Stability of Sunspots to Convective Motions. I. Adiabatic Instability

Resumen

For determining the adiabatic stability of a uniform vertical field in an arbitrary stratification it is sufficient to consider the limit of infinitesimal horizontal wavelength. It is shown how the behavior of the instability can be estimated qualitatively from the dependence of the equipartition field strength on depth. Modes are calculated numerically for analytic stratification models and for a detailed sunspot stratification, including the effects of partial ionization. It is concluded that for the observed field strengths of umbrae the stratification is indeed unstable, with a growth time of about 18 minutes. The unstable eigenfunctions have a maximum at about 2300 km below the surface of the umbra and are about 3900 km deep. Deeper layers may also be unstable depending on unknown details of the stratification. A connection between fluting instability and convective instability is noted.