Bound oscillations on thin magnetic flux tubes - Convective instability and umbral oscillations

The possibility that 'tube waves' can be trapped on slender solar magnetic flux tubes is investigated. For rigid isothermal flux tubes, it is found that the flux tube geometry can by itself lead to waves which are trapped on the part of the tube that expands with height. Some geometries lead to trapped modes with eigenperiods near 180 s, if parameters appropriate to sunspot umbrae are chosen. It is possible that the umbral oscillations are a manifestation of such trapped waves, if sunspot umbrae consist of an assembly of slender flux tubes, as in the spaghetti model of Parker (1979). For flux tubes which have a constant ratio of Alfven speed to sound speed, it is found that it is primarily the variation of temperature with height which determines whether trapped waves can exist. Certain temperature profiles lead to disturbances for which omega squared is less than zero, corresponding to convective instability or Rayleigh-Taylor instability.