SYMMETRIES OF TIME-DEPENDENT MAGNETOCONVECTION

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 Z2 symmetry by sampling solutions at intervals of 1/2P. Then the fundamental steady solution has the spatiotemporal symmetry D2 = Z2 x Z2 and symmetry-breaking yields solutions with Z2 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 D2h = D2 x Z2 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 D2h 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, D2h x Z2, 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.