Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-10T13:58:20.145Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite amplitude convection between stress-free boundaries; Ginzburg–Landau equations and modulation theory

Published online by Cambridge University Press:  26 September 2008

Andrew J. Bernoff
Andrew J. Bernoff
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability theory for rolls in stress-free convection at finite Prandtl number is affected by coupling with low wavenumber two-dimensional mean-flow modes. In this work, a set of modified Ginzburg–Landau equations describing the onset of convection is derived which accounts for these additional modes. These equations can be used to extend the modulation equations of Zippelius & Siggia describing the breakup of rolls, bringing their stability theory into agreement with the results of Busse & Bolton.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 3 , September 1994 , pp. 267 - 282
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arneodo, A., Coullet, P. & Spiegel, E. 1984 Chaos and temporal intermittency in mildly unstable fluids. In: Turbulence and Chaotic Phenomena in Fluids. Tatsumi, T. (Ed.). Elsevier, pp. 215–220.Google Scholar
Arnold, V. I. 1983 Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag.CrossRefGoogle Scholar
Bernoff, A. 1986 Transitions from order in convection. PhD Thesis, University of Cambridge.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.CrossRefGoogle Scholar
Busse, F. H. 1986 Phase-turbulence in convection near threshold. Contemp. Math. 56, 1–8.CrossRefGoogle Scholar
Busse, F. H. & Bolton, E. W. 1984 Instabilities of convection rolls with stress-free boundaries near threshold. J. Fluid Mech. 146, 115–125.CrossRefGoogle Scholar
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D. 1983 Phase-winding solutions in a finite container above the convective threshold. J. Fluid Mech. 127, 155–184.CrossRefGoogle Scholar
Cross, M. C. 1980 Derivation of the amplitude equation for Rayleigh-Benard instability. Phys. Fluids 29, 1727–1731.CrossRefGoogle Scholar
Cross, M. C. & Newell, A. C. 1984 Convection patterns in large aspect ratio systems. Physica 10D, 299–328.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and bifurcations of Vector Fields. Springer-Verlag.CrossRefGoogle Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303.CrossRefGoogle Scholar
Newell, A. C., Passot, T. & Souli, M. 1990 Phase diffusion and mean drift equations for convection. J. Fluid Mech. 220, 187–251.CrossRefGoogle Scholar
Normand, C., Pomeau, Y. & Velarde, M. G. 1977 Convective instability: A physicists's approach. Rev. Mod. Phys. 49, 581–624.CrossRefGoogle Scholar
Schluter, A., Lortz, D. & Busse, F. H. 1965 On the stability of finite amplitude convection. J. Fluid Mech. 23, 129–144.CrossRefGoogle Scholar
Segal, L. A. 1969 Distant sidewalls cause slow amplitude modulation of cellular convection. J. Fluid Mech. 38, 203–224.CrossRefGoogle Scholar
Spiegel, E. A. 1985a Cosmic arrhythmias. In: Chaos in Astrophysics. Perdang, J. & Spiegel, E. G. (Eds.). Reidel, pp. 93–135.Google Scholar
Spiegel, E. A. 1985b Lectures. In: Woods Hole Oceanog. Inst. Tech. Rept. WHOI-85–36. Veronis, G. & Hudon, L. M. (Eds.). Woods Hole Oceanographic Institution, pp. 1–49.Google Scholar
Zippelius, A. & Siggia, E. D. 1982 Disappearance of stable convection between free-slip boundaries. Phys. Rev. A26, 1788–1790.CrossRefGoogle Scholar
Zippelius, A. & Siggia, E. D. 1983 Stability of finite-amplitude convection. Phys. Fluid 26, 2905–2915.CrossRefGoogle Scholar