Letter | Published:

Linking of vortex rings

Naturevolume 354pages5053 (1991) | Download Citation



THE topology of vortex lines, which trace the local vorticity in a fluid flow just as streamlines trace the velocity, is important in attempts to understand, describe and control flows in various applications. Changes in this topology may, for example, affect mixing in flows, and may be significant for the dynamics of turbulence. In particular, the behaviour of closed loops of vortex lines (vortex rings) has been studied ever since Kelvin's 'vortex atom' theory, and contributed to Tait's development of the topological theory of knots. Several recent experimental and computational studiesl–9 have explored the 'reconnection' of initially distinct vortex rings; particularly elegant are Schatzle's10 experiments in which two vortex rings, inclined towards one another, go through two reconnections, after which two new rings, comprising half of each of the originals, emerge. Here we describe three-dimensional numerical simulations which establish a simple mechanism by which the linking of two vortex rings may be achieved starting from an unlinked initial state. Appropriate initial states were identified by simulating the unlinking of two initially linked vortex rings and then reversing the vorticity of the final state and running the simulation backwards. This computational procedure sheds light on why, both in experiment and simulation, linking is not always achieved from an arbitrary initial configuration of unlinked vortex rings set on a collision course.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.


  1. 1

    Kambe, T. & Takao, T. J. phys. Soc. Japan 31, 591–599 (1971).

  2. 2

    Oshima, Y. & Asaka, S. Nat. Sci. Rep. Ochanomizu Univ. 26, 31–37 (1975); J. phys. Soc. Japan 42, 708–713 (1977).

  3. 3

    Ashurst, W. & Meiron, D. I. Phys. Rev. Lett. 58, 1632–1635 (1987).

  4. 4

    Melander, M. & Zabusky, N. J. Fluid Dyn. Res. 3, 247–250 (1988).

  5. 5

    Kida, S. & Takaoka, M. Fluid Dyn. Res. 3, 257–261 (1988).

  6. 6

    Oshima, Y. & Izutsu, N. Phys. Fluids 31, 2401–2403 (1988).

  7. 7

    Melander, M. V. & Hussain, F. in Topological Fluid Mechanics (eds Moffatt, H. K. & Tsinober, A.) 485–499 (Cambridge University Press, 1990).

  8. 8

    Kida, S., Takaoka, M. & Hussain, F. Phys. Fluids A1, 630–633 (1989); (Corrigendum) ibid. 2, 638 (1990); J. Fluid Mech. 230, 583–646 (1991).

  9. 9

    Melander, M. V. & Hussain, F. Phys. Fluids A1, 633–636 (1989).

  10. 10

    Schatzle, P. R. thesis, California Institute of Technology (1987).

  11. 11

    Moffatt, H. K. J. Fluid Mech. 35, 117–129 (1969).

  12. 12

    Leonard, A. Ann. Rev. Fluid Mech. 17, 523–559 (1985).

  13. 13

    Winckelmans, G. & Leonard, A. in Mathematical Aspects of Vortex Dynamics Ch. 2 (ed. Caflisch, R. E.) (Society of Industrial and Applied Mathematics, 1989).

  14. 14

    Zawadzki, I. & Aref, H. Phys. Fluids A3, 1405–1410 (1991); J. comput. Phys. (submitted).

  15. 15

    Siggia, E. D. Phys. Fluids 28, 794–805 (1985).

  16. 16

    Yamada, H. & Matsui, T. Phys. Fluids 21, 292–294 (1978).

Download references

Author information


  1. University of California at San Diego, La Jolla, California, 92093-0225, USA

    • Hassan Aref
    •  & Ireneusz Zawadzki


  1. Search for Hassan Aref in:

  2. Search for Ireneusz Zawadzki in:

About this article

Publication history



Issue Date



Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.