DOES HIGH-LATITUDE SOLAR ACTIVITY LEAD LOW-LATITUDE SOLAR ACTIVITY IN TIME PHASE?

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1 The Astrophysical Journal, 646: , 2006 August 1 # The American Astronomical Society. All rights reserved. Printed in U.S.A. A DOES HIGH-LATITUDE SOLAR ACTIVITY LEAD LOW-LATITUDE SOLAR ACTIVITY IN TIME PHASE? K. J. Li, 1 P. X. Gao, 1,2 and J. Qiu 3 Received 2005 March 30; accepted 2006 February 8 ABSTRACT Using data from the Carte Synoptique solar filaments archive, we investigate whether there is a time lag between high-latitude solar activity and low-latitude solar activity. The cross-correlation analysis of the number of high-latitude filaments per Carrington rotation ( NHF) and that of low-latitude filaments per Carrington rotation ( NLF) shows, although inconclusively, that NLF possibly lags behind NHF. The periodic characteristics of both NHF and NLF clearly indicate that the activity of high-latitude filaments is evidently leading the activity of low-latitude filaments. Thus, the present study suggests that high-latitude solar activity leads low-latitude solar activity in time phase. Subject headinggs: Sun: activity Sun: filaments Sun: general Online material: color figure 1 National Astronomical Observatories, Yunnan Observatory, Chinese Academy of Sciences, Kunming , China; lkj@ynao.ac.cn. 2 Graduate School of the Chinese Academy of Sciences, Beijing, China. 3 Big Bear Solar Observatory, New Jersey Institute of Technology, Big Bear City, CA INTRODUCTION It is an interesting issue whether a certain solar activity indicator respectively at high (over 50 ) and low (below 50 ) latitudes is in phase or not (Tanaka 1964; Makarov & Makarova 1987; Makarov et al. 1989, 2001a, 2001b; Sakurai 1998; Riehokainen et al. 2001; Li et al. 2002a, 2002b). Polar faculae, which were discovered by the German amateur astronomer Weber in the 19th century (Weber 1865), have been observed for more than 100 years. They are easily observed and demonstrated to be a good indicator of solar activity at high latitudes. The phase relation between the activity of polar faculae (solar activity at high latitudes) and the sunspot cycle (solar activity at low latitudes) has been extensively studied and was once a controversial problem in history. Weber was the first to notice that the activity maximum of polar faculae occurred in the year 1867, when the sunspot cycle reached the minimum (Makarov & Makarova 1996). According to Kiepenheuer (1953), observations of polar faculae by Greenwich Observatory show no pronounced relationship with the solar cycle :::andtheydonotappeartobeinanyway associated with the polar prominences. Using the long-term record of polar faculae, Waldmeier (1955) found that the polar facula cycle is closely connected with global magnetic field variations. Based on the long-term record of polar faculae, Saito & Tanaka (1960) showed an evident anticorrelation between the abundance of polar faculae and solar activity in that the maximum of the former seems to take place 1 yr ahead of the minimum of the latter. Sheeley (1964, 1991) found a 90 phase shift between the cycles of the sunspot number and the polar facula number, with the maximum of the sunspot number occurring earlier. However, Makarov & Makarova (1996) obtained the opposite result that the monthly number of polar faculae correlates with the monthly sunspot areas with a time lag of about 6 yr applied to the latter. Recently, using polar facula observations by the National Astronomical Observatory of Japan from 1951 to 1991 and the Mount Wilson Observatory from 1906 to 1990, Li et al. (2002a) found that polar faculae have their own activity cycle, with the maximum occurring during the years of solar minimum. Through studying the phase relation between activities of solar active prominences at low and high latitudes in the period , Li et al. (2002b) found that from the solar equator to the solar poles, the activity of solar active prominences peaks earlier at higher latitudes, and that the cycle of solar active prominences at high latitudes (over 50 ) leads by 4 yr both the sunspot cycle and the corresponding cycle of solar active prominences at low latitudes (below 40 ). In summary, it is basically confirmed that solar activities at high and low latitudes are not in phase. However, merely from correlation analysis, we cannot determine whether high-latitude or low-latitude activity is in the lead, although the dynamo theory infers that solar activity at high latitudes should lead solar activity at low latitudes. In the present study, we attempt to answer this question by comparing the periodicity of the filament numbers at high and low latitudes using data from the Carte Synoptique solar filaments archive ( Mouradian 1998; Coffey & Hanchett 1998a, 1998b). 2. PHASE RELATION OF SOLAR FILAMENTS AT HIGH AND LOW LATITUDES Lucien d Azambuja investigated the behavior of solar filaments and prominences over many years and maintained a synoptic program similar to the Zurich sunspot program (d Azambuja 1923; Coffey & Hanchett 1998b). He published the first Cartes Synoptiques de la Chromosphere Solaire et Catalogue des Filaments de la Couche Superieure, a compendium of reduced solar observations beginning with Carrington rotation 876 (d Azambuja 1928). Since then, data through 1989 have been published in succeeding Carte Synoptiques issues (Coffey & Hanchett 1998b). The World Data Center A for Solar-Terrestrial Physics has digitized the Carte Synoptiques (Coffey & Hanchett 1998b). 4 Here we use the Carte Synoptique solar filaments archive ( Mouradian 1998), namely, the catalog of solar filaments from 1919 March to 1989 December, corresponding to Carrington solar rotations from number 876 to The catalog includes 41,044 filament regions in total (Coffey & Hanchett 1998b). We count the number of low-latitude solar filaments ( NLF) whose latitudes are 50 in each of the Carrington solar rotations from number 876 to The number of high-latitude solar filaments (NHF) whose latitudes are >50 is also counted 4 The data can be accessed via the World Wide Web at ftp://ftp.ngdc.noaa.gov/stp/solar_data/solar_filaments.

2 HIGH- AND LOW-LATITUDE SOLAR ACTIVITY 1393 Fig. 1. Numbers of solar filaments per CR at high latitude (top) and low latitude (bottom). in each Carrington solar rotation. Figure 1 shows that both NHF and NLF wax and wane with an approximately 11 yr Schwabe cycle, as the sunspot number does. Such very nice periodic behavior is called here the activity cycle of solar filaments at high and low latitudes. However, the figure clearly shows that the activities of high- and low-latitude filaments are not in phase, and the former is seemingly ahead of the latter. Figure 2 shows the latitudinal distribution of filaments varying with time, the so-called butterfly diagrams. As shown in the figure, within a cycle, the distribution of filaments drifts from middle latitudes toward both low and high latitudes. From the figure, it cannot yet be inferred whether high-latitude or low-latitude activity is in the lead. Rudiger & Brandenburg (1995) proposed a unique dynamo model to produce butterfly diagrams with two branches: a lowlatitude branch, which propagates toward the equator, and a high- latitude branch, which propagates toward the pole. This kind of branching in the butterfly diagrams, however, had not been observed before the year 2003 ( Hathaway et al. 2003). Here, the solar full-disk butterfly diagrams of filaments give evidence to support the model. To further study the lead issue, we have performed a crosscorrelation analysis of NHF and NLF. Figure 3 shows the result of the cross-correlation analysis, in which the abscissa indicates the shift of NHF with respect to NLF, with negative values representing backward shifts. The figure suggests that the best (positive) correlation, with a correlation coefficient of 0.50, occurs Fig. 2. Butterfly diagram of the solar filaments from 1919 March to 1989 December. Fig. 3. Cross-correlation coefficient between the numbers of the solar filaments respectively at high and low latitudes.

3 1394 LI, GAO, & QIU Vol. 646 Fig. 4. Local wavelet power spectral maps of the NHF (left)and NLF(right). The region below the dashed line indicates the COI. [See the electronic edition of the Journal for a color version of this figure.] when NHF is shifted forward by 19 Carrington solar rotations (CRs). From random data sequences of the same size, one may obtain a correlation coefficient of about 0.2 with a probability less than (Liu 1996). Therefore, the obtained correlation is significant. When NHF is shifted backward by 120 CRs, the second-largest positive coefficient of 0.46 is obtained. The correlation of the activity cycle of solar filaments at high latitudes is slightly larger with the subsequent activity cycle of solar filaments at low latitudes than with the preceding cycle. It seems that NHF should be ahead of NLF in phase. When NHF is shifted forward by 90 CRs, the best (negative) correlation occurs with a correlation coefficient of When NHF is shifted backward by 46 CRs, the second-largest negative coefficient of 0.56 is obtained. Thus, the correlation of the activity cycle of solar filaments at high latitudes is slightly larger with the subsequent activity cycle of solar filaments at low latitudes than with the preceding cycle. It also seems that NHF should be ahead of NLF in phase. In order to determine the significance of the difference between the above two negative coefficient values, a statistical test is made as follows. Let us suppose that two Gaussian-distributed data sets X and Y have a correlation coefficient R.Ifasampleof size N is taken from them, then we obtain the sample correlation coefficient r. Ifwedefinez ¼ 0:5ln½(1 þ r)/(1 r)š, the distribution of z is approximately a Gaussian with a mean of 0:5ln½(1 þ R)/(1 R)Š and a dispersion of 2 ¼ 1/(N 3). The data used here cover 948 CRs, and N ¼ ¼ 858 when NHF is shifted forward by 90 CRs, or N ¼ ¼ 902 when NHF is shifted forward by 46 CRs. Thus, ¼ 0:0342 or 0:0334, correspondingly. We also find z ¼ 0:6338 for r ¼ 0:56 (backward shift of 46 CRs) and z ¼ 0:7028 for r ¼ 0:61 (forward shift of 90 CRs). Now let us try to reject the hypothesis that the difference between the two correlation coefficients is not significant and that they can result from the same value of, say, z ¼ 0:6683 (which is the mean of and ). We see that z ¼ 0:6338 and 0:7028 are at about 1.02 from the center of the Gaussian distribution. Therefore, the difference between the two negative correlation values is significant with a probability of only about 70% (similarly, the difference between the two positive correlation values is found to be significant with a probability of only about 58%). That is to say, NHF is likely to be ahead of NLF in phase. Next, we attempt to address this topic by comparing the periodic characteristics of both NHF and NLF using the wavelet transform. The wavelet transform is a very powerful tool to analyze nonstationary signals. It permits the identification of the main periodicities in a time series and the evolution in time of each frequency, which is then used in the present study, and the complex wavelet transform can be used to do the phase coherence analysis and to show the phase difference of two time series (Bloomfield et al. 2004; Grinsted et al. 2004). In the present study, the complex Morlet wavelet transform with the nondimensional frequency taken to be 6 (Torrence & Compo 1998) is used to show the periodicity in both NHF and NLF, and their local wavelet power spectra are given in Figure 4. In the figure the y-axis ordinate is on a scale of the exponent of 2. The Fourier transform is usually used to speed up the computation in a wavelet analysis program ( Torrence & Compo 1998). However, as the Fourier transform assumes that the data are periodic, and most time series are of finite length, it introduces errors at the edges of the transform (De Moortel et al. 2004). The region in which the transform suffers from these edge effects is known as the cone of influence (COI). As in Torrence & Compo (1998), the COI is defined such that the wavelet power for a discontinuity at the edge decreases by a factor of e 2. Portions of the transform that are outside the area encompassed by the time axis and the COI are subject to these edge effects and are therefore unreliable (De Moortel et al. 2004; Torrence & Compo 1998). In Figure 4, such an area is marked as the area above the dashed line. From the figure, it can be found that for both NHF and NLF, the period belt of the highest power, or the brightest island in the blue sea, is located around the 11 yr Schwabe period. This belt is away from the area above the dashed line, indicating that the belt is hardly affected by the COI and thus reliable. The belt of NHF appears to resemble that of NLF, but the two belts are apparently not in phase. We perform a cross-correlation analysis of the two belts, and the method is the same as that used in the cross-correlation of NHF and NLF. The period widths of the two belts are both taken to range from 9 to 12 yr, including 107 scale (period) values. In

4 No. 2, 2006 HIGH- AND LOW-LATITUDE SOLAR ACTIVITY 1395 Fig. 5. Cross-correlation coefficient between the two brightest belts of Fig. 3. The two belts are both taken to range from the period of 9 yr to that of 12 yr. order to reduce the edge effects on our analyses, the time span of the two belts is taken from CR 1126 to CR 1573, including 447 CR values, and in the following correlation analyses, the time span is the same as that used here. Thus, there are 47,829 values (447 ; 107) in each of the two belts, which are extracted from the wavelet power spectra shown in Figure 4 and used for crosscorrelation. Figure 5 shows the result of the cross-correlation analysis, in which the abscissa indicates the shift of the belt of NHF with respect to that of NLF, with negative values representing backward shifts. The figure indicates that when the belt of NHF is shifted forward by 25 CRs, the best (positive) correlation occurs with a correlation coefficient of 0.90, which is significant. Most importantly, it implies that NHF should be ahead of NLF in phase. Given in Figure 6 is an average over the power at all periods of each of the two belts ranging from 9 to 12 yr at each CR. Similarly, we perform a cross-correlation analysis of the two power-averaged wavelet spectra. Figure 7 gives the result of the cross-correlation analysis, in which the abscissa shows the shift of the time-averaged wavelet spectrum of NHF with respect to that of NLF, with negative values indicating backward shifts. The figure shows that when the power-averaged spectrum of NHF is shifted forward by 45 CRs, the best (positive) correlation occurs with a correlation coefficient of 0.83, which is significant. It implies that NHF should be ahead of NLF in phase. The Schwabe period of NLF, the period whose global power is the highest, is 10.5 yr, and its local power spectrum is shown in Fig. 7. Cross-correlation coefficient between the two power-averaged wavelet spectra shown in Fig. 6. Figure 8. The Schwabe period of NHF is 10.7 yr, which is also the period whose global power is the highest in the global power spectrum of NHF, and its local power spectrum is also shown in Figure 8. Similarly, we also perform a cross-correlation analysis of the two local power spectra. Figure 9 shows the result of the cross-correlation analysis, in which the abscissa is the shift of the local power spectrum of NHF with respect to that of NLF, and negative values represent backward shifts. The figure indicates that when the local power spectrum of the Schwabe period in NHF is shifted forward by 16 CRs, the best (positive) correlation occurs with a correlation coefficient of 0.97, which is very significant. It clearly implies that NHF should be leading NLF in phase. In summary, the periodic characteristics of the filament numbers at high and low latitudes clearly indicate that solar activity of filaments at high latitudes leads solar activity of filaments at low latitudes in time phase. A larger nondimensional frequency of the Morlet wavelet gives better frequency resolution, and a smaller one, better time resolution. In order to check how the chosen value for the nondimensional frequency (! ¼ 6) in the complex Morlet wavelet transform affects the cross-correlation results presented in this study, we repeated the analysis for! ¼ 3 and 12. However, for the! ¼ 3 case we find that the periodicities ( period value and local wavelet power spectrum) of the Schwabe period (the most eminent period) in NHF and NLF are obviously different from Fig. 6. Power-averaged wavelet spectra of both the NHF (solid line) and NLF (dotted line) over the periods from 9.0 to 12.0 yr. Fig. 8. Local power spectra of the Schwabe period in both the NHF (solid line) and NLF(dotted line), whose global power is the highest.

5 1396 LI, GAO, & QIU Vol. 646 Fig. 9. Cross-correlation coefficient between the local power spectra of the Schwabe period in both the NHF and NLF. each other, and the correlation between both is very low (the correlation coefficient is 0.11); the method proposed here does not work. For the! ¼ 12 case the Schwabe period is severely affected by the COI, and the method proposed here still does not work. It seems that the frequency should not be too large or too small. 3. CONCLUSION AND DISCUSSIONS In this paper, we investigate whether high-latitude solar activity is leading low-latitude solar activity in time phase, using data from the Carte Synoptique solar filament archive. First, we count the number [ f 1(t)] of low-latitude solar filaments (NLF) whose latitudes are 50 and the number [ f 2(t)] of high-latitude solar filaments (NHF) whose latitudes are >50 in each Carrington solar rotation (t). In order to know whether the time series f 2(t) leads the time series f 1(t), we do a correlation analysis of both to find the maximum correlation coefficient when shifting one versus the other. The cross-correlation analysis of NHF and NLF shows that NHF is possibly ahead of NLF in phase, the probability being about 70%. Then, the complex Morlet wavelet transform is used to show the periodicity of both NHF and NLF. It is found that for both NHF and NLF, the period belt of the highest power is located around the 11 yr Schwabe period. The belt of NHF appears to correlate with that of NLF if the former is shifted forward by a certain Carrington rotation number. We perform a cross-correlation analysis of the two belts, and the periods of the two belts are both taken to range from 9 to 12 yr. It is found that the best (positive) correlation occurs when the belt of NHF is shifted forward by 25 CRs. We also perform a cross-correlation analysis of the two power-averaged wavelet spectra of the two belts, which are the averages of the local power of all period scales from 9 to 12 yr, and it is found that the best (positive) correlation occurs when the power-averaged spectrum of NHF is shifted forward by 45 CRs. Finally, we conduct a cross-correlation analysis of the local power spectra of the Schwabe period in both NHF and NLF, and it is found that the best (positive) correlation occurs when the local power spectrum of the Schwabe period in NHF is shifted forward by 16 CRs. Results from all the above analyses imply that NHF should be leading NLF in phase. Thus, the periodic characteristics of the filament numbers at high and low latitudes clearly indicate that solar activity of filaments at high latitudes is evidently leading solar activity of filaments at low latitudes in time phase. Of course, the method proposed here only works on the condition that f 1(t) and f 2(t) have similar periodicities, and indeed, the high-latitude activity has a similar periodicity to that of the the low-latitude activity when the nondimensional frequency of the Morlet wavelet is taken to be 6 (the correlation coefficients in the above three cases are statistically significant). However, when the nondimensional frequency of the Morlet wavelet is taken to be 3, a smaller value, the periodicities (period value and local wavelet power spectrum) of the Schwabe period (the most eminent period) in NHF and NLF are obviously different from each other, and the correlation between both is very low. Thus, the method proposed here does not work. Although the cross-correlation analyses of the three kinds of local power spectra of both NHF and NLF give the same result that solar activity of filaments at high latitudes is evidently leading solar activity of filaments at low latitudes in time phase, the shift values are somewhat different from one case to another, and they are also different from the shift value obtained when NHF is shifted forward with respect to NLF. We do not know the reason for such difference but guess that it is related to the following two points: (1) The characteristics of the local power spectrum of NHF are similar to but not exactly the same as those of NLF. (2) When we conduct a cross-correlation analysis of two kinds of data, the shift of one data set forward or backward with respect to the other will lead to a reduction of the number of used data. Therefore, we infer that comparison of the periodic characteristics of two sets of data can only yield some qualitative results, such as which time series leads the other in phase, but cannot give exact quantitative results, such as the amount of the shift in phase. According to the Babcock-Leighton model of the solar cycle (Babcock 1961; Leighton 1964), all magnetic flux emergence from the solar interior takes place at low latitudes in the form of active regions. Evolution of large-scale weak magnetic fields is considered to be exclusively caused by dispersal and surface transport of active region flux (Makarov & Tlatov 1999). The large-scale field patterns are formed by redistribution of old magnetic flux, and consequently the cycle of weak magnetic fields arises from a deep-seated toroidal field that encircles the Sun below the activity belts. Using Mount Wilson magnetograph data, Howard & Labonte (1981) showed that polar magnetic fields are formed entirely by movement of magnetic fields from the sunspot latitudes to the poles. The observed weak polar magnetic fields are a direct manifestation of the poloidal field of the solar dynamo, and the poleward transport of the magnetic flux from solar active regions plays a crucial role in reversing this poloidal field (Leighton 1964; Makarov & Tlatov 1999). Several studies on the evolution of large-scale magnetic fields in solar cycles ( Makarov & Makarova 1996) suggested that the polar activity of the Sun is the beginning of a new global sunspot cycle. Thus, the dynamo theory combined with observations of large-scale magnetic fields suggested that high-latitude solar activity should lead low-latitude solar activity in time phase, which is confirmed by our correlation analysis. On the other hand, it is inferred that highlatitude solar activity should have the same period as low-latitude solar activity. The authors would like to thank the referee for useful and helpful comments. The wavelet transform software was provided by C. Torrence and G. Compo and is available at This work is supported by the Natural Science Fund of China ( ), the 973 projects, and the Chinese Academy of Sciences.

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