Advertisement

KdV ’95 pp 159-172 | Cite as

Coherent Structure Visiometrics:From the Soliton to HEC

  • Norman J. Zabusky

Abstract

In this paper I give a brief personal review of the history of the early soliton days and how they led to the visiometrics and reduced modeling paradigm that has been a part of my approach to nonlinear science in the last three decades. I illustrate it with HEC (Hybrid Elliptic-Contour): a fast, minimal, asymptotically motivated model for unforced, 2-dimensional incompressible weakly dissipative turbulence (U2DIT).

Key words

visiometrics reduced modeling asymptotics fluid dynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benzi, R., Briscolini, M., Colella, M., and Santangelo, P.: A simple point vortex model for two-dimensional decaying turbulence, Phys. Fluids A 4 (1992), 1036–1039.zbMATHCrossRefGoogle Scholar
  2. 2.
    Bitz, F. J. and Zabusky, N. J.: David and ‘visiometrics’: Visualizing and quantifying evolving amorphous objects, Computers in Physics, Nov./Dec. (1990), 603–614.Google Scholar
  3. 3.
    Carnevale, G. F., McWilliam, J. C., Pomeau, Y., Weiss, J. B., and Young, W. R.: Evolution of vortex statistics in two-dimensional turbulence, Phys. Rev. Lett. 66 (1992), 2735–2737.CrossRefGoogle Scholar
  4. 4.
    Deem, G. S., Zabusky, N. J., and Kruskal, M. D.: Formation, propagation and interaction of ‘solitons’ in nonlinear dispersive media (numerical solutions of differential equations describing wave motion in nonlinear dispersive media), Bell Telephone Laboratories Film Library, 1965–1975.Google Scholar
  5. 5.
    Dritschel, D. G.: Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows, Comp. Phys. Rep. 10 (1989), 77–146.CrossRefGoogle Scholar
  6. 6.
    Dritschel, D. G.: Vortex feature tracking in high reynolds number two-dimensional flows, in preparation, 1995.Google Scholar
  7. 7.
    Dritschel, D. G. and Legras, B.: The elliptical model of two-dimensional vortex dynamics, II. Disturbance equations, Phys. Fluids A 3 (1991) 855–869.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dritschel, D. G.: Vortex properties of two-dimensional turbulence, Phys. Fluids A 5 (1993), 984–997.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dritschel, D. G.: A general theory for two-dimensional vortex interaction, submitted to J. Fluid Mech., 1995.Google Scholar
  10. 10.
    Dritschel, D. G. and Waugh, D. W.: Quantification of the inelastic interaction of two asymmetric vortices in two-dimensional vortex dynamics, Phys. Fluids A 4 (1992), 1737–1744.CrossRefGoogle Scholar
  11. 11.
    Dritschel, D. G and Zabusky, N. J.: Toward a reduced model for nearly-inviscid unforced incompressible two-dimensional turbulence, letter submitted to Phys. Fluids, 1995.Google Scholar
  12. 12.
    Korteweg, D. J. and de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39 (1895), 422–443.zbMATHGoogle Scholar
  13. 13.
    Legras, B. and Dritschel, D. G.: The elliptical model of two-dimensional vortex dynamics, I. The basic state, Phys. Fluids A 3 (1991), 845–854.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Mariotti, A., Legras, B., and Dritschel, D. G.: Vortex stripping and the erosion of coherent structures in two-dimensional flows, Phys. Fluids 6 (1994), 3954–3962.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Melander, M., Zabusky, N. J., and Styczek, A. S.: A moment model for vortex interactions of the 2d Euler equations and computational validation of an elliptical representation, J. Fluid Mech. 167 (1986), 95–115.zbMATHCrossRefGoogle Scholar
  16. 16.
    Melander, M. V., McWilliams, J. C., and Zabusky, N. J.: Axisymmetrization and gradientintensification of an isolated 2d vortex through filamentation, J. Fluid Mech. 178 (1987), 137–159.zbMATHCrossRefGoogle Scholar
  17. 17.
    Piva, R., Riccardi, G., and Benzi, R.: A physical model for merging in 2d decaying turbulence, Preprint, 1995.Google Scholar
  18. 18.
    Silver, D. and Zabusky, N. J.: Quantifying visualizations for reduced modeling in nonlinear science: Extracting structures from data sets, J. Visual Comm. Image Representation 4 (1993), 46–61.CrossRefGoogle Scholar
  19. 19.
    Taub, A. (ed.): The Collected Works of John von Neumann, Macmillan, New York, 1963.Google Scholar
  20. 20.
    Yao, H. B., Dritschel, D. G., and Zabusky, N. J.: High-gradient phenomena in 2d vortex interaction, to appear in Phys. Fluids, 1995.Google Scholar
  21. 21.
    Zabusky, N. J.: Coherent structures in fluid dynamics, in The Significance of Nonlinearity in the Natural Sciences, Plenum Press, New York, 1977, pp. 145–205.CrossRefGoogle Scholar
  22. 22.
    Zabusky, N. J.: Computational synergetics and mathematical innovation, 7. Comput. Phys. 43 (1981), 195–249.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Zabusky, N. J.: Computational synergetics, Phys. Today 37 (1984), 36–46.CrossRefGoogle Scholar
  24. 24.
    Zabusky, N. J., Hughes, M., and Roberts, K. V.: Contour dynamics for the Euler equations in two dimensions, J. Comp. Phys. 30 (1979), 96–106.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Zabusky, N. J., Silver, D., Pelz, R., and Vizgroup ’93: Visometrics, juxtaposition, and modeling, Phys. Today (1993), 24–31.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Norman J. Zabusky
    • 1
  1. 1.Laboratory for Visiometrics and Modeling, Department of Mechanical and Aerospace EngineeringRutgers UniversityPiscatawayUSA

Personalised recommendations