The evolution of a horizontal scale for oscillatory magnetoconvection

Oscillatory convective motions have been observed in the umbrae of sunspots and, in the past, the linear theory of overstability has been used for sunspot models. Here, a nonlinear model for oscillatory convection has been used to investigate the possibility of a preferred horizontal cell size for these motions, in the presence of a magnetic field. The integration forward in time, from the conductive state, of the nonlinear multimode equations governing magnetoconvection when the magnetic Prandtl number is less than one portrays a complex interaction between the evolving magnetic and vertical velocity horizontal scales. Preferred horizontal scales for the convective cells have been established by identifying the modes that substantially contribute to the overall convective heat transport. All other modes, although initially perturbed, in time essentially decay to zero through self-interaction.