Spatiotemporal Chaos in Interfacial Waves

  • J. P. Gollub
  • R. Ramshankar

Abstract

Spatiotemporal chaos occurs in many hydrodynamic systems. This state is distinguished from temporal chaos by at least partial loss of coherence between different spatial regions, and from fully developed turbulence by the absence of a cascade. Several experimental and theoretical examples of spatiotemporal chaos are briefly reviewed. Parametrically forced surface waves provide a favorable context for the study of spatiotemporal chaos since the number of active modes can be controlled by varying the driving frequency. In this chapter, we describe the onset and statistical properties of spatiotemporal chaos in surface waves. Of particular note is the presence of phenomena on scales much greater than the wavelength, so that the de-correlation that is one hallmark of spatiotemporal chaos is incomplete.

Keywords

Surface Wave Correlation Length Chaotic Dynamic Excitation Frequency Excitation Amplitude 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahlers, G., Cannell, D.S., and Steinberg, V., 1985, “Time dependence of flow patterns near the convective threshold in a circular cylinder,” Phys. Rev. Lett. 54, 1373–1376.CrossRefGoogle Scholar
  2. Ahlers, G., Cannell, D.S., and Heinrichs, R., 1987, “Convection in a binary mixture,” Nuc. Phys. B (Proc. Suppl.) 2, 77–86.CrossRefGoogle Scholar
  3. Albert, B.S., 1986, “Pattern formation in a nematic liquid crystal undergoing an electrohydrodynamic instability in the presence of a spatially periodic forcing,” Haverford College, B.S. thesis (unpublished).Google Scholar
  4. Aubry, N., Holmes, P., Lumley, J.L., and Stone, E., 1988, “The dynamics of coherent structures in the wall region of a turbulent boundary layer,” J. Fluid Mech. 192, 115–173.MathSciNetMATHCrossRefGoogle Scholar
  5. Benjamin, T.B. and Ursell, F., 1954, “The stability of the plane free surface of a liquid in vertical periodic motion,” Proc. R. Soc. Lond. A225, 505–515.MathSciNetGoogle Scholar
  6. Bodenschatz, E., Kaiser, M., Kramer, L., Pesch, W., Weber, A., and Zimmermann, W., 1989, “Pattern and defects in liquid crystals,” in New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequillibrium (eds. P. Coullet and P. Huerre), Plenum Press.Google Scholar
  7. Chaté, H. and Manneville, P., 1987, “Transition to turbulence via spatiotemporal intermittency,” Phys. Rev. Lett. 58, 112–115.CrossRefGoogle Scholar
  8. Ciliberto, S. and Gollub, J.P., 1985, “Chaotic mode competition in parametrically forced surface waves,” J. Fluid Mech. 158, 381–398.MathSciNetCrossRefGoogle Scholar
  9. Ciliberto, S. and Bigazzi, P., 1988, “Spatiotemporal intermittency in Rayleigh-Bénard convection,” Phys. Rev. Lett. 60, 286–289.CrossRefGoogle Scholar
  10. Coullet, P., Elphick, C., Gil, L., and Lega, J., 1987, “Topological defects of wave patterns,” Phys. Rev. Lett. 59, 884–886.CrossRefGoogle Scholar
  11. Coullet, P., Gil, L., and Repaux, D., 1989, “Defects and subcriticai bifurcations,” Phys. Rev. Lett. 62, 2957–2960.CrossRefGoogle Scholar
  12. Coullet, P., Gil, L., and Lega, L., 1989, “A form of turbulence associated with defects,” Physica D 37, 91–103.MathSciNetMATHCrossRefGoogle Scholar
  13. Crawford, J.D., Knobloch, E., and Riecke, H., 1989, “Mode interactions and symmetry,” in Proceedings of the International Conference on Singular Behavior and Nonlinear Dynamics, Samos, Greece, World Scientific.Google Scholar
  14. Croquette, V., 1989, “Convective pattern dynamics at low Prandtl number,” Contemp. Phys. 30, 113–133 (Part I); Contemp. Phys. 30, 153–171 (Part II).CrossRefGoogle Scholar
  15. Daviaud, F., Dubois, M., and Bergé, P., 1989, “Spatio-temporal intermittency in quasi one-dimensional Rayleigh-Bnard convection,” Europhys. Lett. 9, 441–446.CrossRefGoogle Scholar
  16. Douady, S. and Fauve, S., 1988, “Pattern selection in Faraday instability,” Europhys. Lett. 6, 221–226.CrossRefGoogle Scholar
  17. Ezerskii, A.B., Korotin, P.I., and Rabinovich, M.I., 1985, “Random self-modulation of two-dimensional structures on a liquid surface during parametric excitation,” Pis’ma Zh. Eksp. Teor. Fiz. 41, 129–131; JETP Lett. 41, 157–160.Google Scholar
  18. Ezerskii, A.B., Rabinovich, M.I., Reutov, V.P., and Starobinets, I.M., 1986, “Spatiotemporal chaos in the parametric excitation of a capillary ripple,” Zh. Eksp. Teor. Fiz. 91, 2070–2083; Sov. Phys. JETP 64, 1228–1236.Google Scholar
  19. Ezersky, A.B. and Rabinovich, M.I., 1990, “Nonlinear wave competition and anisotropic spectra of spatio-temporal chaos of Faraday ripples,” Europhys. Lett. 1990, 243–249.Google Scholar
  20. Faraday, M., 1831, “On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces,” Phil. Trans. R. Soc. Lond. 121, 299–340.CrossRefGoogle Scholar
  21. Funakoshi, M. and Inoue, S., 1987, “Chaotic behavior of resonantly forced surface water waves,” Phys. Lett. A. 121, 229–232.CrossRefGoogle Scholar
  22. Gollub, J.P. and Meyer, C.W., 1983, “Symmetry-breaking instabilities on a fluid surface,” Physica 6D, 337–346.Google Scholar
  23. Goren, G., Procaccia, I., Rasenat, S., and Steinberg, V., 1989, “Interactions and dynamics of topological defects”, Phys. Rev. Lett. 63, 1237–1240.CrossRefGoogle Scholar
  24. Gu, X.M., Sethna, P.R., and Narain, A., 1988, “On three-dimensional nonlinear subharmonic resonant surface waves in a fluid. Part I: Theory,” Trans ASME E: J. Appl. Mech. 55, 213–219.CrossRefGoogle Scholar
  25. Henderson, D.M. and Miles, J.W., 1990, “Single-mode Faraday waves in small containers,” J. Fluid Mech. 213, 95–109.CrossRefGoogle Scholar
  26. Heutmaker, M.S., 1986, “A quantitative study of pattern evolution and time dependence on Rayleigh-Bénard convection,” University of Pennsylvania, Ph.D. thesis (unpublished).Google Scholar
  27. Heutmaker, M.S. and Gollub, J.P., 1987, “Wave-vector field of convective flow patterns,” Phys. Rev. A 35, 242–259.CrossRefGoogle Scholar
  28. Hohenberg, P.C. and Shraiman, B.I., 1989, “Chaotic Behavior of an Extended System,” Physica D 37, 109–115.MathSciNetCrossRefGoogle Scholar
  29. Kai, S. and Hirakawa, K., 1978, “Successive transitions in electrohydrodynamic instabilities of nematics,” Prog. Theor. Phys. Suppl. 64, 212–243.CrossRefGoogle Scholar
  30. Kaneko, K., 1984, “Period-doubling of kink-antikink patterns, quasiperiodicity in anti-ferro-like structures and spatial intermittency in coupled logistic lattice,” Prog. Theor. Phys. 72, 480–486.MATHCrossRefGoogle Scholar
  31. Kolodner, P., Bensimon, D., and Surko, C.M., 1988 “Traveling wave convection in an annulus,” Phys. Rev. Lett. 60, 1723–1726.CrossRefGoogle Scholar
  32. Manneville, P., 1981, “Statistical properties of chaotic solutions of a one-dimensional model for phase turbulence,” Phys. Lett. 84A, 129–132.MathSciNetCrossRefGoogle Scholar
  33. Meron, E. and Procaccia, I., 1986, “Low-dimensional chaos in surface waves: theoretical analysis of an experiment,” Phys. Rev. A 34, 3221–3237.CrossRefGoogle Scholar
  34. Meron, E., 1987, “Parametric excitation of multimode dissipative systems,” Phys. Rev. A 35, 4892–4895.MathSciNetCrossRefGoogle Scholar
  35. Miles, J.W., 1984, “Resonantly forced surface waves in a circular cylinder,” J. Fluid Mech. 149, 15–31.MathSciNetMATHCrossRefGoogle Scholar
  36. Miles, J. and Henderson, D., 1990, “Parametrically forced surface waves,” Ann. Rev. Fluid Mech. 22, 143–165.MathSciNetCrossRefGoogle Scholar
  37. Milner, S., 1989, “Square patterns and secondary instabilities in driven capillary waves,” preprint.Google Scholar
  38. Nobili, M., Ciliberto, S., Cocchiaro, B., Faetti, S., and Fronzoni, L., 1988, “Time-dependent surface waves in a horizontally oscillating container,” Europhys. Lett. 7, 587–592.CrossRefGoogle Scholar
  39. Pocheau, A., 1989, “Phase dynamics attractors in an extended cylindrical convective layer,” J. Physique 50, 25–69.Google Scholar
  40. Pocheau, A., Croquette, V., and Le Gal, P., 1985, “Turbulence in a cylindrical container of Argon near threshold for convection,” Phys. Rev. Lett. 55, 1094–1097.CrossRefGoogle Scholar
  41. Pomeau, Y., Pumir, A., and Pelcé, 1984, “Intrinsic stochasticity with many degrees of freedom,” J. Stat. Phys. 37, 39–49.CrossRefGoogle Scholar
  42. Pumir, A., 1985, “Statistical properties of an equation describing fluid interfaces,” J. Physique 46, 511–522.MathSciNetCrossRefGoogle Scholar
  43. Rehberg, I., Rasenat, S., and Steinberg, V., 1989, “Traveling waves and defect-initiated turbulence in electroconvecting nematics,” Phys. Rev. Lett. 62, 756–759.CrossRefGoogle Scholar
  44. Shraiman, B.I., 1986, “Order, disorder and phase turbulence,” Phys. Rev. Lett. 57, 325–328.CrossRefGoogle Scholar
  45. Simonelli, F. and Gollub, J.P., 1989, “Surface wave mode interactions: effects of symmetry and degeneracy,” J. Fluid Mech. 199, 471–494.MathSciNetCrossRefGoogle Scholar
  46. Sirovich, L., 1991, “Empirical Eigenfunctions and Low Dimensional Systems,” in New Perspectives in Turbulence, L. Sirovich, Ed., Springer-Verlag. (This volume.)Google Scholar
  47. Sompolinsky, H., Crisanti, A., and Sommers, H.J., 1988, “Chaos in random neural networks,” Phys. Rev. Lett. 61, 259–262.MathSciNetCrossRefGoogle Scholar
  48. Steinberg, V., Moses, E., and Fineberg, J., 1987, “Spatio-temporal complexity at the onset of convection in a binary fluid,” Nuc. Phys. B (Proc. Suppl.) 2, 109–124.CrossRefGoogle Scholar
  49. Tufillaro, N.B., Ramshankar, R., and Gollub, J.P., 1989, “Order-disorder transition in capillary ripples,” Phys. Rev. Lett. 62, 422–425.CrossRefGoogle Scholar
  50. Waller, I. and Kapral, R., 1984, “ Spatial and temporal structures in systems of coupled nonlinear oscillators,” Phys. Rev. A 30, 2047–2055.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • J. P. Gollub
  • R. Ramshankar

There are no affiliations available

Personalised recommendations