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Astron. Astrophys. 364, 799-815 (2000)

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7. Results

We can utilize the velocity scaling and fitting to examine the solar background velocity variations and the dependence of the rms velocity amplitude on orbital velocity.

7.1. Power spectra

The power spectra for the velocity and the difference velocity are shown in Fig. 14 and Fig. 15 respectively. Also shown on the velocity power spectrum plot is a fit to the pattern based on the formula given by Harvey (1985) which uses only three components. This fit is discussed further in the next paragraph. An alternate treatment of the GOLF data (Robillot et al. 1998) calculates a velocity from a ratio which is related to quantities used in the present approach. The numerator of their ratio is proportional to the velocity derived here. The denominator of their ration at time scales longer than one day is proportional to our difference velocity while at time scales shorter than one day the denominator is essentially constant. One might expect there to be a break in the derived power spectrum related to the denominator smoothing if the numerator and denominator are correlated at time scales shorter than one day. Consequently, it is of interest to examine the components separately. The power spectrum from the velocity signal shown in Fig. 14 is based on an approach which deals with the GOLF data without any frequency dependent factors for all time periods between the 80 day limit from the phase function detrending and the cutoff of 6000 µHz imposed for convenience in this display which emphasizes the low frequency range. It provides an estimate of the solar background spectrum free from the effects of the diurnal cycles and atmospheric transparency problems of ground-based observers and takes advantage of the long and nearly continuous GOLF sequence. As such it represents an important estimation of the solar background spectrum. Previous power spectra including this wide range of frequencies have been published by Jiménez et al. (1988) and Pallé et al. (1995). At the lowest frequencies, the darkening effects of sunspots certainly can introduce a variability which is not a true velocity signal. Indeed, the considerations by Harvey (1985) anticipated that the velocity-like signal at periods comparable to the rotation period would be due to a cross-talk between intensity and velocity. We recognize this and simply refer to the time series as velocity signals even though some of what they include comes from intensity variations.

[FIGURE] Fig. 14. This figure shows the log of the power spectrum versus the log of the frequency for the average of all four data channels (PM's 1 and 2 plus both states of magnetic modulation). The power spectral values have been binned into intervals of constant change in log[FORMULA].

[FIGURE] Fig. 15. This figure shows the log of the power spectrum of the difference in velocity between the plus and minus states of magnetic modulation versus the log of the frequency for the average of the data channels from PM 1 and 2. The power spectral values have been binned into intervals of constant change in log[FORMULA].

For the power spectrum derived from the velocity signal, there is a break in the slope at approximately 25 µHz. Below this frequency the slope matches that of the Harvey (1985) model for active regions with a slope of -2. The shape above the break is steeper than would be expected from the Harvey model for granulation alone although the break point is near where Pallé et al. (1995) showed the granulation and active region contributions crossing over. This change of slope suggests that the effects of active regions should become small in the frequency range of 100 to 1000 µHz where the search for low frequency solar modes is currently concentrated. Apparently the solar background in this frequency range is largely due to convective effects rather than magnetic and activity related effects. Between about 50 µHz and 250 µHz the power spectral slope is about [FORMULA] while between 600 µHz and 1000 µHz the slope is steepest at -1.15. A change in slope of this type with a decrease toward lower frequency indicates a convective phenomenon with a lifetime comparable to the inverse of the frequency where the negative curvature in the power spectrum is greatest. This suggests the lifetime of the convective elements most responsible for the background noise is due to long-lived granulation. Based on these considerations, the fit to the Harvey formula utilizes only three components: an active region (AR) contribution with [FORMULA] m s-1 and [FORMULA] s, a long-lived granulation (GL) contribution with [FORMULA] m s-1 and [FORMULA] s, and a short-lived granulation (GH) contribution with an amplitude of [FORMULA] m s-1 and [FORMULA] s. These two contributions represent parts of a distribution rather than distinct populations with well-defined properties. The amplitude and life-time parameters have the usual meaning of the Harvey power spectrum form where the power is [FORMULA]. Due to the steepness of the low frequency part of the power spectrum, there can be no contribution from supergranulation or mesogranulation unless the contribution hypothesized for the active regions is steeper than the -2 slope adopted by the Harvey model.

We caution against a physical interpretation of the power spectrum shape in terms of the processes of granulation and supergranulation. The fact that a component having a range of lifetimes near that of the granulation should be present in the power spectrum while there can be no contribution from a component having the lifetime of supergranulation is very hard to explain on the basis of a physical model. If conservation of matter can cause the contribution from supergranulation to drop out, it should apply equally well to granulation and prevent the features with that lifetime from appearing in Fig. 14. It is probable that some other process must be invoked to explain the features seen in this power spectrum. Ulrich (1999) has suggested recently that atmospheric gravity waves have a resonance at the temporal frequency shown in Fig. 14 when driven by structures having a spatial wave number appropriate for supergranulation. This resonance response can alter the balance between the intensity and velocity components of the signal. The nature of the power spectrum shown in Fig. 14 clearly deserves additional attention.

The strongest peaks at the lowest frequencies correspond to 27 and 13 days and are the well-known rotational modulation associated with the active region signal (Jiménez 1988). The power spectrum of the difference velocity has a lower slope of [FORMULA] and is dominated by shot noise for frequencies above 11 µHz. When the difference signal is used to form a ratio prior to the calculation of a velocity, a low-pass filter is used to eliminate those frequency components above 11 µHz. Due to this shot noise, it is not possible to distinguish between convective and active region effects in the 100 to 1000 µHz frequency band in the case of the power spectrum of the difference velocity.

Distinct peaks or groups of peaks are found in the power spectrum of the velocity at periods of 26.9, 18.0, 9.0, 6.8 and 3.5 days. The peaks at 26.9, 18.0 and 3.5 days are distinct while those at 9.0 and 6.8 days are clusters. For the power spectrum of the difference in velocities, peaks at 27.0 and 9.0 days are clear while bands of enhanced power are seen near 13 and 7 days.

7.2. Amplitude in the 5-minute band

The velocity scaling derived in this paper depends on the determined line profile factors and even though the orbital velocity moves the GOLF working point on the solar line by [FORMULA]nm, the scale factor included in [FORMULA] should properly account for this variation. The scale factor [FORMULA] changes by 30% during the year and differs between the two states of instrumental magnetic modulation by 4%. According to line contribution functions provided to us by Severino (1993), the height of formation using the Magain (1986) formulation varies between 242 and 310 km on the Vernazza et al. (1981) model C scale. These altitudes differ in density by 75% so that if the velocity scales as [FORMULA], the rms velocity should differ by 32% with the rms velocity being largest when the GOLF working point is closest to the line core at the most negative velocity. In addition, two states of magnetic modulation are offset from each other by an amount of 7% the orbital velocity range and should consequently show a difference in rms velocity of 2%.

In order to study the altitude dependence, the full time series was band-pass filtered to include only variations in the five-minute band having frequencies between 2000 µHz and 5000 µHz. The time sequence was divided into segments of 27.28 days, a period corresponding to one synodic Carrington rotation, and the rms variation in velocity was calculated for each. The results are shown in Fig. 16 for both states of magnetic modulation. One period including the time of the sodium cell cooldown in September of 1996 is left out because the gap during the four days of no data has been filled with a noiseless auto-recursive representation which reduced the rms artificially for that Carrington rotation. It is clear from Fig. 16 that the observed dependence on altitude is much less than the estimate given here. By extracting the lowest four harmonic components from the time series of [FORMULA] and comparing these components with the first four harmonic components from the orbital velocity, we find that variation in [FORMULA] correlated with the change in the spectral line working point is only 3.3% instead of the anticipated 32%. In a similar fashion the average value of [FORMULA] for the plus state of magnetic modulation is only [FORMULA]% smaller than that for the minus state instead of the expected 2%. Such a slight increase in the rms amplitude with altitude indicates that the modes are damped in this part of the sun's atmosphere.

[FIGURE] Fig. 16. This figure shows the rms velocity variability calculated from segments one synodic Carrington rotation in duration using a velocity time series filtered to remove components with periods longer than 4 hours. The solid and dotted lines are for the negative and positive magnetic modulation states respectively.

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Online publication: January 29, 2001