The Temporal and Spatial Evolution of Magnetohydrodynamic Wave Modes in Sunspots

Through their lifetime, sunspots undergo a change in their area and shape and, as they decay, they fragment into smaller structures. Here, for the first time we analyze the spatial structure of the magnetohydrodynamic (MHD) slow-body and fast-surface modes in the observed umbrae as their cross-sectional shape changes. The proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) techniques were used to analyze 3 and 6 hr Solar Dynamics Observatory/Helioseismic and Magnetic Imager time series of Doppler velocities at the photospheric level of approximately circular and elliptically shaped sunspots. Each time series was divided into equal time intervals to evidence the change in the shape of the sunspots. To identify the physical wave modes, the POD/DMD modes were cross-correlated with a slow-body mode model using the exact shape of the umbra, whereas the shape obtained by applying a threshold level of the mean intensity for every time interval. Our results show that the spatial structure of MHD modes are affected, even by apparently small changes in the umbral shape, especially in the case of the higher-order modes. For the data sets used in our study, the optimal time intervals to consider the influence of the change in the shape on the observed MHD modes is 37-60 minutes. The choice of these intervals is crucial to properly quantify the energy contribution of each wave mode to the power spectrum.