Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems pp 197-209 | Cite as

# Phase-Mean Drift Equation for Convection Patterns in Large Aspect Ratio Containers

- 146 Downloads

## Abstract

We present the phase diffusion and mean drift equation which describe the convection pattern in large aspect ratio containers for arbitrary large Rayleigh numbers. An exact agreement is found with the borders of the Busse balloon, concerning the long wavelength instabilities. We propose a calculation of the selected wavenumber which agrees closely with experiments and we predict a new instability which appears to be important in initiating time dependence. We predict also the Rayleigh numbers at which loss of spatial correlation due to global defect nucleation will occur.

## Keywords

Prandtl Number Rayleigh Number Phase Equation Convection Pattern Circular Patch## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Stuart, J. T., “On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. I. The basic behavior in plane Poiseuille flow,” J. Fluid Mech.,
**9**, 353 (1960).MathSciNetzbMATHCrossRefGoogle Scholar - [2]Malkus, W. V. R. and Veronis, G., “Finite amplitude convection,” J. Fluid Mech.,
**4**, 225 (1958).MathSciNetADSzbMATHCrossRefGoogle Scholar - Schulter, A., Lortz, D. and Busse, F. H., “On the stability of steady finite amplitude convection,” J. Fluid Mech.,
**23**, 129 (1965).MathSciNetADSCrossRefGoogle Scholar - [3]Lorentz, E. N., “Deterministic non-periodic flow,” Journal of Atmospheric Sciences,
**20**, 130 (1963).ADSCrossRefGoogle Scholar - [4]Newell, A. C. and Whitehead, J. A.,”Finite Bandwidth, Finite Amplitude Convection,” J. Fluid Mech.,
**38**, 179 (1969).CrossRefGoogle Scholar - [5]Segel, L. A., 1969, “Distant side-walls cause slow amplitude modulation of cellular convection,” J. Fluid Mech.,
**38**, 203 (1969).ADSzbMATHCrossRefGoogle Scholar - [6]Busse, F. H., “On the stability of two-dimensional convection in a layer heated from below,” J. Math. Phys.,
**46**, 149–150 (1967).Google Scholar - [7]Eckhaus, A. W., Studies in Nonlinear Stability Theory, (New York: Springer-Verlag) (1965).Google Scholar
- [8]Siggia, E. D. and Zippelius, A., “Pattern selection in Rayleigh-Bénard convection near threshold,:” Phys. Rev. Lett.,
**47**, 835 (1981b).ADSCrossRefGoogle Scholar - [9]Cross, M. C. and Newell, A. C, “Convection Patterns in Large Aspect Ratio Systems,” Physica,
**10D**, 299–328 (1984).MathSciNetADSGoogle Scholar - [10]Bensoussan A., Lions, J. L. and Papanicolaou, G., “Asymptotic Analysis for Periodic Structures,” Studies in Mathematics and its Applications, Vol. 5, North-Holland (1978).zbMATHGoogle Scholar
- [11]Whitham, G. B., “Linear and Nonlinear Waves,” Wiley-Interscience (1974).zbMATHGoogle Scholar
- [12]Busse, F. H. and Whitehead, J. A., “Instabilities of convection rolls in a high Prandtl number fluid,” J. Fluid Mech.,
**47**, 305–320 (1971).ADSCrossRefGoogle Scholar - [13]Busse, F. H. and Whitehead, J. A. “Oscillatory and collective instabilities in large Prandtl number convection,” J. Fluid Mech.,
**66**, 67–79 (1974).ADSCrossRefGoogle Scholar - [14]Busse, F. H., “Nonlinear properties of convection,” Rep. Prog. Phys.,
**41**, 1929–1967 (1978).ADSCrossRefGoogle Scholar - [15]Busse, F. H., “Transition to turbulence in Rayleigh-Bénard convection,” Hydrodynamic Instabilities and the Transition to Turbulence, edited by H. L. Swinney and J. P. Gollub (Berlin: Springer-Verlag), 97 (1981).CrossRefGoogle Scholar
- [16]Clever, R. M. and Busse, F. H., “Transition to time dependent convection,” J. Fluid Mech.,
**65**, 625–645 (1974).ADSzbMATHCrossRefGoogle Scholar - Clever, R. M. and Busse, F. H., “Instabilities of convection rolls in a fluid of moderate Prandtl number,” J. Fluid Mech.,
**91**, 319–335 (1979).ADSCrossRefGoogle Scholar - [17]Clever, R. M. and Busse, F. H., “Large wavelength convection rolls in low Prandtl number fluid,” J. Appl. Math. Phys. Z. angew. math. Phys.,
**29**, 711–714 (1978).CrossRefGoogle Scholar - [18]Steinberg, V., Ahlers, G., Cannell, D. S., “Pattern formation and wavenumber selection by Rayleigh-Bénard convection in a cylindrical container,” Physica Scripta,
**T13**, 135 (1985).Google Scholar - [19]Heutmaker, M. S. and Gollub J. P., “Wave-vector field of convective flow patterns,” Phys. Rev. A,
**35**, 242 (1987).ADSCrossRefGoogle Scholar - [20]Croquette, V., Le Gal, P. and Pocheau A., Physica Scripta,
**T13**, 135 (1986).ADSCrossRefGoogle Scholar - [21]Bodebschatz, E., Pesch, W., and Kramer, L., “Structure and dynamics of dislocations in anisotropic pattern forming systems,” Physica D,
**32**, 135 (1988).ADSCrossRefGoogle Scholar