Symmetries of time-dependent magnetoconvection

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Abstract
In the presence of a magnetic field, convection may set in at a stationary or an oscillatory bifurcation, giving rise to branches of steady, standing wave and travelling wave solutions. Numerical experiments provide examples of nonlinear solutions with a variety of different spatiotemporal symmetries, which can be classified by establishing an appropriate group structure. For the idealized problem of two-dimensional convection in a stratified layer the system has left-right spatial symmetry and a continuous symmetry with respect to translations in time. For solutions of period P the latter can be reduced to Z 2 symmetry by sampling solutions at intervals of ½P. Then the fundamental steady solution has the spatiotemporal symmetry D 2 = Z 2 ⊗ Z 2 and symmetry-breaking yields solutions with Z 2 symmetry corresponding to travelling waves, standing waves and pulsating waves. A further loss of symmetry leads to modulated waves. Interactions between the fundamental and its first harmonic are described by the group D 2h = D 2 ⊗ Z 2 and its invariant subgroups, which describe solutions that are either steady or periodic in a uniformly moving frame. For a Boussinesq fluid in a layer with identical top and bottom boundary conditions there is also an up-down symmetry. With fixed lateral boundaries the spatiotemporal symmetries, again described by D 2h and its invariant subgroups, can be related to results obtained in numerical experiments and analysed by Nagata et al. (1990). With periodic boundary conditions, the full symmetry group, D 2h ⊗Z 2, is of order 16. Its invariant subgroups describe pure and mixed-mode solutions, which may be steady states, standing waves, travelling waves, pulsating waves or modulated waves.