Oscillatory convection in sunspot umbrae

Summary Subphotospheric convection is partially inhibited by the strong vertical magnetic field at the centre of a sunspot. We investigate the effects of stratification on non-linear magnetoconvection in a fully compressible perfect gas by means of two-dimensional numerical experiments. Behaviour depends critically on the ratio ξ of the magnetic to the thermal diffusivity, which increases with depth. If ξ > 1 throughout the layer we find steady overturning convection with an asymmetry between rising and falling plumes. For ξ > 1 we obtain oscillatory convection with reversals of the flow. When ξ > 1 at the top of the layer but ξ > 1 at its base, convection sets in as steady motion but there is a transition (involving several bifurcations) to mixed-mode periodic solutions as the Rayleigh number is increased. The flow at the bottom of the layer does not reverse but adjacent rising plumes alternate between active and quiescent states. In the umbral photosphere t, > 1 but ξ > 1 at depths below 1500 km owing to the increase in opacity. Our results imply that time-dependent convection immediately below the photosphere is coupled to motion at levels where ξ > 1. They also explain the existence of umbral dots with a lifetime corresponding to the Alfven time for the converting layer.