Pattern selection in three-dimensional magnetoconvection

Abstract

Weakly nonlinear Boussinesq convection in an imposed vertical magnetic field is studied. Stress-free boundary conditions are used at top and bottom of the layer where the magnetic field is assumed to remain vertical. Both steady and oscillatory convection are considered. Among all the spatially periodic patterns steady rolls are the preferred pattern when the onset of convection is steady state; in the overstable regime a variety of spatially and temporally periodic patterns may be stable depending on the parameters. These include in particular a pattern called alternating rolls; the location of stable alternating rolls appears to be consistent with recent numerical simulations of compressible magnetoconvection where such patterns have been found. In certain cases the conduction solution loses stability directly to spatially periodic patterns with nonperiodic time dependence. The analysis of the possible patterns and their stability takes advantage of existing results from equivariant bifurcation theory and uses the critical wavenumber selected by the linear stability calculation as a function of the imposed magnetic field.