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The Language of Pattern and Form

  • N.J. Balmforth
  • A. Provenzale
  • J.A. Whitehead
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 582)

Abstract

Geology and geomorphology deal with some of the most striking patterns of Nature. From mountain ranges and mid-ocean ridges, to river networks and sand dunes, there is a whole family of forms, structures, and shapes that demand rationalization as well as mathematical description. In the various chapters of this volume, many of these patterns will be explored and discussed, and attempts will be made to both unravel the mathematical reasons for their very existence and to describe their dynamics in quantitative terms. In this introductory chapter, we discuss some of the methods that can be adopted in the study of patterns, and use the specific examples of convection - an evergreen classic in nonlinear fluid dynamics—and of the formation of aeolian ripples—another phenomenon that strikes the imagination of anybody who has been travelling in a sand desert.

Keywords

Solitary Wave Rayleigh Number Hopf Bifurcation Dynamical System Theory Heteroclinic Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford 1961) 643 pp.zbMATHGoogle Scholar
  2. 2.
    J.S. Turner: Buoyancy E.ects in Fluids (Cambridge University Press, New York 1973) 367 pp.Google Scholar
  3. 3.
    G. Schubert: Ann. Rev. Fluid Mech. 24, 395–397 (1992)CrossRefGoogle Scholar
  4. 4.
    M.C. Cross, P.C. Hohenberg: Rev. Mod. Phys. 65(3), 851–1112 (1993)CrossRefADSGoogle Scholar
  5. 5.
    J.R. de Bruyer, E. Bodenschatz, S.W. Morris, S.P. Traino., Y. Hu, et al.: Rev. Sci. Inst. 67(6), 2043–2067 (1996)CrossRefADSGoogle Scholar
  6. 6.
    E. Bodenschatz, W. Pesch, G. Ahlers: Ann. Revs. Fluid Mech. 32, 709–778 (2000)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    W.V.R. Malkus, G. Veronis: J. Fluid Mech. 4, 225–260 (1958)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    M. Colombini, G. Seminara, M. Tubino: J. Fluid Mech. 181, 213–232 (1987)zbMATHCrossRefADSGoogle Scholar
  9. 9.
    A.C. Newell, J.A. Whitehead: J. Fluid Mech. 28, 279–303 (1969)CrossRefADSGoogle Scholar
  10. 10.
    L.A. Segel: J. Fluid Mech. 38, 203–224 (1969)zbMATHCrossRefADSGoogle Scholar
  11. 11.
    C.S. Bretherton, E.A. Spiegel: Phys Lett. A 96, 152 (1983)CrossRefADSGoogle Scholar
  12. 12.
    R. Schielen, A. Doelman, H.E. de Swart: J. Fluid Mech. 252, 325–356 (1993)zbMATHCrossRefADSGoogle Scholar
  13. 13.
    M.J. Ablowitz, H. Segur: Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics (SIAM, Philadelphia 1981)Google Scholar
  14. 14.
    D.R. Scott, D.J. Stevenson, J.A. Whitehead, Jr.: Nature, 319, 759–761 (1986)CrossRefADSGoogle Scholar
  15. 15.
    P. Olson, U. Christensen: J. Geophys. Res. 91(B), 6367 (1986)ADSCrossRefGoogle Scholar
  16. 16.
    J.A. Whitehead: Am. J. Phys. 55(11), 998–1003 (1987)CrossRefADSGoogle Scholar
  17. 17.
    K.R. Helfrich, J.A. Whitehead: Geophys. Astrophys. Fluid Dyn. 51, 35–52 (1990)CrossRefADSGoogle Scholar
  18. 18.
    J.A. Whitehead, K.R. Helfrich: Geophys. Res. Letts. 13, 545–546 (1986)ADSCrossRefGoogle Scholar
  19. 19.
    R.A. Bagnold: The Physics of Blown Sand and Desert Dunes (Chapman and Hall, London 1941)Google Scholar
  20. 20.
    D.A. Rumpel: Sedimentology 32, 267–275 (1985)CrossRefADSGoogle Scholar
  21. 21.
    B.B. Willetts, M.A. Rice: ‘Inter-saltation collisions’. In: Proc. Int. workshop on the physics of blown sand, Dept. of theoretical statistics, Unversity of Aarhus 1, 83–100 (1986)Google Scholar
  22. 22.
    J. Ungar, P.K. Ha.: Sedimentology 34, 289–299 (1987)CrossRefADSGoogle Scholar
  23. 23.
    R.S. Anderson: Sedimentology 34, 943–956 (1987)CrossRefADSGoogle Scholar
  24. 24.
    H. Nishimori, N. Ouchi: Phys. Rev. Lett. 71, 197 (1993)CrossRefADSGoogle Scholar
  25. 25.
    R. Hoyle, A. Woods: Phys. Rev. E 56, 6861 (1997)CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    O. Terzidis, P. Claudin, J.-P. Bouchaud: Eur. Physics Journal B 5, 245 (1998)CrossRefADSGoogle Scholar
  27. 27.
    A. Valance, F. Rioual: Eur. Physics Journal B 10, 543 (1999)CrossRefADSGoogle Scholar
  28. 28.
    L. Prigozhin: Phys. Rev. E, 60, 729–733 (1999)CrossRefADSGoogle Scholar
  29. 29.
    Z. Csahok, C. Misbah, F. Rioual, A. Valance. Dynamics of aeolian sand ripples. Preprint cond-mat/0001336 (2000)Google Scholar
  30. 30.
    J.-P. Bouchaud, M.E. Cates, J.R. Prakash, S.F. Edwards: J. Phys. France I 4, 1383 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • N.J. Balmforth
    • 1
  • A. Provenzale
    • 2
    • 3
  • J.A. Whitehead
    • 4
  1. 1.Department of Applied Mathematics and Statistics School of EngineeringUniversity of CaliforniaSanta CruzUSA
  2. 2.Istituto di CosmogeofisicaTorinoItaly
  3. 3.ISI FoundationTorinoItaly
  4. 4.Woods Hole Oceanographic InstitutionWoods HoleUSA

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