Published online by Cambridge University Press: 21 April 2006
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.