Astron. Astrophys. 346, 633-640 (1999)

4. Power spectra

The next step of the data analysis was the computation of power spectra, or "diagnostic diagrams", for each of the 3D-arrays. The Fourier transform of the time-dependent velocity signal is defined as

and is calculated as a discrete Fourier transform.

The power spectrum is

Since we have removed the solar rotation from the data, we can assume that the velocity signals do not have a preferred horizontal direction, and that, therefore, the power spectrum depends only on the horizontal wave number . The diagnostic diagrams are obtained by azimuthally averaging the power spectra in the -plane. The size of the investigated fields is 243 Mm 243 Mm on the Sun, about 1/3 of the solar radius, so the use of cartesian coordinates is justified.

From the total duration ( s) and the horizontal dimension ( Mm) we have a frequency resolution of mHz and a wave number resolution of Mm^-1. (Note that we use the cycle frequency instead of the angular frequency for the analysis of the observations. Consequently, the power spectra are indeed -diagrams.)

Fig. 1 shows the -diagram corresponding to disk center. About a dozen modes are visible. The lowermost ridge corresponds to global surface waves (the f mode) with the dispersion relation

with ms^-2. The dotted line in Fig. 1 marks the dispersion relation of the f mode. The horizontal wave number is related to the degree l of the spherical surface harmonic by , with the solar radius Mm. As an example of an off-center diagnostic diagram the power spectrum at is shown in Fig. 2. There is considerably less power at high frequencies than in Fig. 1, in particular for large . Moreover, the noise level is higher in the off-center diagram.

Fig. 1. The -diagram of the disk center, showing about a dozen modes. The intensity scale is logarithmic, as shown in Fig. 3. The dotted line is the dispersion relation of the f mode. The vertical bars indicate l-values of 200, 400, and 800, respectively

Fig. 2. The -diagram at . Otherwise as Fig. 1

© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
helpdesk.link@springer.de