Localised Oscillatory States in Magnetoconvection

Localised states are found in many pattern forming systems. The aim of this paper is to investigate the occurrence of oscillatory localised states in two-dimensional Boussinesq magnetoconvection. Initially considering an idealised model, in which the vertical structure of the system has been simplified by a projection onto a small number of Fourier modes, we find that these states are restricted to the low $\zeta$ regime (where $\zeta$ represents the ratio of the magnetic to thermal diffusivities). These states always exhibit bistability with another non-trivial solution branch, in other words they show no evidence of subcritical behaviour. This is due to the weak flux expulsion that is exhibited by these time-dependent solutions. Using the results of this parameter survey, we locate corresponding states in a fully-resolved two-dimensional system, although the mode of oscillation is more complex in this case. This is the first time that a localised oscillatory state, of this kind, has been found in a fully-resolved magnetoconvection simulation.