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Large Scale Structures in Nonlinear Physics pp 1–20Cite as

The phase-diffusion equation and its regularization for natural convective patterns

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Part of the book series: Lecture Notes in Physics ((LNP,volume 392))

Abstract

We first present the phase diffusion and mean drift equation which describe convective patterns in large aspect ratio containers and for arbitrary Rayleigh and Prandtl numbers. Some applications are presented such as the prediction of the selected wavenumber or the instability of foci. We propose in a second step a regularized form of the phase diffusion equation able to reproduce the formation and dynamics of defects.

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References

  1. A.C. Newell and J.A. Whitehead, “Finite bandwidth, finite amplitude convection”, J. Fluid Mech. 38 (1969), 279. and L.A. Segel, J. Fluid Mech. 38 (1969) 203.

    Google Scholar 

  2. M. Bestehorn and H. Haken, “Traveling waves and pulses in a two-dimensional large-aspect-ratio system”, Phys. Rev. A 42 (1990) 7195.

    Article  PubMed  Google Scholar 

  3. J. Swift and P.C. Hohenberg, “Hydrodynamic fluctuations at the convective instability”, Phys. Rev. A 15, (1977) 319.

    Article  Google Scholar 

  4. P. Coullet, L. Gil and J. Lega, “Defect mediated turbulence”, Phys. Rev. Lett. 62 (1989) 1619.

    Article  PubMed  Google Scholar 

  5. M.C. Cross and A.C. Newell, “Convection Patterns in Large Aspect Ratio Systems”, Physica 10D (1984) 299.

    Google Scholar 

  6. A.C. Newell, T. Passot and M. Souli, “Convection at Finite Rayleigh Numbers in Large-Aspect-Ratio Containers”, Phys. Rev. Lett. 64 (1990) 2378.

    Article  PubMed  Google Scholar 

  7. A.C. Newell, T. Passot and M. Souli, “The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers”, J. Fluid. Mech. 220, (1990) 187–252.

    Google Scholar 

  8. G.B. Whitham, “Linear and nonlinear waves”, Wiley-Interscience, New-York (1974).

    Google Scholar 

  9. L.N. Howard and N. Kopell, “Slowly Varying Waves and Shocks Structures in Reaction-Diffusion Equations”, S. Appl. Math. 56 (1977) 95.

    Google Scholar 

  10. Y. Kuramoto, “Chemical Oscillations, Waves and Turbulence”, Springer Series in Synergetics 19, (1984) H. Haken Ed., Springer-Verlag.

    Google Scholar 

  11. Y. Pomeau and P. Manneville, “Stability and fluctuations of a spatially periodic convective flow”, Journal de Physique-Lettres 23 (1979), L609.

    Google Scholar 

  12. F.H. Busse, “On the stability of two-dimensional convection in a layer heated from below”, J. Math. Phys. 46 (1967) 149.

    Google Scholar 

  13. F.H. Busse and J.A. Whitehead, “Instabilities of convection rolls in a high Prandtl number fluid”, J. Fluid Mech., 47 (1971) 305.

    Google Scholar 

  14. F.H. Busse and J.A. Whitehead, “Oscillatory and collective instabilities in large Prandtl number convection”, J. Fluid Mech., 66 (1974) 67.

    Google Scholar 

  15. F.H. Busse, “Nonlinear properties of convection”, Rep. Prog. Phys., 41 (1978) 1929.

    Article  Google Scholar 

  16. F.H. Busse, “Transition to turbulence in Rayleigh-Bénard convection", Hydrodynamic Instabilities and the Transition to Turbulence, edited by H.L. Swinney and J.P. Gollub (1981) 97, (Berlin: Springer Verlag).

    Google Scholar 

  17. R.M. Clever and F.H. Busse, “Transition to time dependent convection”, J. Fluid Mech., 65 (1974) 625.

    Google Scholar 

  18. R.M. Clever and F.H. Busse, “Instabilities of convection rolls in a fluid of moderate Prandtl number”, J. Fluid Mech., 91 (1979) 319.

    Google Scholar 

  19. R.M. Clever and F.H. Busse, “Large wavelength convection rolls in low Prandtl number fluid”, J. Appl. Math. Phys. Z. angew. math. Phys. 29 (1978) 711.

    Article  Google Scholar 

  20. Y. Pomeau and P. Manneville, “Wavelength selection in axisymmetric cellular structures”, J. Physique Lett. 42 (1981) 1067.

    Google Scholar 

  21. V. Steinberg, G. Ahlers and D.S. Cannell, “Pattern formation and wavenumber selection by Rayleigh-Bénard convection in a cylindrical container”, Physica Scripta, T13 (1985) 135.

    Google Scholar 

  22. Y. Pomeau and P. Manneville, “Wavelength selection in axisymmetric cellular structures”, J. Physique Lett., (1981), 42, 1067.

    Google Scholar 

  23. A.C. Newell, “Two Dimensional Convection Patterns in Large Aspect Ratio Systems”, Lecture Notes in Num. Appl. Anal. 5 (1982), 205.

    Google Scholar 

  24. M.S. Heutmaker and J.P. Gollub, “Wave-vector field of convective flow patterns”, Phys. Rev. A 35 (1987) 242.

    Article  PubMed  Google Scholar 

  25. S. Fauve, “Large Scale Instabilities of Cellular Flows”, in ”Instabilities and Nonequilibrium Structures”, (1987) 63–88, Eds. E. Tirapegui and D. Villaroel, D. Reidel Publishing Company.

    Google Scholar 

  26. A.C. Newell, “Solitons in Mathematics and Physics”, CBMS-NSF vol. 48 (1985), SIAM.

    Google Scholar 

  27. J.C. Luke, “A perturbation Method for Nonlinear Dispersive Wave Problems”, Proc. Roy. Soc. London Ser. A 292 (1966), 403.

    Google Scholar 

  28. T. Passot and A.C. Newell, “The regularization of the phase diffusion equation for natural convective patterns“, submitted to J. Physique.

    Google Scholar 

  29. P. Manneville, “A two-dimensional model for three-dimensional connective patterns in wide containers”, J. Physique 44 (1983) 759.

    Google Scholar 

  30. H.S. Greenside, M.C. Cross and W.M. Coughran, Jr., “Mean Flows and the Onset of Chaos in Large-Cell Convection”, Phys. Rev. Lett. 60 (1988), 2269.

    Article  PubMed  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, 85721, Tucson, AZ

    T. Passot & A. C. Newell

Authors
  1. T. Passot
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  2. A. C. Newell
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Jean-Daniel Fournier Pierre-Louis Sulem

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© 1991 Springer-Verlag]

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Passot, T., Newell, A.C. (1991). The phase-diffusion equation and its regularization for natural convective patterns. In: Fournier, JD., Sulem, PL. (eds) Large Scale Structures in Nonlinear Physics. Lecture Notes in Physics, vol 392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54899-8_29

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  • DOI: https://doi.org/10.1007/3-540-54899-8_29

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54899-7

  • Online ISBN: 978-3-540-46469-3

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