. Fusion 41, B167–B175 (1999).

    Article  ADS  Google Scholar 

  12. Freedman, M. H. & Berger, M. A. Combinatorial relaxation of magnetic fields. Geophys. Astrophys. Fluid Dynam. 73, 91–96 (1993).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  14. Moffatt, H. K. Some developments in the theory of turbulence. J. Fluid Mech. 106, 27–47 (1981).

    Article  ADS  Google Scholar 

  15. Ricca, R. L. & Berger, M. A. Topological ideas and fluid mechanics. Phys. Today 49, 28–34 (1996).

    Article  ADS  Google Scholar 

  16. Barenghi, C. F. Knots and unknots in superfluid turbulence. Milan J. Math. 75, 177–196 (2007).

    Article  MathSciNet  Google Scholar 

  17. Babaev, E. Non-Meissner electrodynamics and knotted solitons in two-component superconductors. Phys. Rev. B 79, 1–6 (2009).

    Article  Google Scholar 

  18. Bouligand, Y. Recherches sur les textures des états mésomorphes. 6 - Dislocations coins et signification des cloisons de Grandjean-Cano dans les cholestériques. J. Physique 35, 959–981 (1974).

    Article  Google Scholar 

  19. Kambe, T. & Takao, T. Motion of distorted vortex rings. J. Phys. Soc. Jpn 31, 591–599 (1971).

    Article  ADS  Google Scholar 

  20. Kida, S. A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  21. Ricca, R. L., Samuels, D. & Barenghi, C. Evolution of vortex knots. J. Fluid Mech. 391, 29–44 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  22. Proment, D., Onorato, M. & Barenghi, C. Vortex knots in a Bose–Einstein condensate. Phys. Rev. E 85, 1–8 (2012).

    Article  Google Scholar 

  23. Ricca, R. New developments in topological fluid mechanics. Il Nuovo Cimento. 32, 185–192 (2009).

    ADS  Google Scholar 

  24. Thomson, W. On vortex atoms. Phil. Mag. 34, 94–105 (1867).

    Article  Google Scholar 

  25. Moffatt, H. K., Kida, S. & Ohkitani, K. Stretched vortices—the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241–264 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  26. Acheson, D. J. Elementary Fluid Dynamics (Clarendon, 1990).

    MATH  Google Scholar 

  27. Saffman, P. G. Vortex Dynamics (Cambridge Univ. Press, 1992).

    MATH  Google Scholar 

  28. Calini, A., Keith, S. F. & Lafortune, S. Squared eigenfunctions and linear stability properties of closed vortex filaments. Nonlinearity 24, 3555–3583 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  29. Kida, S. & Takaoka, M. Reconnection of vortex tubes. Fluid Dynam. Res. 3, 257–261 (1988).

    Article  ADS  Google Scholar 

  30. Donnelly, R. J. Vortex rings in classical and quantum systems. Fluid Dynam. Res. 41, 051401 (2009) 1–31.

    Article  ADS  MathSciNet  Google Scholar 

  31. Douady, S. & Couder, Y. Direct observation of the intermittency of intense vorticity filaments in turbulence. Phys. Rev. Lett. 67, 983–986 (1991).

    Article  ADS  Google Scholar 

  32. Thomson, W. XXIV. Vibrations of A Columnar Vortex 155–168 (Philosophical Magazine Series 5, Vol. 10, 1880).

    MATH  Google Scholar 

  33. Sethian, J. A. A fast marching level set method for monotonically advancing fronts. Proc. Natl Acad. Sci. USA 93, 1591–1595 (1996).

    Article  ADS  MathSciNet  Google Scholar 

  34. Deschamps, T. & Cohen, L. D. Minimal paths in 3D images and application to virtual endoscopy. Med. Image Anal. 5, 281–299 (2001).

    Article  Google Scholar 

  35. Dengler, R. Self inductance of a wire loop as a curve integral. Preprint at http://arxiv.org/abs/1204.1486 (2012).

  36. Oshima, Y., Kambe, T. & Asaka, S. Interaction of two vortex rings moving along a common axis of symmetry. J. Phys. Soc. Jpn 38, 1159–1166 (1975).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors acknowledge useful discussions with J. Burton, P. M. Chaikin, E. Efrati, H. M. Jaeger, H. K. Moffatt, S. R. Nagel, M. Scheeler, T. Witten and W. Zhang. We acknowledge MRSEC Shared Facilities at the University of Chicago (DMR-0820054) for the use of their instruments. This work was supported by the National Science Foundation (NSF) Materials Research and Engineering Centers (MRSEC) Program at the University of Chicago (DMR-0820054). W.T.M.I. further acknowledges support from the A.P. Sloan Foundation through a Sloan fellowship, and the Packard Foundation through a Packard fellowship.

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Authors and Affiliations

  1. Department of Physics, James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA

    Dustin Kleckner & William T. M. Irvine

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  1. Dustin Kleckner
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  2. William T. M. Irvine
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Contributions

W.T.M.I. conceived and supervised research. W.T.M.I. and D.K. designed experiments and developed vortex generation and analysis techniques. D.K. developed the tomography apparatus and software, performed experiments and processed data. D.K. and W.T.M.I. analysed results and wrote the manuscript.

Corresponding authors

Correspondence to Dustin Kleckner or William T. M. Irvine.

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The authors declare no competing financial interests.

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Kleckner, D., Irvine, W. Creation and dynamics of knotted vortices. Nature Phys 9, 253–258 (2013). https://doi.org/10.1038/nphys2560

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