The Contribution of Large Recurrent Sunspot Groups to Solar Activity: Empirical Evidence

Abstract

We identify large sunspot nestlets (SN) mostly containing recurrent sunspot groups and investigate the indices of solar activity defined as the 11- or 22-year moving average of the daily areas of the SN. These nestlets, 667 in total, are constructed from the daily 1874–2020 RGO/SOON catalogue, which contains 41,394 groups according to their IDs, with a machine-learning technique. Within solar cycles 15–19, the index contributed disproportionately strongly to the overall solar activity: the index is normalized to a quasi-constant shape by a power function of the activity, where the exponent is approximately $1.35$. Large SN contribute to solar activity even more in cycle 22, underlying the second largest peak of solar activity within the last Gleissberg cycle in ∼1985. Introducing another composite, moderate SN normalized by the overall activity, we observe its quasi-constant shape in cycles 15–19 and a general anti-correlation with the first normalized composite. The constructed sunspot nestlets constitute a modified catalogue of solar activity. We define the average lifetime per day in 22-year windows for the modified catalogue, in line with Henwood et al. (SoPhys 262, 299, 2010), and reproduce the dynamics of this quantity they revealed for 1900–1965. The average lifetime derived from the moderate SN is found to form a wave with minima at the beginning of the 20th and 21st centuries, resembling the Gleissberg cycle with long minima. The average lifetime characterizing large SN exhibited a deeper minimum at the beginning of the 20th century than 100 years later.

1. Introduction

The largest sunspots are characterized by a particular contribution to solar activity [1,2,3,4,5,6]. Identifying recurrent sunspot groups and adding their sizes to the statistical analysis, Nagovitsyn et al. [7] established that the coefficient of the proportionality in the Gnevyshev–Waldmeier rule is greater than earlier known estimates. The classification of sunspots into large, moderate, and small should ideally be universal going beyond the scope of individual papers. Obridko and Badalyan [2], splitting the range of sunspot areas into three intervals: up to 100 microhemispheres (MH), 100–500 MH, and >500 MH, found that the solar cycle “period” is most clearly detected with large sunspots, whereas the smaller sunspots somewhat unexpectedly follow ∼20 and ∼60 year quasi-periodicities. The contribution of large sunspots to the whole activity can also be related to the properties of quasi-biennial oscillations [8,9]. Mandal and Banerjee [3] worked with the solar cycle strength defined through the total area of specific sunspots emerging during the cycle. According to [3], analyzing cycles 16 to 23, sunspots with areas between 200 and 500 MH contribute more to odd cycle numbers. Nevertheless, Mandal and Banerjee [3] argued that only large sunspots represent the main indicators of solar activity and help quantify the asymmetry between solar hemispheres. On the decadal-to-centennial scale, the secular Gleissberg cycle (see [10,11,12] among others), modulating the amplitude of the solar cycle, affects the long-term properties of sunspots in a different way depending on their size and governs the sunspot formation in general [2,3].

Large and small sunspots are thought to represent the large- and small-scale components of the solar dynamo [3,13]. Obridko and Badalyan [2] conjectured that the formation of large sunspots occurs in the subsurface layers of the Sun, whereas small sunspots are connected to the deep dynamo. The state of the art in dynamo modeling allows model sequences of the sunspots to be generated. These artificial sunspots follow not only the basic regularities but also some of the long-term irregularities of solar activity [14]. The suggested algorithms producing artificial sunspots differ in where the flux originated and how it is transferred through the convection zone [15,16,17]. As the algorithms are driven by the observations of real sunspots, a better understanding of their relevant features, to be reproduced by the models, is required.

The identification of recurrent sunspot groups performed by Nagovitsyn et al. [7] to uncover the relationship between their area and lifetime complements earlier efforts to produce the catalogue of long-lived groups [18,19] and follows the opinion of Ringnes [20] who argued that “a revised catalogue of recurrent spots …would …be very desirable”.

This paper tackles the contribution of large sunspots to general solar activity on annual-to-multi-decadal scales in quantitative and qualitative ways. Addressing this challenge, we construct an algorithm aimed to identify large and moderate recurrent sunspot groups from the Royal Greenwich Observatory (RGO, 1874–1975) and Solar Optical Observing Network (SOON, 1976–2020) daily databases. Designing the details of the identification algorithm, we are motivated to trace the manifestation of strong magnetic field in the same location for several solar rotations. The identified groups, i.e., the groups selected by the algorithm, form so called nestlets (first, introduced by Henwood et al. [19] for a similar purpose). The identification leads us to proxies of strong magnetic field, which are the composites of the solar activity defined as the 11- or 22-year smoothing of the daily area of the sunspot groups included into the constructed nestlets. The temporal variations of the composites compared with general solar activity and the lifetime of the sunspot nestlets constitute the content of the paper, which is structured in a standard way. Section 2 and Section 3 describe the data and the method. Results are presented in Section 4. The discussion is in Section 5. Section 6 concludes. Technical details of the identification algorithm and its detailed comparison with the expert identification of the recurrent groups by [18] and earlier algorithms by Henwood et al. [19] and Nagovitsyn et al. [7] are relegated to the Appendix A, Appendix B, Appendix C and Appendix D.

2. Data

We use the daily series compiled by the Royal Greenwich Observatory (RGO) from 1874 to 1976 and the US Air Force from its Solar Optical Observing Network (SOON) from 1976 onward [10]. The joint catalogue contains 41394 different groups according to their IDs.

The change in the place and facilities of the observation brings inconsistency into the joint RGO/SOON database ([21,22]). The methods of the data processing including the SOON practice of rounding down the limb-area correction factors also affect the consistency of the joint catalog. The web sites https://solarscience.msfc.nasa.gov/greenwch.shtml (accessed on 5 January 2022) and http://solarcyclescience.com/activeregions.html (accessed on 5 January 2022) ([23]) contain RGO/SOON sunspot group data from 1874 onward. Hathaway et al. [24] defined the correction factor $1.4$ for the SOON areas (i.e., the areas associated with 1976 and later years). We denote $\mathbf{D}$ the set of the sunspot groups from the joint RGO/SOON catalogue.

The choice of the databases is worth commenting on. Great many efforts have been recently made to improve the indices of solar activity (see, e.g., [25,26,27]). The homogeneity of the RGO time series in 1880–1920 were recently questioned and, as a result, the corrections were proposed by Willis et al. [28], Cliver [29]. Clette et al. [30] produced the second version of ISSN adjusting so called observer factor in the definition of the Wolf numbers and working through the discontinuity in the time series, thought to be performed with the Wolf numbers in 1945. Several composites have been built on the sunspot groups: the group sunspot number by Hoyt and Schatten [31] and its extension by Lockwood et al. [32], the “backbone” group number reconstruction by Svalgaard and Schatten [33], and the group number derived by Usoskin et al. [34]. Our choice of the RGO/SOON database is motivated by the requirement to deal with a long time series which contain the position of the spots together with the characteristic of the activity strength.

3. Method

3.1. Focus of the Identification

Large recurrent sunspot groups give a relevant proxy to the strong magnetic field of the Sun manifested at specific locations for weeks. However, the identification of the recurrent groups is uncertain since they are not observed at the “back”side of the Sun (at least, until Solar Terrestrial Relations Observatory, STEREO, [35] and Solar Dynamics Observatory [36] started recording the solar surface invisible from the Earth). Henwood et al. [19] referred to the results of their identification of recurrent groups as nestlets. In contrast to Henwood et al. [19], we focus on the identification of large and moderate recurrent groups and end up with large and moderate nestlets. As we will show later in detail, the design of our identification can lead to errors when a genuine “old” group is vanishing at the appearance on the solar disk but another larger group emerging nearby is included to the nestlets or no groups are identified at all. In the first case, the identification modifies the notion of nestlets introduced by Henwood et al. [19] through an increase of their size. In the second case, the vanishing part of sunspot groups (or the whole group if it returns to the solar disk for the first time) is not included into our nestlets. The identification of disappearing recurrent groups seems to rise uncertainties whereas their contribution to the solar activity associated with the nestlets is minor. As a result, we are biased to omit a minor contribution but avoiding potential erroneous identifications.

3.2. Identification of the Recurrent Groups

The rotation period of the Sun is approximately 27 days. The exact value depends on the latitude, as the Sun exhibits the differential rotation. If a sunspot group leaving the solar disk appears on it once more this occurs ∼14 days later in a neighborhood of the point with the same Carrington coordinates (i.e., the coordinates related to the Sun). The explicit position is affected primarily by a slow drift of the sunspots across the Sun estimated in up to $0.01$–$0.03$ degrees per day [37,38]. Rare chaotic jumps in the records of specific sunspot groups are mainly explained by errors [39]. However, the appearance or disappearance of sunspots withing a group also affects the reported group position. As a result, a new occurrence of recurrent groups are expected in an ellipse centered at the longitude and latitude of the previous observation. Referring to the ellipse, we mean the points on the two-dimensional plane consisting of the latitudes and the longitudes, which are not related to the embedding of the Sun in the real three-dimensional space. We specify the ellipse in the following way.

Let $P=27.2757$ be the Carrington rotation period that represents a certain average of the periods observed because of the differential rotation of the Sun. Then we put $\Delta =13.2\approx 360/P$. This $\Delta$ roughly represents the move of each group per day across the solar disk caused by the (differential) solar rotation.

Further, we consider the sunspot groups with the longitude located in the range $S_r=[{90}^{\circ }-2\Delta ,{90}^{\circ }-\Delta ]$ and focus on those that the maximum of their recorded areas exceeds some relatively large threshold A. These groups are going to leave the disk two days later (if they still exist). Let $G_1$ be one of these groups observed at some time $t_1$ (measured in days) with the longitude $x_1$ from the range $S_r$, the latitude $y_1$, and the area $a_1$. For the sake of simplicity in the notation, the identification procedure is explain just for $G_1$. We intend to look for its new appearance at the left of the solar disk $16.5$–$17.5$ days later. Let $G_2$ be a group observed at $t_2\in [t_1+16.5,t_1+17.5]$ days. Then we introduce the ellipse centered at the Carrington coordinates $(x_1,y_1)$ of the group $G_1$ and endowed with some semi-axes $l_x$ and $l_y$, which are the parameters of the procedure that has to be adjusted and fixed. If $G_2$ is located within the ellipse, it is a candidate to be identified with the group $G_1$. Each group $G_1$ can have either 0, or 1, or more such candidates. The group $G_1$ is claimed to be non-recurrent in the first case. A single candidate is identified as $G_1$ in the second case. The group with the area being closest to the area $a_1$ of $G_1$ is identified with $G_1$ in the third case.

Eventually, we choose the semi-axes to minimize the errors of the identification. To this end, we use the sunspot groups that drift 7–10 days along the solar disk. With the threshold $A=950$ MH and the groups observable at least 10 consecutive days, the semi-axes $l_x$ and $l_y$ are adjusted via the maximization of the $F_1$-score: $F_1=\frac{tp}{tp+0.5(fp+fn)}$, where $tp$, $fp$, and $fn$ are the numbers of the true positive, false positive, and false negative outcomes of the identification. The adjusted values ${12}^{\circ }$ and $5^{\circ }$ of the longitudinal and latitudinal widths of the semi-axes are stable with respect to the drop of T from 10 to 7 days. This stability in T favors the application of the adjusted values to the real identification when $T\approx 17$. The details of the algorithm are described in the Appendix A, Appendix B, Appendix C and Appendix D.

3.3. Samples of Large and Moderate Nestles

We extract two sub-catalogues of the groups included into the nestlets with our identification algorithm. Let $\mathbf{L}$ be extracted sunspot groups from $\mathbf{D}$ such that the maximum of their recorded areas is more than 950 MH. Let $\mathbf{M}$ be extracted sunspot groups from $\mathbf{D}$ such that the maximum of their recorded areas is between 700 and 950 MH. Then the sub-catalogues $\mathbf{L}$ and $\mathbf{M}$ are our large and moderate nestlets. Controlling the necessity to deal with just large recurrent sunspot groups, we also consider two other sets $\mathbf{H}$ and $\mathbf{LH}$ of large sunspot groups. The set $\mathbf{H}$ consists of all (non-necessary recurrent) sunspot groups from $\mathbf{D}$ such that the maximum of their recorded areas is more than 950 MH. Note that big non-recurrent groups from $\mathbf{H}$ do not belong to $\mathbf{L}$. Regarding the opposite exclusion, if a recurrent group $G\in \mathbf{L}$ is obtained as the identification of two groups $G_1$ and $G_2$ and, for example, $G_2$ does not show up an area greater than 950 MH, then only $G_1$ is in H, but $G_2$ is not. The set $\mathbf{LH}$ is the union of $\mathbf{H}$ and $\mathbf{L}$. It is defined to exhibit the transition between conclusions obtained with $\mathbf{L}$ and $\mathbf{H}$.

We explore the time series which consist of consequent daily areas of sunspot groups averaged over 11-year sliding windows. The daily areas are taken from the samples $\mathbf{D}$, $\mathbf{L}$, $\mathbf{M}$, $\mathbf{H}$, and $\mathbf{LH}$. Their averages are measured in MH per day. Formally, let $\Omega \subseteq \mathbf{D}$ be the set of sunspot groups and $a_{\Omega }\left(d\right)$ be the total area of all sunspot groups from $\Omega$ associated with day d. Then we put

$$R_{\Omega }\left(t\right)=\frac{1}{N}\sum _{d\in [t-(N-1)/2,t+(N-1)/2]}{a}_{\Omega }\left(d\right),\Omega \in \{\mathbf{M},\mathbf{L},\mathbf{H},\mathbf{LH}\},$$

(1)

where $N=4017$ is the length of the 11-year moving window, and denote $R_{\mathrm{ISSN}}\left(t\right)$ the 11-year moving average of index ISSN (International Sunspot Numbers obtained from WDC-SILSO, Royal Observatory of Belgium, Brussels, [40], http://www.sidc.be/SILSO/(accessed on 5 January 2022)). With these composites, we discuss the contribution of the relatively large sunspot groups to the solar activity. The composite $R_{LH}$ is used to describe the change in the dynamics of the composites as $R_L$ substitutes $R_H$.

The time series $R_M$ and $R_L$ constitute a part of the series $R_D$. According to Figure 1, these parts are well correlated with the full series. As is well known, the smoothed ISSN are also correlated with $R_D$ (red and blue curves). In particular, all four curves attain a global minimum, a global maximum, and a second maximum at cycles 13–14, 18–19, and 21–22, respectively (Figure 2). We note that the variability of $R_M$ (orange curve) is smaller than that of $R_L$ (magenta curve).

Estimating the contribution of the different components of $R_D$ to the full index, we normalize the series $R_M$ and $R_L$ into two new series:

$$r_{\Omega }=\frac{R_{\Omega }}{R_D},\Omega =\mathbf{L},\mathbf{M},r_{\Omega ,\beta }^{*}=\frac{R_{\Omega }}{R_D^{\beta }}\Omega =\mathbf{H},\mathbf{LH},\mathbf{L},$$

(2)

where the exponent $\beta$ is adjusted for each composite separately to obtain the flattest graph of $r_{\Omega ,\beta }^{*}$ with the data of cycles 16–19 (the deviation from the mean value for these data is minimized with the step of $0.01$ in the values of $\beta$). The adjusted values of $\beta$ are $1.20$, $1.19$, and $1.35$ for the composites $\mathbf{H}$, $\mathbf{LH}$, and $\mathbf{L}$, respectively.

4. Results

4.1. Indices of Recurrent Sunspot Groups

The normalized large selected sunspot groups $r_L$ (the magenta curve in Figure 2) still followed $R_D$ after 1915, thus, contributing disproportionately strongly to the overall solar activity. This disproportionality can be assessed quantitatively by using the time series $r_{L,\beta }^{*}\left(t\right)$ defined through the normalization by the power function, Equation (2). This $r_{L,\beta }^{*}\left(t\right)$ (green curve in Figure 2 obtained with $\beta =1.35$ and shifted via the multiplication by an appropriate factor to uplift it to the range of the values of the other curves) exhibits quasi-constant behavior when t is located in 1925–1965, cycles 16–19. The adjusted value of $\beta$ is not important itself. We emphasize here the very possibility to quantify the specific contribution of the large selected sunspot groups during 4 cycles in a row in a simple way. The variability of $r_{L,\beta }^{*}$ was larger before 1925 and after 1965. Nevertheless, the values of $r_{L,\beta }^{*}$ equaled to ∼0.25 (in the units of Figure 2) were attained in cycles 14, 15, 20, and 21, which is not far from the values $0.28$–$0.30$ observed during cycles 16–19.

The weak variability of the normalized composite $r_{L,\beta }^{*}$ is observed with the range of $\beta$ from $[0.3,0.4]$, where $1.35$ is the best estimate. This property characterizes just the large selected sunspot groups. The variability of the normalized composite $r_{H,\beta }^{*}$ built on the large (but not necessarily recurrent) groups is stronger (the black curve on Figure 3). The composite $\mathbf{LH}=\mathbf{L}\cup \mathbf{H}$ collects features of both datasets. As a result, the deviation from the mean value of $r_{LH,\beta }^{*}$ is between that of $r_{L,\beta }^{*}$ and $r_{H,\beta }^{*}$. The standard deviation computed with $\mathbf{L}$ is less than that found with $\mathbf{LH}$ and $\mathbf{H}$ on the cycles 16–19. The shift of the optimal exponent of $1.35$ toward 1 for the normalized composites $r_{H,\beta }^{*}$ and $r_{LH,\beta }^{*}$ emphasizes the role of just recurrent groups (selected to the nestlets) in their disproportionate contribution to solar activity.

The normalized moderate selected sunspot groups $r_M$ demonstrated quasi-constant behavior in 1915–1975 followed by slow variations from 1975 onward (the orange curve in Figure 2). In general, the $r_M$ and $r_{L,1.35}^{*}$ curves exhibited an anti-correlation (most clearly observed with the 1965–1997 data). Interestingly, the change from $r_{L,1.35}^{*}$ to $r_L$ weakens the anti-correlation despite the fact that the composites $r_M$ and $r_L$ correspond to each other by definition. During the short interval from 1898 till 1903 and later from 1997 till 2010, $r_M$ and $r_{L,1.35}^{*}$ varied co-directionally. New data allow us to estimate a further agreement between $r_{L,1.35}^{*}$ and $r_M$.

4.2. Recurrent Sunspot Group Lifetime

Blanter et al. [41] claimed that the lifetime of sunspots increased by factor $1.4$ during 1915–1940. They obtained this result indirectly when simulating the solar activity with a modulated AR-1 process. Henwood et al. [19] confirmed this prediction identifying the recurrent sunspot groups from the catalogue known as Greenwich Photo-heliographic Result (GRP), https://www.ngdc.noaa.gov/stp/solar/greenwich.html (accessed on 5 January 2022). We are able to reproduce the dynamics of the lifetime exposed by Henwood et al. [19] with our recurrent sunspot groups. This supports the reliability of our methodology and allows us to make an additional conjecture regarding the lifetime of large recurrent sunspot groups.

We note that the identification of large recurrent groups leads to the modification of the initial catalogue D: the components of the large selected groups are combined, whereas the rest of the groups are left as they are. Let $D^{\prime }$ be the modified catalogue. Further, following Henwood et al. [19], if several groups from $D^{\prime }$ share the day of the first record, only the group with the largest lifetime is kept and the other groups are eliminated. We denote $D^{″}\subset D^{\prime }\subset D$ the resulting catalogue. The lifetime of each group from $D^{″}$ with the first and the last records separated from the limb on at least $2{\Delta }^{\circ }$ is the time difference between them increased by 1 day. If the first or the last records are offset from the limb on less than $2{\Delta }^{\circ }$ then 8 days are added to the time difference between these records (or 7 days if the offset is less than ${\Delta }^{\circ }$). For any 22-year window the lifetime ${\mathcal{T}}_g$ of each group g associated with this window is summed up and divided by $22·365.25$. The result introducing the lifetime per day is denoted ${\tau }_{D^{″}}$ and assigned to the center of the window:

$${\tau }_{D^{″}}\left(d\right)=\frac{1}{22·365}\sum _{d_{0g}\in [d-11·365,d+11·365-1]}{\mathcal{T}}_g,$$

where $d_{0g}$ represents the day of the group g’s first record. In words, ${\tau }_{D^{″}}$ represents the average number of the days such that the groups from $D^{″}$ exist in the Sun. The 22-year windows are used to follow Blanter et al. [41] and Henwood et al. [19]. The same definition of the lifetime applied to the catalogues M and L returns us the normalized lifetime per day naturally called ${\tau }_M$ and ${\tau }_L$ (i.e., the average number of the days such that the groups from M and L, respectively, exist in the Sun).

The normalized lifetime per day ${\tau }_{D^{″}}$ shown in Figure 4 in blue and obtained with the approach of paper [19] but with our identification of the recurrent groups repeats the graph constructed by Henwood et al. [19]. The right endpoint of the graph from paper [19] corresponds to 1965, 11 year prior to the move of the observation place occurred in 1976. As a result of this move, the number and the area of the sunspots increased. Corrections were introduced to calibrate the indices of solar activity based on the sunspot groups [21]. However, these corrections do not calibrate the lifetime. Therefore, the comparison of the parts located before 1965 and after 1987 are impeded. We can note based on the right part of the graph itself, that a decrease in ${\tau }_{D^{\prime }}$ started when the sliding window reached the descending phase of cycle 23. This decrease turned to the fall when the data of cycle 24 were substituting for the data of cycle 22 in the sliding window.

The lifetime composites ${\tau }_L$ and ${\tau }_M$ derived from only large and, respectively, only moderate recurrent sunspot groups follow the rises and falls of ${\tau }_{D^{″}}$. The graphs of ${\tau }_L$ and ${\tau }_M$ look like a scaled version of ${\tau }_{D^{″}}$. The ${\tau }_M$ graph contains a secular wave with the maximum at cycles 18–19, which resembles the Gleissberg cycle. The anomalous 20th cycle affects mostly the composite ${\tau }_L$ that characterizes the large selected sunspot groups, causing a drop followed by a rise which violates a steady decrease of the wave of the Gleissberg cycle. The adequate calibration of the lifetime has to shift the right (i.e., after 1976) part of the composites downward, but the exact position has not been determined yet. Nevertheless, the peculiarities of our identification algorithm, which matches the components with similar areas, makes the inconsistency related to large recurrent groups smaller. Therefore, the calibrated right part of the (blue) ${\tau }_{D^{″}}$ graph might fall at the right end to the level of the beginning of the 20th century, thus staying in line with the existence of the Gleissberg-cycle wave (which seems to be the case for ${\tau }_M$). On the contrary, the fall of the (magenta) ${\tau }_L$ at the right to the level attained a century ago seems very unlikely.

5. Discussion

This paper reveals new regularities and irregularities of solar activity on the decadal-to-multi-decadal scale. Applying a machine learning technique to identify large recurrent groups from the daily 1874–2020 RGO/SOON catalogue we select the nestlets of sunspot groups and examine a proxy $R_L$ of solar activity defined as the daily area of these groups averaged over 11 years. This proxy $R_L$ is a part of the index $R_D$ which represents the 11-year moving average of the daily areas of all sunspot groups. The index $R_D$ is related to various strongly correlated proxies of solar activity, discussed in details, for example, by Lockwood et al. [26], which are inferred from the sunspot groups. The construction of the nestlet catalogue complements efforts performed to produce and calibrate records of solar activity [26,27,33,34,42,43].

We posit that the indices $R_L$ and $R_D$ exhibit a strong correlation (Figure 1). The contribution of large sunspot nestlets to solar activity is established to be disproportionately strong in terms of the relationship between $R_L$ and $R_D$ (Figure 2). The disproportionality is estimated quantitatively by the normalization of the index $R_L$ by the power function $R_D^{\beta }$ of the full index. The proportional contribution is given by the exponent $\beta =1$. However the normalization with $\beta =1$ does not exclude a positive correlation with solar activity. A larger exponent is required, which is $\beta \approx 1.35$. The normalized time series $r_L^{*}$ exhibited a fixed level during cycles 15–19 and attained the values close to this level in cycles 14 and 20–23 (Figure 2). We stress that the preliminary selection of the recurrent groups is important to quantify the properties of the large sunspot groups. The composite $R_H$ built on the large sunspot groups from the initial catalogue still contributes to the overall activity disproportionately strongly, but weaker than $R_L$ does. The normalization of $R_H$ to $r_{H,\beta }^{*}$ by a power function of the full index $R_D$ that intends to flatten the graph within cycles 16–19 requires the exponent $\beta =1.19$ being closer to 1 than $1.35$ and results in a less flatter part of $r_{H,1.19}^{*}$ than that of $r_{L,1.35}^{*}$ (Figure 3). The peculiar role of the largest sunspots found here is in line with the conclusions of other papers [3,44,45,46].

Our additional empirical findings are based on the fraction $\tau$ of days such that the sunspot nestlets exist in the Sun. These fractions are computed within 22-year sliding windows. The strongest contribution of the large sunspot nestlets to the overall solar activity occurred during its second largest maximum in cycles 21–22 (see $R_L$ and $r_L^{*}$ curves in Figure 1 and Figure 2, respectively) when the fraction ${\tau }_L$ of the days with these large nestlets exhibited rather regular values (Figure 4). Thus, the maximum of $R_L$ occurred in cycles 21–22 was driven primarily by the growth in the area of the largest groups rather than in their number per day.

The moderate nestlets are characterized by the indices $R_M$ and ${\tau }_M$, which represent the 11-year average of their daily areas and 22-year average of their number per day, respectively. The ratio of $R_M$ to the full index $R_D$ exhibited a constant level in cycles 15–19 and a strong anti-correlation with the normalized index $r_{L,1.35}^{*}$ (Figure 2). The deep minima of solar activity observed in ∼1900 and 2008 are characterized by the short termination of the anti-correlation between large and moderate long-lived sunspot group composites and co-directional changes in the two series. These episodes may signal the beginning of a new Gleissberg cycle. We note that moderate non-recurrent sunspot groups attained the main maximum in cycles 21–22 [2], in contrast to the recurrent groups. We support the conjecture by Henwood et al. [19] regarding the existence of the Gleissberg cycle in the dynamics of the lifetime (with the quantity that they introduced), arguing that the index ${\tau }_M$, which represents the average daily number of the moderate nestlets observed in the Sun, followed a secular wave with minima in approximately 1906 and 2005. Moreover, the index ${\tau }_M$ is characterized by rather extent previous minimum of the Gleissberg cycle around 1900 (Figure 4). Therefore, the presented results suggest that the minimum of the centennial cycle started in solar cycle 23 and followed by solar cycle 24 may be long without actually entering a Grand Minima epoch. In this case, we should not expect a quick return of solar activity to the high level recorded in the mid of 20th century. The study of the average lifetime of different recurrent sunspot groups, which is more natural characteristic than the fraction $\tau$, can shed more light on the properties of the Gleissberg cycle. But this is worth doing in a separate study.

Our identification of the recurrent groups is similar to that performed by Nagovitsyn et al. [7], but we focus on moderate and large groups ending up with a more accurate identification of just these groups with at least the training set related to the visible part of the Sun, Figure A1. With 1944–1976 year data, a turn from ${12}^{\circ }\times 2^{\circ }$ (related to [7]) to ${12}^{\circ }\times 5^{\circ }$ ellipse reduces the number of errors from 15 to 3 (see

Table A2. The errors in the identification of large sunspot groups with the ellipses of the sizes ${12}^{\circ }\times 2^{\circ }$ (roughly corresponding to [7]) and ${12}^{\circ }\times 5^{\circ }$ used in this paper on the training set (A3) defined with the 1944–1976 RGO/SOON catalogues and $T=10$. The RGO/SOON ID are given; the pairs of the ID show the two groups that are erroneously united into a single group.

$\mathit{A}>950$ MH	${12}^{\circ }\times 2^{\circ }$	${12}^{\circ }\times 5^{\circ }$
Error type I	(18872, 18882),	(19061, 19062),
	(18959, 18971)
Error type II	18559, 21356, 19200, 21506	17782, 18756
	14838, 16486, 18756, 20125, 23119
	14717, 15638, 17161, 17782

with the complete list of the IDs of the groups related to the errors). As the drift of sunspot groups depends on their area [10], one may adjust the dependence of the ellipse axes on the area of the recurrent sunspots when identifying all recurrent sunspot groups. The direct comparison between the results obtained with our and Henwood et al. [19]’s identification procedures is not well defined, as nobody knows what groups are indeed recurrent. As an example, we take 8 groups which are observed in 1986, called recurrent by Henwood et al. [19], and satisfied our criteria of the moderate/large group identification. All of them are indeed identified as recurrent by our algorithm (see Appendix C ). Furthermore, the conclusions regarding the solar proxies built on the identified groups are comparable, and we completely reproduce the dynamics of the lifetime found by Henwood et al. [19] with the 1874–1976 data (Figure 4). The reproduction of the time variability of the lifetime found by Henwood et al. [19] and Blanter et al. [41] gives additional credibility to our identification mechanism.

Our large and moderate nestlets are related to but differ from sunspot nests, active regions, active longitudes, and complexes of activity. The nests were defined by Castenmiller et al. [47] as the groups of sunspots that keep their location during 6–15 solar rotations. The sunspot nests and the other terms reported above are introduced to describe the persistence of the strong magnetic field at fixed location (region, longitude) in the Sun (see [48,49,50] among others). This phenomenon can be associated with several sunspot groups of different sizes located within a broad region, in contrast to a single group found in smaller regions and selected to the nestlets.

6. Conclusions

We have emphasized that large recurrent sunspot groups contributed disproportionately strongly to the overall solar activity in 1915–2005, probably except a neighborhood of 1975. The 11-year moving average $R_L$ of the areas of these groups normalized to $r_{L,\beta }^{*}=R_L/R_D^{\beta }$ by the 11-year moving average $R_D$ of the areas of all groups with the exponent $\beta =1.35$ exhibited a weak variability around a constant level during cycles 15–19, thus, giving evidence for the estimate $R_L\sim R_D^{1.35}$. The quasi-constant behavior of $r_{L,\beta }^{*}$ was followed by a drop in the anomalous cycle 20 and a rise to its global maximum in cycles 21–22. The average fraction ${\tau }_L$ of the days with the large nestlets in the Sun kept exhibiting regular values at the time of this maximum of $r_{L,\beta }^{*}$.

In contrast to the large selected sunspot groups, the moderate ones followed the solar activity as it was: $R_M\sim R_D$, whereas the pattern of their active days given by ${\tau }_M$ resembled the Gleissberg cycle with minima at the beginning of the 20th and 21st centuries.

The particular role of the long-lived sunspot groups highlighted in this paper may be explained by interactions of two multi-scale processes which contribute to solar dynamo. One process is explicitly connected to the global component of the dynamo being responsible for the sunspot formation. The other is related to the turbulent diffusion destroying the sunspots. One may assume that the sunspot formation process continues with the development of the sunspots in the Sun. The sunspot area enlarges when the spots rise into the surface layers from the base of the convective zone. The growth in the area remains regular while the moderate recurrent sunspot groups are created. The regularities of this creation found in this paper are probably governed by the Gleissberg cycle. However, the largest sunspot groups, which are extreme events in the probability distribution of the sunspot groups with respect to their areas [46], exhibit more complex behavior, as derived here and by other authors (see [2,3,13]). Better understanding of the processes related to the amplification of the magnetic field in the surface layers may shed light on the anomalous contribution of the largest recurrent sunspot groups to the overall activity.