  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.
