Skip to main content Accessibility help
Hostname: page-component-55597f9d44-2qt69 Total loading time: 0.369 Render date: 2022-08-12T16:12:03.050Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

The symmetry of convective transitions in space and time

Published online by Cambridge University Press:  21 April 2006

Dan Mckenzie
Department of Earth Sciences, Bullard Laboratories, Madingley Road, Cambridge CB3 0EZ, UK


There is a close relationship between the symmetry changes in the plane and space groups of convective systems and the type of the bifurcation. To explore this relationship the plane- and space-group symmetry of many convective circulations is first classified using standard crystallographic notation. The transitions that occur between these patterns can be described by the loss of either a translational or a point-group symmetry element. These are referred to as klassengleiche and translationengleiche or k and t, transitions respectively. Any transition can be decomposed into a series of k and t transitions. The symmetry of the governing differential equations is most easily discussed when these are written in terms of potentials, and allows transitions to be classified as pitchfork, transcritical or Hopf bifurcations. Such classification can be carried out from symmetry alone, without any consideration of the functional form of the solutions, the Rayleigh number or the importance of the nonlinear terms. For this purpose it is convenient to define a factor group, the irreducible representations of which transform as do the variables and differential operators in the equations.

Analysis of planform transitions observed in convective flows when the viscosity is temperature dependent using plane groups shows that all except the transition from the conductive solution to hexagons are pitchfork bifurcations. The factor groups involved are the cyclic group Z2, and the dihedral groups D3 and D5. When the viscosity is constant, space groups are needed, and symmetry arguments show that all except the Eckhaus instability are pitchfork bifurcations, including that from the conductive solution. Hopf transitions to solutions periodic in time, in double-diffusive, in small- and in large-Prandtl-number convection, all involve loss of reflection symmetry in time and the factor group is D2. The same approach suggests how transitions to circulations that are not periodic in either space or time may occur by period doubling in space or in time or in both.

Research Article
© 1988 Cambridge University Press

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.)


Arnol'd, V. I. 1983 Geometrical Methods in the Theory of Ordinary Differential Equations. Springer.
Bolton, E. W., Busse, F. H. & Clever, R. M. 1986 Oscillatory instabilities of convection rolls at intermediate Prandtl numbers. J. Fluid Mech. 164, 469485.Google Scholar
Bradley, C. J. & Cracknell, A. P. 1972 The Mathematical Theory of Symmetry in Solids. Oxford University Press.
Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. & Zassenhaus, H. 1978 Crystallographic Groups of Four-Dimensional Space. Wiley.
Bülow, R., Neubüser, J. & Wondratschek, H. 1971 On crystallography in higher dimensions. II, Procedure of computation in R4. Acta Crystallogr. A27, 520523.Google Scholar
Busse, F. H. 1967a The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625649.Google Scholar
Busse, F. H. 1967b On the stability of two-dimensional convection in a layer heated from below. J. Maths & Phys. 46, 140150.Google Scholar
Busse, F. H. 1972 The oscillatory instability of convection rolls in a low Prandtl number fluid. J. Fluid Mech. 52, 97112.Google Scholar
Busse, F. H. & Bolton, E. W. 1984 Instabilities of convection rolls with stress-free boundaries near threshold. J. Fluid Mech. 146, 115125.Google Scholar
Busse, F. H. & Clever, R. M. 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319335.Google Scholar
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305320.Google Scholar
Buzano, E. & Golubitsky, M. 1983 Bifurcation on the hexagonal lattice and the planar Bénard problem. Phil. Trans. R. Soc. Lond. A 308, 617667.Google Scholar
Cotton, F. A. 1963 Chemical Applications of Group Theory. Interscience.
Curry, J. H., Herring, J. R., Loncaric, J. & Orsag, S. A. 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 138.Google Scholar
Di Bartolo, B. 1968 Optical Interactions in Solids. Wiley.
Golubitsky, M. & Schaeffer, D. 1985 Singularities and Groups in Bifurcation Theory. Springer.
Golubitsky, M. & Stewart, I. 1985 Hopf bifurcation in the presence of symmetry. Arch. Rat. Mech. Anal. 87, 107165.Google Scholar
Golubitsky, M., Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh-Bénard convection. Physica 10D, 249276.Google Scholar
Grossman, I. & Magnus, W. 1964 Groups and their Graphs. Washington: Mathematical Association of America.
Hahn, T. (ed.) 1983 International Tables for Crystallography, vol. A. Space Group Symmetry. Dordrecht: Reidel.
Heesch, H. 1930 Über die vierdimensionalen Gruppen des driedimensionalen Raums. Z. Kristallogr. 73, 325345.Google Scholar
Heine, V. & McConnel, J. D. C. 1984 The origin of incommensurate structures in insulators. J. Phys. C: Solid State Phys. 17, 11991220.Google Scholar
Henry, N. F. M. & Londsdale, K. (eds) 1952 (revised 1977) International Tables for X-ray Crystallography. Birmingham: Kynoch.
Hermann, C. 1929 Zur systematischen strukturtheorie. IV, Untergruppen. Z. Kristallogr. 69, 533555.Google Scholar
Hill, V. E. 1975 Groups, Representations and Characters. Hafner.
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Appl. Maths 1, 303323.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Addison-Wesley.
Landau, L. D. & Lifshitz, E. M. 1980 Statistical Physics. Pergamon.
Ledermann, W. 1973 Introduction to group theory. Longman.
Lennie, T., McKenzie, D. P., Moore, D. R. & Weiss, N. O. 1988a A numerical investigation of three dimensional convection driven by fixed heat flux boundary conditions. J. Fluid Mech. (to be submitted).Google Scholar
Lennie, T. B., McKenzie, D. P., Moore, D. R. & Weiss, N. O. 1988b The breakdown of steady convection. J. Fluid Mech. 188, 4785.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225260.Google Scholar
Moore, D. R. & Weiss, N. O. 1973 Two dimensional Rayleigh-Bénard convection. J. Fluid Mech. 58, 289312.Google Scholar
Neubüser, J., Wondratschek, H. & Bülow, R. 1971 On crystallography in higher dimensions, 1. General definitions. Acta Crystallogr. A 27, 517520.Google Scholar
Richter, F. M. 1978 Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature. J. Fluid Mech. 89, 553560.Google Scholar
Ruelle, D. & Takens, F. 1971a On the nature of turbulence. Commun. Math. Phys. 20, 167192.Google Scholar
Ruelle, D. & Takens, F. 1971b Note concerning our paper ‘On the nature of turbulence’. Commun. Math. Phys. 23, 343344.Google Scholar
Sattinger, D. H. 1977 Group representation theory and branch points of nonlinear functional equations. SIAM J. Math. Anal. 8, 179201.Google Scholar
Sattinger, D. H. 1978 Group representation theory, bifurcation theory and pattern formation. J. Fluid Mech. 28, 58101.Google Scholar
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129144.Google Scholar
Shubnikov, A. V. & Koptsik, V. A. 1974 Symmetry in Science and Art. Plenum.
Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer.
Stern, M. E. 1960 The salt fountain and thermohaline convection. Tellus 12, 172175.Google Scholar
Veronis, G. 1965 On finite amplitude instability in thermohaline convection. J. Mar. Res. 23, 117.Google Scholar
White, D. B. 1981 Experiments with convection in a variable viscosity fluid, Ph.D. thesis, Cambridge University.
White, D. B. 1988 The planforms and onset of convection with a temperature-dependent viscosity. J. Fluid Mech.Google Scholar
Wigner, E. P. 1959 Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic.
Willis, G. E. & Deardorff, J. W. 1970 The oscillatory motions of Rayleigh convection. J. Fluid Mech. 44, 661672.Google Scholar
Wondratschek, H., Bülow, R. & Neubüser, J. 1971 On crystallography in higher dimensions, III. Results in R4. Acta Crystallogr. A 27, 523535.Google Scholar
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The symmetry of convective transitions in space and time
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

The symmetry of convective transitions in space and time
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

The symmetry of convective transitions in space and time
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *