Temporal Variation of the Rotation of the Solar Mean Magnetic Field

The solar mean magnetic field (SMMF) is the value of the magnetic field strength averaged over the entire solar disk (Scherrer et al. 1977a; Haneychuk 1998). That is, the SMMF is the magnetic field of the Sun as a star (Kotov 2008; Xiang & Qu 2016). The majority of the SMMF signal seems to be caused by the large-scale background fields of the quiet photosphere, while the contribution of the fields of sunspots and active regions is less significant (Demidov et al. 2002; Kotov et al. 2002; Kotov 2003, 2008, 2009). Thus, the SMMF seems to be an important characteristic of the evolution of the solar large-scale magnetic field (Kotov 2008).

As the SMMF is an integrated characteristic of the Sun, it should be related to other general parameters, such as solar rotation (Grigor'ev & Demidov 1987). Many authors have detected the periodicity of the SMMF and found the rotation period from the temporal variation of the SMMF. Kotov et al. (1977) and Kotov & Demidov (1980) showed the first evidence for the SMMF varying with solar rotation cycle (Haneychuk 1998). The results of Kotov & Levitskii (1983, p. 23) showed that the power spectrum of the SMMF data exhibits a set of discrete lines within the period of 27–29 days. Grigor'ev & Demidov (1987) used the correlation periodgram analysis to investigate the short-time periodicity of the SMMF data at the Sanyan observatory and found a main period of 26.81 days as well as a period of 13.48 days. Rivin & Obridko (1992) claimed that the SMMF during the years 1975–1990 has periods of 27, 13.5, and 9 days. Haneychuk (1998) determined the rotation period by analyzing the SMMF data from 1968–1996 and found that the main rotation period is about 26.92 ± 0.01 days. He also pointed out that the rotation of magnetic patterns exhibits different behaviors before and after the magnetic polarity reversal. The fourier power spectrum analysis of the SMMF for the epochs of solar maxima (the years of 1980 and 1991) and solar minima (the years of 1986 and 1996) made by Das & Nag (1999) indicated that an average periodicity of 26.7 days (with an uncertainty below 0.35 days) can be seen. The results of Ye et al. (2012) indicate that the main periods around the rotation period can be seen both in the solar activity maximum and minimum times, but they differ in that the periods for the maximum times are generally shorter than those for the minimum times. Mordvinov & Plyusnina (2000) found that the modes with periods 27.8–28.0 days dominate the rising phase of solar activity cycles, whereas the 27-day rotational mode dominates the declining phase. Haneychuk et al. (2003) found that the synodic period of the rotation of the solar magnetic field is about 26.929 ± 0.015 days, but concluded that it does not vary with time across the years 1968–2001. Because authors have adopted different periodicity analysis methods, the frequency resolutions and uncertainties obtained are, accordingly, found to be slightly different. Data sets from different observatories may be obtained by different data processes and will also be affected by the instrumental weight functions. In our opinion, different methods applied to different data sets may cause small differences in results. It is generally accepted that the rotation of the SMMF varies with time. Previous studies have found the rotation period by performing a periodicity analysis of the SMMF during a particular time interval. Thus, the results of these studies will have suffered the influence of differing periods, which we believe may be a primary reason why the results vary.

Heristchi & Mouradian (2009) suggested a method called global rotation, which was applied to structures of solar activity. This method is a global measure of rotation rate and gives an average value for the structures of the same nature (Mouradian & Heristchi 2005). This method does not take into account the latitudinal variation of structures, and thus, it can not directly reflect the differential rotation (Mouradian et al. 2002). Li et al. (2011a, 2011b) used a continuous complex Morlet wavelet transformation to investigate the temporal variation of the rotational cycle lengths of daily sunspot areas and daily sunspot numbers from a global point of view. Xie et al. (2012) studied general hemispheric rotation by analyzing the hemispheric sunspot numbers through the wavelet transformation. The rotation of the solar magnetic field has been studied by using magnetic structures (Brajša et al. 2002; Javaraiah et al. 2005; Gigolashvili et al. 2013; Xiang et al. 2014) as well as the pattern of the surface magnetic field (Snodgrass 1983; Stenflo 1989; Chu et al. 2010; Shi & Xie 2013, 2014). The SMMF is one of the magnetic patterns that can reflect the global characteristics of the magnetic field of the Sun. In this work, the rotation of the SMMF will be investigated from a global view through the application of the continuous wavelet transformation analysis to the daily SMMF. Previous investigations have mainly focused on searching the periodicity of the SMMF during a certain time interval and usually obtained a mean result over a specific time or solar-cycle phase. Our work will focus on the temporal variation of the rotation of the SMMF, which will show a detail solar-cycle-related variation and enable us to better study the relationship between the rotation of the SMMF and solar activity. Such study can help achieve a better understanding of the cyclic behavior of the rotation of the Sun as a star and provide useful information for the study of the solar activity cycle.

2.1. Data

The data used in this study are the daily SMMF from 1975 May 16 to 2014 July 31,⁶ which are measured by the Wilcox Solar Observatory (WSO). The daily SMMF observations at WSO are shown in Figure 1, covering more than three solar cycles. The data gaps occurring in the considered time interval have been filled in using the linear interpolation method. From the figure, we see that the absolute values of the SMMF are relatively smaller around the minimum times of sunspot activity. This shows an obvious 11-year solar-cycle variation and indicates that the SMMF is an important characteristic of solar magnetic variability and cyclic activity.

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2.2. Rotational Modulation in the SMMF

A simple way to detect 27-day recurrences is to utilize the 27-day recurrent plot introduced by Bartels (Svalgaard & Wilcox 1975; Wagner 1976; Scherrer et al. 1977a, 1977b; Kane 2011). Figure 2 depicts the 27-day recurrent plot of all SMMF observations throughout the entirety of our study. The plot is organized by individual 27-day Bartels rotations. The vertical axis indicates calendar year, and the colors represent the range of values of the SMMF. The maximum and minimum times of solar cycles 21–24 are also marked in the figure. It is evident that similar colors usually concentrate in the same areas. This indicates that the quasi-27 days should exist in some intervals. Some structures with obvious recurrent periods longer or shorter than 27 days are marked with bold lines in the figure. It can be observed that from 1979.0–1979.5 there are structures that slant to the right, which corresponds to a recurrence period of ~28 days. From 1982.5–1984.6, the large features are moving to the left, corresponding to a recurrence period close to 26.75 days. In the years 2002.6–2003.8, the large features are moving to the left, corresponding to a recurrence period of about 26.5 days. It seems that the structure with recurrent period longer than 27 days appears in the rising phase, while the structures with recurrent periods shorter than 27 days appear in the declining phase. The Bartels-like plot of the SMMF during the year 1984 is shown in Figure 3. In the figure, each row depicts 27 daily values in succession. It can be seen that some maxima and minima of the SMMF appear in the same vertical position in each row (the day number of the corresponding Bartels rotation). This also suggests recurrences of 27 days. Both figures indicate that the SMMF shows a 27-day recurrence tendency. This suggests that rotational modulation exists in the SMMF.

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The continuous wavelet transformation is also employed to study the short-term periodicity of the change in the daily SMMF. As we known, a one-dimensional time series can be decomposed into a two-dimensional time–frequency space through the wavelet analysis (Torrence & Compo 1998; Li 2008; Xie et al. 2012; Deng et al. 2016). Thus, this method not only determines the periodicity of the dominant modes of variability, but also displays how the modes vary in time (Torrence & Compo 1998; Chowdhury & Dwivedi 2011). The details about the wavelet analysis can be found in Boberg et al. (2002) and Li et al. (2005). Because the Morlet wavelet can provide a good balance between time and frequency localization, it is a wise choice to use it for feature-extraction (Torrence & Compo 1998; Grinsted et al. 2004; Li et al. 2011a; Kong et al. 2014). Hence, we choose the Morlet wavelet in this study as in Boberg et al. (2002). In the wavelet transformation analysis, the statistical significance test is carried out by assuming that the noise has a red spectrum, as is described by Torrence & Compo (1998).

Before the wavelet transformation is performed, the original data are normalized—a procedure that subtracts the mean value and then divides by the variance in a set of data. The resulting local wavelet power spectrum and global power spectrum of the daily SMMF are presented in Figure 4. The local wavelet power spectrum clearly shows a time–frequency behavior of the SMMF over a wide range of timescales. From the local wavelet power spectrum, it can be seen that the relative higher power belt appears around the rotational cycle of the Sun as well as its half, and this can be seen more clearly around the maximum times of solar cycles. Moreover, from the global power spectrum, which shows the time-averaged wavelet spectrum over all of the local wavelet spectrum, it is found that the global wavelet spectrum has one pronounced peak that clearly shows the dominant period. This indicates the value of the rotation period is 27.5 ± 0.1 days and that the wavelet analysis can detect the rotational signal from the change in the daily SMMF. It can also be seen that the rotation period is not the only period in the timescale shorter than 64 days (at the 95% confidence level); the half rotation period can also be seen, whose value is about 13.7 ± 0.1 days. This period, appearing as a harmonic of rotational modulation, may be related to 180° oppositely directed active longitudes (Donnelly & Puga 1990; Das & Nag 1999; Chowdhury et al. 2013) or caused by the four-sector structure of the large-scale solar magnetic field (Mordvinov & Plyusnina 2000).

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2.3. Temporal Variation of the Rotation of the SMMF

Mordvinov & Plyusnina (2000) have pointed out that the rotational modulation of the SMMF dominates the wavelet spectrum at 27–30-day and 13.5-day timescales. From Figure 4, we can also see that, at a specific time, the rotation period has the highest spectral power among the period scales of 26–30 days in the local wavelet power spectrum. Using that observation, the rotation period at each time can be determined. The uncertainty of the rotational cycle length at each time point is between 0.09 and 0.11 days. Next, a one-year smoothing is performed on the obtained temporal variation of the rotational cycle lengths (the rotational cycle length is the length of the rotation period). The new time series is given in Figure 5, which displays how the lengths of the rotation period change with time. From Figure 5, we can find that the rotational cycle lengths vary in a solar cycle. The longest rotational cycle length of each solar cycle seems to always appear in the rising phase, while the shortest one seems to appear in the declining phase. The mean rotational cycle lengths for the rising phase of solar cycles 21–24 are 28.13 ± 0.66, 28.23 ± 0.37, 28.39 ± 0.81, and 28.33 ± 0.69 days, respectively. For the declining phase of solar cycles 21 to 23, they are 27.08 ± 0.41, 27.75 ± 0.67, and 27.13 ± 0.57 days, respectively. The mean rotational cycle length for the rising phase of all the solar cycles in the considered time is 28.28 ± 0.67 days, while for the declining phase of all the solar cycles, it is 27.32 ± 0.64 days. The difference of the mean rotational cycle length between the rising phase and the declining phase is 0.96 days.

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For further study, the auto-correlation analysis is used to investigate the periodicity in the temporal variation of the rotational cycle lengths of the SMMF, as  was done in Chandra & Vats (2011), and the result is shown in Figure 6. As the figure shows, the auto-correlation coefficients peak when the phase shift is about 10.1 years (the uncertainty is less than 0.1 year), which is statistically significant at the 99% confidential level. This indicates that the variation of the rotational cycle lengths may have some relation with the 11-year solar activity cycle. The periods about 5.1 and 7.5 years can also be seen in the figure but are not statistically significant.

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To determine the relationship between the rotational cycle lengths and the phase of solar cycle, we further investigate the solar-cycle-related variation of the rotational cycle lengths. First, the rotational cycle lengths of the SMMF are averaged for each year during the entire time considered. To further clarify the variation, the mean rotational cycle length of the considered time is subtracted from the obtained yearly mean rotational cycle lengths. Then, the variation of the residual yearly mean rotational cycle lengths with solar-cycle phase is plotted relative to the nearest preceding sunspot minimum, which is shown in Figure 7. It can be found that the rotational cycle lengths generally seem to be longer during the rising phase of the solar activity cycle and shorter during the declining phase of the solar activity cycle.

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As we known, the sunspot number is usually used as an index of solar activity (Chernosky & Hagan 1958; Altschuler & Sastry 1965; Cole 1973; Legrand & Simon 1981; Bogart 1982; Hathaway et al. 1994; Li et al. 2003, 2005, 2014; Gao et al. 2011; Gao 2016).  To better understand the relationship between solar rotation and solar activity, we perform a cross-correlation analysis between the rotational cycle lengths and the corresponding one-year smoothed daily sunspot numbers, which is shown in Figure 8. The daily sunspot numbers can be downloaded from National Oceanic and Atmospheric Administration (NOAA).⁷ The detailed description about the cross-correlation analysis can be found in Deng et al. (2013a, 2013b). In the figure, the abscissa indicates the shift of the rotational cycle lengths with respect to the daily sunspot numbers, with negative values representing backward shifts. From the figure, one can find that the cross-correlation coefficients have a maximum positive correlation of 0.41 at about 2.5 years and a maximum negative correlation of −0.48 at about −1.9 years. Both the positive lag of 2.5 years and the negative lag of −1.9 years suggest the existence of the phase difference between the rotational cycle lengths and solar activity. As is shown in the figure, there is another peak, which corresponds to the phase shift of about −7.8 years. Thus, the phase shift between the two peaks is about 10.3 years. Additionally, another valley can be seen, which corresponds to the phase shift of about 8.3 years. Thus, the phase shift between the two valleys is about 10.2 years. These phase shifts between the two peaks and two valleys may also indicate the existence of a quasi-11-year period in the variation of the rotational cycle lengths.

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In this paper, the temporal variation of the global rotation of the SMMF is studied based on the daily SMMF from 1975 May 16 to 2014 July 31 through use of a continuous wavelet transformation method. The results obtained are as follows:

1. The continuous wavelet transformation analysis suggests the periods of 27.5 and 13.7 days existing in the temporal variation of the SMMF, which has also been shown by many previous research, e.g., Rivin & Obridko (1992) and Ye et al. (2012). The local wavelet power spectrum displays that the highest power continuously appears around 27.5 days during the considered time interval. This means that the rotation modulation is persistent. And the 27-day recurrent plots in Bartels form also confirm the existence of quasi-27-day rotation modulation. Both analyses indicate that the apparent changes in the SMMF at short-timescale are mainly caused by the rotational modulation.

2. A significant period appearing at the 11-year Schwabe cycle can be found in the variation of rotational cycle lengths. This implies the existence of a relationship between the rotational cycle lengths and solar activity. Brajša et al. (2006) reported the existence of a similar period of 10.6 years in the variation of the solar rotation during the years 1874–1981. Javaraiah & Komm (1999) found an 11-year period in the Sun's mean rotation by analyzing the Greenwich sunspot group data in the years 1879–1976. Javaraiah (2013) also indicated that there exists a quasi-11-year cycle in the variation of equatorial rotation rates. Our result based on analyzing the SMMF data confirms their findings.

3. The analysis concerning the dependence of the rotational cycle lengths on solar cycle phase shows that the rotation period generally seems to be longer than the mean rotation period during the rising phase of the solar cycle but shorter during the declining phase. Haneychuk (1998) found that during the years 1971–1980 (before the magnetic field polarity reversal in 1980), the rotation period is about 27.39 days, while during the years 1981–1990 (after the polarity reversal), the rotation period is about 26.89 days. Figure 5 in our paper also indicates that the rotational cycle length before 1980 is longer than that during 1981–1990 in general. Svalgaard & Wilcox (1975) used the interplanetary field as a proxy for the large-scale solar field, and they reported both 27-day structure and 28.5-day structure present on the Sun. The 28.5-day structure is usually most prominent during the years near solar maximum. Our result, showing an increase in rotational cycle lengths in the interval between solar minimum and maximum, does not seem to be inconsistent with the appearance of the 28.5-day feature around solar maximum. Svalgaard (1972) studied the long-term behavior of the interplanetary sector structure from 1926–1970 and found that the rotation period of the sector structure varies sharply from about 27.1 to 28.3 days during the beginning of a solar cycle and then decreases and remains rather constant at about 27.2 days during the last half of the solar cycle. Our result seems to be in accordance with the finding of Svalgaard (1972). Our result is also in close agreement with Mordvinov & Plyusnina (2000), who studied the cyclic change of solar rotation also based on SMMF data and found that the rotational modes with periods of 27.8–28.0 days are dominant in the rising phase of solar cycles, whereas the 27-day rotational mode dominates the declining phase of the solar cycles. However, we show a more detailed inter-cycle variation of the rotational cycle lengths.

4. There exists a phase difference between the rotational cycle lengths and solar activity, indicating that the variation of solar rotation and the variation of solar activity are not in phase, although they both have a period of about 11 years.

According to our results, the rotation of the SMMF varies with time within a solar cycle. This can explain some previous results concerning periodicity around the rotational period of the SMMF. For instance, the time interval studied by Haneychuk (1998) is from the years 1968–1996, covering the complete solar cycles 21 and 22, as well as the declining phase of solar cycle 20. Based on our result, the rotational cycle length is relatively shorter during the declining phase. Thus, the mean rotation period during 1968–1996 should be relatively shorter, and it is about 26.92 days, as obtained by Haneychuk (1998). The rotation period obtained by Grigor'ev & Demidov (1987) for the years 1982–1984 is about 26.81 days. This time interval is in the declining phase of solar cycle 21; thus, the mean rotational cycle length should be shorter than 27 days according to our results, which is consistent with the result of Grigor'ev & Demidov (1987). The quasi-11-year period directly indicates that the rotation of the SMMF is related to solar activity. The solar-cycle phase dependence of the rotational cycle lengths also implies the relationship between the rotation of the SMMF and solar activity. As demonstrated by the gross amplitude modulation shown in Figure 1, there must be some relationship between the SMMF and the appearance of new flux on the Sun during the solar cycle. Moreover, early in the cycle, when new flux emerges at higher latitudes, the corresponding differential rotation rate of those features is smaller, while later in the cycle, more flux emerges at lower latitudes where the differential rotation rate is higher. This is consistent with the pattern found in this study: the rotational cycle lengths are longer during the rising phase than they are after solar maximum. It seems that the solar-cycle variation of the rotational cycle lengths of the SMMF should be related to latitudinal variation of the new flux emergence and the differential rotation feature of new flux. However, the phase analysis indicates that the relationship between the rotation of the SMMF and solar activity is much more complex, and further study is needed in the future.

We thank the anonymous referee very much for a careful reading of the manuscript and constructive comments. The authors are indebted to Professor Ke-Jun Li for his constructive ideas and helpful suggestions on the manuscript. The data used here are all downloaded from various web sites, and the authors express their deep thanks to the staff of these web sites. Wilcox Solar Observatory data used in this study was obtained via the web site http://wso.stanford.edu courtesy of J.T. Hoeksema. The Wilcox Solar Observatory is currently supported by NASA. The wavelet software was provided by C. Torrence and G. Compo. It is available at URL: http://paos.colorado.edu/research/wavelets/. This work is funded by the National Natural Science Foundation of China (11573065, 11673061, and 11633008), the Applied Basic Research Foundation of Yunnan Province, China (Grant No. 2014FB190), the Collaborating Research Program of CAS Key Laboratory of Solar Activity (KLSA201613), the Specialized Research Fund for State Key Laboratories, Center for Astronomical Mega-Science, and the Chinese Academy of Sciences.