4 Total velocity power variations

The first observations of the variation of the TVP for all measured p-modes was reported by Pallé et al. (1990a,b) who found an increase of 30 to $40\%$, anti-correlated with the solar activity cycle. Afterwards, Anguera Gubau et al. (1992) found similar changes using different analysis techniques and more data of the same type; they interpreted these results as a decrease in the efficiency of the excitation of such modes at solar activity maximum, since absorption of mode power by local magnetic structures (see e.g. Bogdan et al. 1993) is a small influence and cannot explain such a high ratio.

We have also calculated the TVP in the spectrum which is proportional to the area under the main peak of the cross-correlation function. Once the main peak is fitted to a Lorentzian profile, the TVP is calculated as the amplitude times the width (i.e. ${\it TVP}^i=A^i\cdot \Gamma^i$). We show in Fig. 3, the percentile changes of the TVP compared to its minimum value. Again, the radio flux is shown here as an index of the solar activity cycle. From the figure, we are able to see that the variation of the TVP between minimum and maximum of the solar cycle is around 20$\%$. Moreover, the lower figure shows clearly the anti-correlation with the solar cycle, the TVP decreases when activity increases. The separated contributions of the odd and even pairs of modes were also studied (see the inset box in Fig. 3) as we did in the analysis of the frequency shifts. The results are shown in the sub panel in Fig. 3. Notice that the variation of the TVP between extreme phases of the cycle for even and odd degrees is approximately 20$\%$, in good agreement with the amplitude of the integrated measurement. Moreover, both contributions are in phase, which contrasts with the results found for the frequency shifts, where even modes seems to respond later than the odd ones. Odd and even TVP appear also better correlated around the maximum than during the low activity phases: both present exactly the same bump in the middle of the maximum of activity close to 1990, anti-correlated with decreasing activity during the same period, but their short term variations are different around 1986 and 1996 during the activity minima

Figure 4: TVP increase in percent against the frequency shift. The solid line goes from 1984 to 1989 and the dashed line goes from 1990 to 1999. The corresponding correlation coefficients are shown in Table 3.

 

 

Table 3: Correlation coefficients between the TVP and the solar indices where the TVP is defined as the product between the width and amplitude of the cross-correlation function between a spectrum and the 1986 reference spectrum. TVP refers to the average over the whole observed $\ell$ range ($\ell = 0$, 1, 2, 3) whereas TVP_0,2 and TVP_1,3 refer to the average for even and odd degrees respectively, all quantities being frequency integrated values between 2.5 and 3.7 mHz. The last line gives the correlation coefficients between the TVP and frequency shift. $r_{\rm P}$ is the Pearson linear correlation coefficient, $r_{\rm S}$ the Spearman rank correlation coefficient and $P_{\rm s}$ is the probability of having no correlation.

Index		TVP				TVP0,2				TVP1,3
	$r_{\rm P}$	$r_{\rm S}$	$P_{\rm s}$		$r_{\rm P}$	$r_{\rm S}$	$P_{\rm s}$		$r_{\rm P}$	$r_{\rm S}$	$P_{\rm s}$
$R_{\rm I}$	-0.84	-0.82	4 $\times$ 10-8		-0.87	-0.88	1 $\times$ 10-10		-0.79	-0.71	1 $\times$ 10-5
F10	-0.83	-0.79	2 $\times$ 10-8		-0.86	-0.87	5 $\times$ 10-10		-0.78	-0.69	3 $\times$ 10-5
KPMI	-0.78	-0.75	2 $\times$ 10-6		-0.81	-0.81	7 $\times$ 10-8		-0.74	-0.67	6 $\times$ 10-5
MPSI	-0.82	-0.80	3 $\times$ 10-7		-0.83	-0.85	9 $\times$ 10-9		-0.80	-0.69	5 $\times$ 10-5
TSI	-0.76	-0.67	1 $\times$ 10-4		-0.80	-0.76	2 $\times$ 10-6		-0.70	-0.58	1 $\times$ 10-3
He	-0.84	-0.82	2 $\times$ 10-8		-0.87	-0.87	4 $\times$ 10-10		-0.77	-0.72	8 $\times$ 10-6
$\Delta\nu$	-0.75	-0.73	4 $\times$ 10-6

Linear and rank correlation coefficients were calculated with the same activity indicators than before and they are shown in Table 3. The values found are large, showing a good correlation, but they are systematically lower than those for the frequency shift (range [0.67-0.84] against [0.86-0.94]). Moreover, the correlations are bigger for $\ell = 0$, 2 than for $\ell = 1$, 3 contrary to the results found for the frequency shifts. The linear correlation coefficient between frequency shift and TVP change ( $r_{\rm P} = -0.75$, see also Table 3) is well below the 0.9 reached in average between frequency shift and the solar activity indices suggesting that they are not linearly correlated and Fig. 4 shows indeed that they tend to follow an hysteresis shape, rather than a strict line, when plotted against each other.

This is important because if the decrease in TVP is due to the presence of local or surface activity, as we believe is the cause for the frequency shift, then they should be well correlated with almost no hysteresis. Therefore, the fact that the TVP is less well correlated with the surface activity indices and shows an hysteresis behavior when plotted against the frequency shift, may indicate that indeed its variation is due to a decrease in the excitation efficiency or an increase of the damping rate at maximum which reflects a change in the convection zone structure that does not have to be correlated or strictly in phase with the surface magnetic features. The process of absorption and damping of p-modes by an increasing number of rising flux tubes during the period of high activity explored by Bogdan et al. (1996) is qualitatively compatible with our results. On the other hand, if the TVP variations are only due to geometrical effects, this parameter would probably show a better correlation with the total area covered by the magnetic structures than with just the number of them. Although it is beyond the scope of this work, this should probably be further investigated to get a better picture of the possible sources of this phenomena. A more quantitative work including separate observations of the damping rate and amplitude variations of individual modes (Chaplin et al. 2000) is certainly needed to be more conclusive. These inferences from individual mode fits are potentially more informative but remains however less robust than those obtained from the fit of the cross-correlation of the power spectra.