Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T08:26:25.691Z Has data issue: false hasContentIssue false
  • English
  • Français

Planform evolution in convection–an embedded centre manifold

Published online by Cambridge University Press:  17 February 2009

A. J. Roberts
A. J. Roberts
Affiliation:
Dept. of Applied Math., The University of Adelaide, Adelaide 5001, South Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The new motion of embedding a centre manifold in some higher-dimensional manifold leads to a practical approach to the rational low-dimensional approximation of a wide class of dynamical systems; it also provides a simple geometric picture for these approximations. In particular, I consider the problem of finding an approximate, but accurate, description of the evolution of a two-dimensional planform of convection. Inspired by a simple example, the straightforward adiabatic iteration is proposed to estimate an embedding manifold and arguments are presented for its effectiveness. Upon applying the procedure to a model convective planform problem I find that the resulting approximations perform remarkably well–much better than the traditional Swift-Hohenberg approximation for planform evolution.

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 2 , October 1992 , pp. 174 - 198
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Arneodo, A. and Thual, O., “Direct numerical simulation of a triple convection problem versus normal form predictions”, Phys. Lett. A 109 (1985) 367–372.CrossRefGoogle Scholar
[2]Bestehorn, M., Friedrich, R. and Haken, H., “Traveling waves in nonequilibrium systems”, Physica D 37 (1989) 295–299.CrossRefGoogle Scholar
[3]Brand, H. R., “Phase dynamics—a review and a perspective”, pp. 206–224 in Propagation in systems far from equilibrium ed. by Wespeid, et al. , Springer-Verlag (1988).CrossRefGoogle Scholar
[4]Carr, J., “Applications of centre manifold theory”, Applied Math. Sci. 35 (1981).CrossRefGoogle Scholar
[5]Coullet, P. H. and Spiegel, E. A., “Amplitude equations for systems with competing instabilities”, SIAM J. Appl. Math. 43 (1983) 776–821.CrossRefGoogle Scholar
[6]Cox, S. M. and Roberts, A. J., “Centre manifolds of forced dynamical systems”, J. Austral. Maths. Soc. B 32 (1991) 401–436.CrossRefGoogle Scholar
[7]Craik, A., “Interaction of a short-wave field with a dominant long wave in deep water: derivation from Zakharov's equation”, J. Austral. Math. Soc. B 29 (1988) 430–439.CrossRefGoogle Scholar
[8]Cross, M. C. and Newell, A. C., “Convection patterns in large aspect ratio systems”, Physica D 10 (1984) 299–328.CrossRefGoogle Scholar
[9]Cross, M. C. and Tesauro, G., “Wavenumber selection and persistent dynamics in models of convection”, Physica D 23 (1986), 12–18.CrossRefGoogle Scholar
[10]Eckmann, J.-P. and Wayne, C. E., “Propagating fronts and the center manifold theorem”, preprint (1990).CrossRefGoogle Scholar
[11]Greenside, H. S. and Coughran, W. M. Jr, “Nonlinear pattern formation near the onset of Rayleigh-Bénard convection”, Physical Review A 30 (1984), 398–428.CrossRefGoogle Scholar
[12]Haken, H., Synergetics, an introduction, (Springer, Berlin, 1983).CrossRefGoogle Scholar
[13]Hathaway, D. H. and Somerville, R. C. J., “Thermal convection in rotating shear flow”, Geophys. Astrophys. Fluid Dyn. 38 (1987) 43–68.CrossRefGoogle Scholar
[14]Howard, L. N. and Kopell, N., “Slowly varying waves and shock structures in reaction-diffusion equations”, Stud. Appl. Math. 56 (1977) 95–145.CrossRefGoogle Scholar
[15]Mercer, G. N. and Roberts, A. J., “The application of centre manifold theory to the dispersion of contaminant in channels with varying flow properties”, SIAM J. Appl. Math. 50 (1990), 1547–1565.CrossRefGoogle Scholar
[16]Muncaster, R. G., “Invariant manifolds in mechanics I: The general construction of coarse theories from fine theories”, Arch Rat. Mech. 84 (1983) 353–373.CrossRefGoogle Scholar
[17]Newell, A. C. and Whitehead, J., “Finite bandwidth, finite amplitude convection”, J. Fluid Mech. 38 (1969) 279–303.CrossRefGoogle Scholar
[18]Pomeau, Y. and Manneville, P., “Stability and fluctuations of a spatially periodic convective flow”, J. Phys. Lett. 40 (1979) 609–612.CrossRefGoogle Scholar
[19]Proctor, M. R. E., “Planform selection by finite amplitude thermal convection between poorly conducting slabs”, J. Fluid Mech. 113 (1981) 469–485.CrossRefGoogle Scholar
[20]Roberts, A. J., “Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations”, J. Austral. Math. Soc. Ser. B 27 (1985) 48–65.CrossRefGoogle Scholar
[21]Roberts, A. J., “The application of centre manifold theory to the evolution of systems which vary slowly in space”, J. Austral. Math. Soc. Ser. B 29 (1988) 480–550.CrossRefGoogle Scholar
[22]Roberts, A. J., “Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems”, J. Austral. Math. Soc. Ser. B 31 (1989) 48–75.CrossRefGoogle Scholar
[23]Roberts, A. J., “The utility of an invariant manifold description of the evolution of dynamical systems”, SIAM J. of Math. Anal. 20 (1989) 1447–1458.CrossRefGoogle Scholar
[24]Roberts, A. J., “Boundary conditions for approximate differential equations”, J. Austral. Math. Soc. Ser. B (1992), to appear.CrossRefGoogle Scholar
[25]Roberts, A. J., “The invariant manifold of beam theory. Part 1: the simple circular rod”, J. Elas 29 (1992), to appear.Google Scholar
[26]Roberts, A. J., “A sub-centre manifold description of the evolution and interaction of nonlinear dispersive waves”, in Nonlinear dispersive waves, (ed. Debnath, L.), (World Sci., 1992).Google Scholar
[27]Schuster, H. G., Deterministic chaos. An introduction. Physik-verlag, 1984.Google Scholar
[28]Segel, L. A., “Distant side-walls cause slow amplitude modulation of cellular convection”, J. Fluid Mech. 38 (1969), p. 203.CrossRefGoogle Scholar
[29]Sijbrand, J., “Properties of center manifolds”, Trans. Amer. Math. Soc. 289 (1985) 431–469.CrossRefGoogle Scholar
[30]Swift, J. and Hohenberg, P. C., “Hydrodynamic fluctuations at the convective instability”, Physical Review A 15 (1977) 319–328.CrossRefGoogle Scholar
[31]Van Kampen, N. G., “Elimination of fast variables”, Physics Reports 124 (1985) 69–160.CrossRefGoogle Scholar
[32]Yuen, H. C. and Lake, B. M., “Nonlinear dynamics of deep-water gravity waves”, Adv. in Appl. Math. 22 (1982) 67–229.Google Scholar
[33]Zakharov, V. E., “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, J. Appl. Mech. Tech. Phys. 2 (1968) 190–194.Google Scholar