Subsurface filamentation of sunspot magnetic fields has been postulated as a source of the visible small-scale structure of sunspot umbrae. We examine this possibility by investigating the magnetohydrodynamic structure of thin, vertical magnetized gas columns embedded in sunspot umbrae. These umbral fluxtubes are assumed to have weaker field strengths than the surrounding umbral atmosphere. The steady-state magnetohydrodynamic equations are solved numerically in the slender fluxtube approximation, thus allowing for stationary internal mass flows. We include the radiative exchange of heat between the umbral fluxtube and the ambient medium using a simple relaxation time approach.
The geometric shape of the steady flow solution is a gas column converging with height. We discuss the relationship of our results to observed properties of umbral brightenings (umbral dots). We show that, even if there is a large difference in magnetic field strength between the dot and the ambient medium in deeper layers, the field strengths are nearly equal in the observable layers, a result required by the observations. We also show that either high temperatures at the lower boundary of the dots or strong upflows are needed in order to produce bright continuum structures.