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

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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 [FORMULA]. The diagnostic diagrams are obtained by azimuthally averaging the power spectra in the [FORMULA]-plane. The size of the investigated fields is 243 Mm [FORMULA] 243 Mm on the Sun, about 1/3 of the solar radius, so the use of cartesian coordinates is justified.

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

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

[FIGURE] Fig. 1. The [FORMULA]-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

[FIGURE] Fig. 2. The [FORMULA]-diagram at [FORMULA]. Otherwise as Fig. 1

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999