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Localised plumes in three-dimensional compressible magnetoconvection

Within the umbrae of sunspots, convection is generally inhibited by the presence of strong vertical magnetic fields. However, convection is not completely suppressed in these regions: bright features, known as umbral dots, are probably associated with weak, isolated convective plumes. Motivated by observations of umbral dots, we carry out numerical simulations of three-dimensional, compressible magnetoconvection. By following solution branches into the subcritical parameter regime (a region of parameter space in which the static solution is linearly stable to convective perturbations), we find that it is possible to generate a solution which is characterised by a single, isolated convective plume. This solution is analogous to the steady magnetohydrodynamic convectons that have previously been found in two-dimensional calculations. These results can be related, in a qualitative sense, to observations of umbral dots.