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Transport in 3D volume-preserving flows

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Summary

The idea of surfaces of locally minimal flux is introduced as a key concept for understanding transport in steady three-dimensional, volume-preserving flows. Particular attention is paid to the role of the skeleton formed by the equilibrium points, selected hyperbolic periodic orbits and cantori and connecting orbits, to which many surfaces of locally minimal flux can be attached. Applications are given to spheromaks (spherical vortices) and eccentric Taylor-Couette Flow.

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References

  1. Abraham R., Shaw C., Dynamics: The Geometry of Behavior, Part 3: Global Behavior, Aeriel Press, Santa Cruz, CA, 1985.

    Google Scholar 

  2. Angenent S., A variational interpretation of Melnikov's function and exponentially small separatrix splitting, in Proceedings of the Workshop on Symplectic Geometry (Warwick, Aug. 1991), Salamon D. A., ed., London Mathematics Society, in press.

  3. Aref H., Stirring by Chaotic Advection, J. Fluid Mech. 143 (1984), 1–21.

    Article  MATH  MathSciNet  Google Scholar 

  4. Arnol'd V. I., Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.

    Google Scholar 

  5. Arnol'd V. I., The Asymptotic Hopf invariant and its applications, Sel. Math. Sov. 5 (1986), 327–345.

    MATH  Google Scholar 

  6. Bajer K., Moffatt H. K., On a Class of Steady Confined Stokes Flows with Chaotic Streamlines, J. Fluid Mech. 212 (1990), 337–363.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bakker P. G., in Topological Fluid Mechanics, Moffatt H. K., Tsinober A., eds., Cambridge University Press, London, 1989.

    Google Scholar 

  8. Bakker P. G., On the topology of 3D separations: A guide for classification, in Continuation and Bifurcations—Numerical Techniques and Applications, Roose D. et al., eds., Kluwer, Dordrecht, Netherlands, 1990, 297–318.

    Google Scholar 

  9. Bensimon D., Kadanoff L. P., Extended chaos and disappearance of KAM trajectories, Physica D 13 (1984), 82–89.

    Article  MathSciNet  MATH  Google Scholar 

  10. Broer H. W., Formal normal forms for vector fields and some consequences for bifurcations in the volume-preserving case, in Dynamical Systems and Turbulence, Rand D. A., Young, L.-S., eds., Springer Lecture Notes in Mathematics 898 (1981), 54–74.

  11. Broer H. W., Vegter G., Subordinate Silnikov bifurcations near some singularities of vector fields having low codimension, Erg. Theory Dyn. Sys. 4 (1984), 509–525.

    MathSciNet  MATH  Google Scholar 

  12. Broomhead D. S., Ryrie S., Particle paths in wavy vortices, Nonlinearity 1 (1988), 409–434.

    Article  MathSciNet  MATH  Google Scholar 

  13. Cary J. R., Littlejohn R. G., Noncanonical Hamiltonian mechanics and its application to magnetic field line flow, Ann. Phys. 151 (1983), 1–34.

    Article  MathSciNet  Google Scholar 

  14. Castillo D. del, Morrison P. J., Hamiltonian chaos and transport in quasi-geostrophic flows, in Chaotic Dynamics and Transport in Fluids and Plasmas, Prigogine 1, ed., American Institute of Physics, College Park, MD, 1992.

    Google Scholar 

  15. Chaiken J., Chevray R., Tabor M., Tan Q. M., Experimental study of Lagrangian turbulence in a Stokes flow, Proc. Royal Soc. London A 408 (1986), 165–174.

    Article  Google Scholar 

  16. Chaiken J., Chu C. K., Tabor M., Tan Q. M., Lagrangian turbulence and spatial complexity in a Stokes flow, Phys. Fluids 30 (1987), 687–699.

    Article  MathSciNet  Google Scholar 

  17. Chance M., Greene J. M., Jensen T. H., The field line topology of a uniform magnetic field superposed on the field of a distributed ring current, Geophys. Astrop. Fl. Dyn. 65 (1992) 203.

    Google Scholar 

  18. Cheng C. Q., Sun Y. S., Existence of invariant tori in 3D measure-preserving mappings, Cel. Mech. 47 (1990), 275–292.

    Article  MathSciNet  MATH  Google Scholar 

  19. Cowley S. W. H., A qualitative study of the reconnection between the earth's magnetic field and an interplanetary field of arbitrary orientation, Radio Sci. 8 (1973), 903–913.

    Google Scholar 

  20. Dai R.-X., Dong Q., Szeri A. Z., Flow between eccentric rotating cylinders: Bifurcation and stability, Int. J. Engng. Sci. 30 (1992), 1323–1340.

    Article  MATH  Google Scholar 

  21. Dallmann U., Schultze-Werning B., Topological changes of axisymmetric and non-axisymmetric vortex flows, in Topological Fluid Mechanics, Moffatt H. K., Tsinober A., eds., Cambridge University Press, London, 1989, pp. 373–383.

    Google Scholar 

  22. Danckwerts P. V., Continuous flow systems: Distribution of residence times, Chem. Eng. Sci. 2 (1953), 1–13.

    Article  Google Scholar 

  23. Davis M. J., Gray S. K., Unimolecular reactions and phase space bottlenecks, J. Chem. Phys. 84 (1986), 5389.

    Article  Google Scholar 

  24. Delshams A., Llave R. de la, private communication.

  25. Delshams A., Martinez-Seara M. T., On the existence of homoclinic orbits in a family of vector fields near an equilibrium point, in European Conference on Iteration Theory, Alsina C., Llibre J., Mira Ch., Simo C., Targonski G., Thibault R., eds., World Scientific Publishers, 1989, 167–177.

  26. Dombre T., Frisch U., Greene J. M., Hénon M., Mehr A., Soward A. M., Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986), 353–391.

    Article  MathSciNet  MATH  Google Scholar 

  27. Feingold M., Kadanoff L. P., Piro O., Passive scalars, 3D volume-preserving maps, and chaos, J. Stat. Phys. 50 (1988), 529–565.

    Article  MathSciNet  MATH  Google Scholar 

  28. Greene J. M., Geometrical properties of 3D reconnecting magnetic fields with nulls, J. Geophys. Res. 93 (1988), 8583–8590.

    Article  Google Scholar 

  29. Greene J. M., MacKay R. S., Stark J., Boundary circles for area-preserving maps, Physica D 21 (1986), 267–295.

    Article  MathSciNet  MATH  Google Scholar 

  30. Guckenheimer J., Holmes P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

    MATH  Google Scholar 

  31. Holmes P. J., Some remarks on chaotic particle paths in time-periodic, 3D swirling flows, in Fluids and Plasmas—Geometry and Dynamics, Marsden J. E., ed., Contemp. Math. 28 (1984), 393–404.

    MATH  Google Scholar 

  32. Kaper T. J., Wiggins S., An analytical study of transport in Stokes flows exhibiting large-scale chaos: the eccentric journal bearing, J. Fluid Mech. 253 (1993), 211–243.

    Article  MathSciNet  MATH  Google Scholar 

  33. Kent P., Elgin J., A Shilnikov-type analysis in a system with symmetry, Phys. Lett. A 152 (1991), 28–32.

    Article  MathSciNet  Google Scholar 

  34. King G. P., Banas K., MacKay R. S., Shaw C. T., Stability and chaotic advection in the eccentric Taylor problem (abstract), in Symposium on Fluid Mechanics of Stirring and Mixing, Phys. Fluids 3, no. 5, pt. 2, (May 1991), 1435.

    Article  Google Scholar 

  35. Lamb H., Hydrodynamics, Dover, 1945.

  36. Lao Y.-T., Finn J. M., Dynamics of a 3D incompressible flow with stagnation points, Physica D 57 (1992), 283–310.

    Article  MathSciNet  Google Scholar 

  37. Lazutkin V. F., Width of stochastic layers, preprint.

  38. Leibovich S., The structure of vortex breakdown, Ann. Rev. Fluid Mech. 10 (1978), 221–246.

    Article  Google Scholar 

  39. MacKay R. S., Introduction to the dynamics of area-preserving maps, in Physics of Particle Accelerators, Month M., Dienes M., eds., American Institute of Physics Conference Proceedings 153 (1987), vol. 1, 534–602.

    Google Scholar 

  40. MacKay R. S., Flux over a saddle, Phys. Lett. A 145 (1990), 425–427.

    Article  MathSciNet  Google Scholar 

  41. MacKay R. S., A variational principle for odd-dimensional invariant submanifolds of an energy surface for Hamiltonian systems, Nonlinearity 4 (1991), 155–157.

    Article  MATH  MathSciNet  Google Scholar 

  42. MacKay R. S., Greene's residue criterion, Nonlinearity 4 (1992), 161–187.

    Article  MathSciNet  Google Scholar 

  43. MacKay R. S., Meiss J. D., Flux and differences in action for continuous time Hamiltonian systems, J. Phys. A 19 (1986), L225–229.

    Article  MathSciNet  Google Scholar 

  44. MacKay R. S., Meiss J. D., The relation between quantum and classical thresholds for multiphoton ionisation, Phys. Rev. A 37 (1988), 4702–4706.

    Article  Google Scholar 

  45. MacKay R. S., Meiss J. D., Percival I. C., Transport in Hamiltonian systems, Physica D 13, (1984), 55–81.

    Article  MathSciNet  MATH  Google Scholar 

  46. MacKay R. S., Meiss J. D., Percival I. C., Resonances in area-preserving maps, Physica D 27 (1987), 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  47. MacKay R. S., Muldoon M. R., Diffusion through spectres: ridge curves, ghost circles and a partition of phase space, Phys. Lett. A 178 (1993), 245–250.

    Article  MathSciNet  Google Scholar 

  48. Meiss J. D., Ott E., Markov tree model of transport in area-preserving maps, Physica D 20 (1986), 387–402.

    Article  MathSciNet  MATH  Google Scholar 

  49. Moffatt H. K., Tsinober A., eds., Topological Fluid Mechanics, Cambridge University Press, London, 1989.

    Google Scholar 

  50. Nijhof J. H. B., A model for steady flow between eccentric cylinders, NSL Report, Univ. of Warwick, Sept. 1990.

  51. Ottino J. M., The Kinematics of Mixing: Stretching, Chaos and Transport, Cambridge University Press, London, 1989.

    MATH  Google Scholar 

  52. Symposium on Fluid Mechanics of Stirring and Mixing, Phys. Fluids A 3, no. 5, pt. 2 (May 1991).

  53. Polterovich L. V., On transport in dynamical systems, Russian Math. Survey 43, no. 1 (1988), 251–252.

    Article  MATH  MathSciNet  Google Scholar 

  54. Robinson R. C., Generic properties of conservative systems I and II, Am. J. Math. 92 (1970), 562–603 and 897–906.

    MATH  Google Scholar 

  55. Rom-Kedar V., Leonard A., Wiggins S., An analytical study of transport, mixing and chaos in an unsteady vortical flow, J. Fluid Mech. 214 (1990), 347–394.

    Article  MathSciNet  MATH  Google Scholar 

  56. Rom-Kedar V., Wiggins S., Transport in two-dimensional maps, Arch. Rat. Mech. Anal. 109 (1990), 229–268.

    Article  MathSciNet  Google Scholar 

  57. Ryrie S., Mixing by chaotic advection in spatially periodic flows, J. Fluid Mech. 236 (1992), 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  58. Shub M., Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987.

    MATH  Google Scholar 

  59. Sinai Ya G., ed., Dynamical Systems II, Springer-Verlag, 1989.

  60. Sun Y. S., On the measure-preserving mappings with ood dimension, Cel. Mech. 30 (1983), 7–19; Invariant manifolds in the measure-preserving mappings with three dimensions, Cel. Mech. 33 (1984), 111–125.

    Article  MATH  Google Scholar 

  61. Vallet B., A version of the analytic KAM theorem for volume-preserving vector fields, preprint (1993).

  62. Weiss J. B., Knobloch E., Mass transport and mixing by modulated travelling waves, Phys. Rev. A 40 (1989), 2579–2589.

    Article  Google Scholar 

  63. Wiggins S., Chaotic Transport in Dynamical Systems, Springer-Verlag, New York, 1992.

    MATH  Google Scholar 

  64. Wigner E., Calculation of the rate of elementary association reactions, J. Chem. Phys. 5 (1937), 720–725.

    Article  Google Scholar 

  65. Winkel M. E. M. de, Bakker P. G., On the topology of 3D viscous flow structures near a plane wall, Univ. of Delft Aero. Engng. Report LR-541 (1988).

  66. Winkel M. E. M. de, Flow structures around non-hyperbolic signularities on the wall with one vanishing eigenvalue, Univ. of Delft Aero. Engng. Report LR-586 (1989).

  67. Xia Z., Existence of invariant tori in volume-preserving diffeomorphisms, Erg. Theory Dyn. Sys. 12 (1992), 621–631.

    MATH  Google Scholar 

  68. Yoccoz J. C., Travaux de Herman sur les tores invariantes, Asterisque 206 (1992), 311–344.

    MATH  MathSciNet  Google Scholar 

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  1. Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

    R. S. MacKay

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Communicated by Stephen Wiggins

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MacKay, R.S. Transport in 3D volume-preserving flows. J Nonlinear Sci 4, 329–354 (1994). https://doi.org/10.1007/BF02430637

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