Nonlinear Effects in Three-minute Oscillations of the Solar Chromosphere. II. Measurement of Nonlinearity Parameters at Different Atmospheric Levels

Recent theoretical studies suggest that the nonlinearity of three-minute velocity oscillations at each atmospheric level can be quantified by the two independent parameters-the steepening parameter and the velocity amplitude parameter. For the first time, we measured these two parameters at different atmospheric levels by analyzing a set of spectral lines formed at different heights of sunspots ranging from the temperature minimum to the transition region. The spectral data were taken by the Fast Imaging Solar Spectrograph of the Goode Solar Telescope, and by the Interface Region Imaging Spectrograph. As a result, from the wavelet power spectra of the velocity oscillations at different heights, we clearly identified the growth of the second harmonic oscillations associated with the steepening of the velocity oscillation, indicating that higher-frequency oscillations of periods of 1.2 to 1.5 minutes originate from the nonlinearity of the three-minute oscillations in the upper chromosphere. We also found that the variation of the measured nonlinearity parameters is consistent with the theoretical expectation that the nonlinearity of the three-minute oscillations increases with height, and shock waves form in the upper chromosphere. There are, however, discrepancies as well between theory and observations, suggesting the need to improve both theory and the measurement technique.