A model of umbral oscillations inherited from subphotospheric fast-body modes

Recently, complex horizontal patterns of umbral oscillations have been reported, but their physical nature and origin are still not fully understood. Here we show that the two-dimensional patterns of umbral oscillations of slow waves are inherited from the subphotospheric fast-body modes. Using a simple analytic model, we successfully reproduced the temporal evolution of oscillation patterns with a finite number of fast-body modes. In this model, the radial apparent propagation of the pattern is associated with the appropriate combination of the amplitudes in radial modes. We also find that the oscillation patterns are dependent on the oscillation period. This result indicates that there is a cutoff radial mode, which is a unique characteristic of the model of fast-body modes. In principle, both internal and external sources can excite these fast-body modes and produce horizontal patterns of umbral oscillations.