The Astrophysical Journal, 730:49 (5pp), 2011 March 20 doi:10.1088/0004-637X/730/1/49C© 2011. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

VARIATIONS OF SOLAR ROTATION AND SUNSPOT ACTIVITY

K. J. Li1,2

, X. J. Shi1,3

, H. F. Liang4, L. S. Zhan

5, J. L. Xie

1,3, and W. Feng

61 National Astronomical Observatories/Yunnan Observatory, CAS, Kunming 650011, China; lkj@ynao.ac.cn

2 Key Laboratory of Solar Activity, National Astronomical Observatories, CAS, Beijing 100012, China3 Graduate School of CAS, Beijing 100863, China

4 Department of Physics, Yunnan Normal University, Kunming 650093, China5 Jingdezhen Ceramic Institute, Jingdezhen 333001, Jiangxi, China

6 Research Center of Analysis and Measurement, Kunming University of Science and Technology, Kunming 650093, ChinaReceived 2010 April 25; accepted 2011 January 18; published 2011 March 3

ABSTRACT

The continuous wavelet transformation is used to study the temporal variations of the rotational cycle length of dailysunspot numbers from 1849 January 1 to 2010 February 28, from a global point of view. The rotational cycle lengthof the Sun is found to have a secular trend, which statistically shows a linear decrease by about 0.47 days during thetime interval considered. The empirical mode decomposition analysis of the temporal variations of the rotationalcycle length shows an acceleration trend for the surface rotation rate from cycles 11 to 19, but a deceleration trendfrom the beginning of cycle 20 onward. We cannot determine whether the rotation rate around the maximum timesof the Schwable cycles should be faster or slower than that around the minimum times, implying no Schwablecycle in the long-term variations of rotation. The results obtained are compared to those from the literature. It isinferred that the variation of the rotational cycle length may be related to the variation of sunspot activity in thelong run.

Key words: Sun: activity – Sun: rotation – sunspots

Online-only material: color figure

1. INTRODUCTION

The discovery of solar differential rotation happened in 1630when Christoph Scheiner, after Galileo’s sunspot observations,first noticed that the equatorial region of the Sun rotatesfaster than higher-latitude regions: 26 days at the equator and30 days at 60◦ latitude (Howard et al. 1984; Sheeley et al.1992; Rybak 1994; Howe et al. 2000a; Le Mouel et al. 2007).Since the basic work of Carrington on solar rotation, studieshave been performed in the framework of differential rotation,and two methods are mainly used: the tracer method and thespectroscopic method (Balthasar & Woehl 1980; Gilman &Howard 1984; Brajsa et al. 2000, 2002; Howe et al. 2000b;Woehl & Schmidt 2000; Antia & Basu 2001). The classic reviewpapers of Howard (1984), Schroter (1985), Snodgrass (1992),Beck (1999), and Paterno (2010) compared and discussed theresults of different measurements of the Sun’s rotation rate,showing great achievement in the observations and analysesof solar surface rotation. However, there are still many aspectsto explore; for example, the long-term variations of the solarrotation rate are still unknown (Komm et al. 1993; Ulrich& Bertello 1996; Stix 2002). An interesting property wasfound: the general trend of the rotation rate of the solar cyclesseemingly increases in the long run (Lustig 1983; Howard1984; Heristchi & Mouradian 2008), but the contrary resultis obtained by Balthasar et al. (1986), Kitchatinov et al. (1999),Zuccarello & Zappala (2003), and Brajsa et al. (2006). In thisstudy, the continuous wavelet transformation is used to givethe long-term variations of the rotational cycle length of dailysunspot numbers (SNs) from 1849 January 1 to 2010 February28, and the empirical mode decomposition (EMD) method(Huang et al. 1998; Feldman 2009) is utilized to investigatethe general trend in the temporal variations of the Sun’s rotationrate.

2. WAVELET ANALYSES OF DAILY SUNSPOT NUMBERS

2.1. Data

Utilized here and shown in Figure 1 are daily SNs from 1849January 1 to 2010 February 28, downloaded from the Web site ofthe Solar Influences Data Analysis Center.7 Since the MaunderMinimum, especially from cycle 6 onward, there seems to havebeen a steady increase in sunspot cycle amplitudes, which iscalled “the secular trend” (Wilson 1988; Solanki et al. 2004). Alinear regression is done to fit the data, and the obtained result isgiven in the figure. The regression line indicates that daily SNsincrease by 31.4, from 39.6 at the beginning to 71.0 at the endof the time interval considered.

2.2. Continuous Wavelet Transformation Analysis

Wavelet analysis involves a transform from a one-dimensionaltime series to a diffuse two-dimensional time–frequency image,localized in frequency and time domains, to detect the localizedand (pseudo-) periodic fluctuations by using the limited timespan of the data (Torrence & Compo 1998). We employ theMorlet wavelet, which is defined as

Ψ0(η) = π−1/4eiω0ηe−12 η2

, (1)

where ω0 is the dimensionless frequency and η is the dimension-less time. When using wavelets for feature-extraction purposes,the Morlet wavelet (with ω0 = 6) is a good choice, since it pro-vides a good balance between time and frequency localization(Torrence & Compo 1998; Grinsted et al. 2004). Figure 2 showsthe continuous wavelet local-power spectrum of daily SNs. Thehighest power belt appears around the rotation cycle of the Sun.

7 http://sidc.oma.be/sunspot-data/

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The Astrophysical Journal, 730:49 (5pp), 2011 March 20 Li et al.

1850 1900 1950 20000

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Figure 1. Daily SNs (the thin solid line) from 1849 January 1 to 2010 February28 and their straight-line regression (the thick solid line). The dashed verticallines indicate the minimum times of sunspot cycles.

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Figure 2. Continuous wavelet power spectrum of daily SNs.

(A color version of this figure is available in the online journal.)

Figure 3 shows the global wavelet power spectrum of the timeseries, which is the average of local “components” over time.From a global point of view, the rotation period is 27.4 days,which is statistically significant at the 95% confidence level(related to red noise; for details, see Torrence & Compo 1998).

3. LONG-TERM VARIATIONS OF SOLAR ROTATION

Figure 4 displays the period length of the rotation cyclevarying with time, related to the mean rotational cycle lengthof 27.4 days. At a certain time point, the rotation period (scale)of an SN has the highest spectral power among the timescalesof 25–31 days (the differential rotation cycle of sunspots islocated within this length range) in its local wavelet powerspectrum. Then, the obtained temporal variations of the solarrotational cycle length are smoothed with a 183-day (a half-year) sliding window, and the smoothed temporal series isshown in the figure. A linear regression is performed to fitthe temporal variations of the rotational cycle length, and theobtained result is also given in the figure. The regression lineindicates that the Sun’s rotation cycle decreases in length byabout 0.47 days during the time interval considered. The Sun

5 10 15 20 25 30 35 40 451.6

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log1

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)

Figure 3. Global wavelet power spectrum of daily SNs (the solid line). Thedashed line shows the 95% confidence level.

1850 1900 1950 2000−3

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Figure 4. Period lengths (the thin solid line) of the rotation cycle, related tothe mean rotational cycle length of 27.4 days. The thick dashed line is theirregression line.

seems to accelerate its surface rotation rate in the long run. Itis noted here that the sliding windows chosen should slightlyaffect the straight-line regression obtained. For example, whena 365-day sliding window is chosen, the above factor changesfrom 0.47 to 0.48 days.

Heristchi & Mouradian (2008) investigated the rotation rateof solar activity from a global point of view, consideringthe daily index or flux instead of individual structures. Theydetermined the rotational rate variations inside the solar cycleby two-year sequences and a comprehensive study of the cycle.Based on their results, we first transform the temporal siderealsunspot rotation rates of cycles 9–23, namely, from the years1849.449 to 2004.112, which were originally given in theirTable 1, into rotational cycle lengths and then plot them inFigure 5. A linear regression is performed to fit the rotationalcycle lengths (related to the mean rotational cycle length of27.4 days), and the regression line is also given in the figure.The correlation coefficient between the rotational cycle lengthsand their corresponding regression values is −0.235, whichis statistically significant at the 95% confidence level. Thus,a secular trend is also found to be statistically prominent inHeristchi & Mouradian’s (2008) results. The regression line

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The Astrophysical Journal, 730:49 (5pp), 2011 March 20 Li et al.

1850 1900 1950 2000−2

−1.5

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Heristchi and MouradianSunspot Number

Figure 5. Temporal sidereal sunspot rotational cycle lengths (the asterisks) ofcycles 9–23 from the years 1849.449 to 2004.112, originally given by Heristchi& Mouradian (2008). The solid line is their linear regression, while the dashedline is the linear regression of the rotational cycle lengths given by us in thepresent study for the same time interval as that of Heristchi & Mouradian.

indicates that the Sun’s rotational cycle length decreases byabout 0.66 days during the time interval considered. A linearregression is performed again to fit the temporal variationsof the rotational period length inferred above to the wavelettransformation analysis (shown in Figure 4), but in the sameinterval as Heristchi & Mouradian’s (2008); then, the regressionline shows a decrease in the rotational cycle length by about0.51 days during the time interval considered.

Brajsa et al. (2006) used the extended Greenwich data setconsisting of positions of sunspot groups to investigate cycle-related variations of solar rotation in the years 1874–1981. Theyfound a secular deceleration of solar rotation in the time intervaland a higher than average rotation velocity in the minimumof activity. Then, they gave a possible interpretation: whenmagnetic fields are weaker, one can expect a more pronounceddifferential rotation, yielding a higher rotation velocity at lowlatitudes on average.

As Brajsa et al. (2006) did, in the following analysis weinvestigate the cycle-related variation of the solar rotationrate. The rotational cycle lengths obtained above are averagedwithin the time interval considered within the same solarcycle phase relative to the nearest preceding sunspot minimum.Consequently, Figure 6 shows the dependence of the rotationalcycle lengths on the phase of the solar cycle relative to the nearestpreceding sunspot minimum, as well as their correspondingstandard errors. Different solar cycles have different periodlengths; thus, also shown in the figure is the dependence ofthe rotational cycle lengths on the phase of the solar cyclerelative to the sunspot maximum, as well as their correspondingstandard errors. As the figure displays, a lower than averagerotation velocity should seemingly appear around the maximumtime of activity, and around the minimum time the rotationvelocity is very close to the average. However, due to largestandard errors shown in the figure, we cannot determinewhether the rotation rate around the maximum time of theSchwable cycle should be faster or slower than that aroundthe minimum time, seemingly implying no Schwable cyclein the long-term variations of the rotational cycle length. Thewavelet transformation analysis of the temporal variations of therotational cycle length indicates that there is indeed no Schwablecycle in the long-term variations.

−6 −4 −2 0 2 4−1

−0.5

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le L

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ays)

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le L

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ays)

Figure 6. Dependence of the period length (the solid lines) of the rotation cycleon the phase of the solar cycle, relative to the nearest preceding sunspot minimum(the bottom panel) and the sunspot maximum (the top panel), respectively. Thedashed lines show their corresponding standard errors.

1860 1880 1900 1920 1940 1960 1980 2000 202020

30

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spot

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← Rotational Cycle Length

← Sunspot Number

Rotational C

ycle Length

1860 1880 1900 1920 1940 1960 1980 2000 202027.2

27.3

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Figure 7. Secular trend in the temporal variations of the Sun’s rotational cyclelength (right coordinate) and the trend in daily SNs (left coordinate).

4. EMD ANALYSIS OF THE TEMPORAL VARIATIONS OFTHE ROTATIONAL CYCLE LENGTH

The EMD (Huang et al. 1998) is a nonlinear time–frequencyanalysis method that decomposes a data set into a finite and oftensmall number of intrinsic mode functions (IMFs; for details,please see Gao et al. 2011). Here, the EMD method is used toextract the general trend in the above temporal variations of theSun’s rotational cycle length. The obtained result, displayed inFigure 7, is the so-called secular trend. As the figure shows, theSun indeed displays a secular acceleration of the rotation ratefrom the beginning of the time interval considered to about 1965,the last year of cycle 19, but since then a secular decelerationof the rotation rate has appeared at the solar surface, althoughits rotational cycle length is still less than the mean value of27.4 days.

The general trend in daily SNs is also picked up through theEMD analysis, which is also given in the figure. The relationbetween the two kinds of trend is very close, which is indicatedin Figure 8. As Figure 7 displays, the general trend of daily SNs(Trend I) seems to lag that of the rotational cycle lengths (TrendII), because Trend I peaks around the year 1980, but Trend II

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The Astrophysical Journal, 730:49 (5pp), 2011 March 20 Li et al.

27.2 27.3 27.4 27.5 27.6 27.7 27.8 27.9

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Figure 8. Relationship between the secular trend in the daily SN (Trend I) and the temporal variations of the rotational cycle length (Trend II), when there is no shiftbetween Trends I and II (the thin line) and when Trend II is shifted forward 9.91 years with respect to Trend I (the bold line).

0 2 4 6 8 10 12 14 16−0.998

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Figure 9. Cross-correlation coefficient between Trends I and II. The abscissaindicates the shift of Trend II with respect to Trend I along the calendar timeaxis, with positive values representing forward shifts.

reaches its minimum in 1965. Figure 9 shows the result of thecross-correlation coefficient (ccc) between them. In the figure,the abscissa indicates the shift of Trend II with respect to TrendI along the calendar time axis, with positive values representingforward shifts. Even if there is no shift between the two, theabsolute ccc is still as high as 0.980, indicating that the twoshould be highly correlated. When Trend II is shifted forward

9.91 years, the absolute value of ccc peaks at 0.9975, and at thistime the relation between Trends I and II is also indicated inFigure 8.

5. CONCLUSIONS AND DISCUSSION

Daily SNs from 1849 January 1 to 2010 February 28 havebeen utilized to study the long-term variations of the solarrotational cycle length; the rotational cycle length of the Sunis found to statistically decrease by 0.47 days in the long runduring the time interval considered, while the daily SN itselfalso shows a linear increase of about 31.4. The EMD analysisindicates that there is a trend for both the daily SN and the dailyrotational cycle length. Sunspot activity displays a trend, calledTrend I: a secular increase of SNs occurs from the beginningof the time interval considered to about 1980, but since then asecular decrease has appeared. Solar rotation also shows a trend,called Trend II: a secular acceleration of the rotation rate occursfrom the beginning of the time interval considered to about1965, but since then a secular deceleration has appeared at thesolar surface. There is a very close relationship between the twotrends. On the one hand, based on the dynamo theory of Babcock(1961), the increase in the solar rotation rate should give riseto the enhancement of sunspot activity. Thus, the secular trendin sunspot activity is inferred, probably caused by the long-term variations of the solar rotational cycle length. On the otherhand, it is found that the surface torsional pattern and perhapsthe magnetic activity as well are only the shadows of anotherunknown phenomenon occurring within the convection zone

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The Astrophysical Journal, 730:49 (5pp), 2011 March 20 Li et al.

(Snodgrass 1987; Li et al. 2008). Are the two secular trendsonly the shadow of the same phenomenon? It is an open issuefor the relation between solar rotation and activity.

Heristchi & Mouradian (2008) investigated the rotation rate ofsolar activity from a global point of view, considering daily SNsof cycles 9–23 from the years 1849.449–2004.112. They deter-mined the rotational rate variations inside a Schwable cycle bytwo-year sequences and the comprehensive study of the cycle.Their results are used here again and also indicate that the Sun’srotational cycle length presents a secular trend, linearly decreas-ing by about 0.66 days during the years 1849.449–2004.112. Ifthe temporal variations of the solar rotational cycle lengths, in-ferred here from the wavelet transformation analysis, are used toperform a linear regression at the same time interval as Heristchi& Mouradian’s, the Sun’s rotational cycle length is found to lin-early decrease by about 0.51 days in the time interval. Thedifference of the obtained values, 0.51 and 0.66 days, is in-ferred to be caused by the different analysis methods used. Dataare divided into 730-day sequences by Heristchi & Mouradian(2008), and each sequence gives a rotation rate through the fastFourier transform and the maximum entropy analysis method.

We do not know the reason why the same data set, theextended Greenwich data set, gives two opposite long-termtrends for the solar surface rotation rate: a secular decelerationfound by Javaraiah et al. (2005a, 2005b), Brajsa et al. (2004,2006), etc., and a secular acceleration found by Heristchi &Mouradian (2008), this paper, and so on. It is possibly related tothe following aspects: (1) rotation, not the differential rotationconsidered by Heristchi & Mouradian (2008) and this paper, butthe differential rotation considered by Javaraiah et al. (2005a,2005b) and Brajsa et al. (2004, 2006); and (2) different data-analysis methods are used. Time–frequency analysis is usedto determine the rotation cycle by Heristchi & Mouradian(2008) and this paper, which is mainly caused by large-scalesolar active regions or active longitudes (Usoskin et al. 2005).Rotation rate is determined through temporal variation of solarsurface tracers’ positions by Javaraiah et al. (2005a, 2005b)and Brajsa et al. (2004, 2006). As pointed out by Brajsa et al.(2006), the main problem of interpreting the results obtainedby various tracers is whether the observed changes representa global variation of the solar rotation or can be caused bysome specific property of the tracer used (e.g., a consequence ofsampling effects of different tracer subtypes during the cycle).The relation between the rotation of the Sun and sunspot activityis complex; further study is needed in the future.

The cycle-related variation of the rotation rate has beenstatistically investigated, and a lower than average rotationvelocity is found to seemingly appear around the maximumtime of activity; around the minimum time, the rotation velocityis seemingly very close to the average. However, due to largestandard errors in temporal variations of the rotation rateaveraged within a Schwable cycle, we cannot determine whetherthe rotation rate around the maximum times of the Schwablecycles should be faster or slower than that around the minimumtimes, seemingly implying no Schwable cycle in the long-termvariations of rotation. The wavelet transformation analysis of thetemporal variations of the rotational cycle length indicates thatthere is indeed no Schwable cycle in the long-term variations.

The EMD analysis shows that there seems to be a phase lagbetween Trends I and II. However, when Trend II reaches itsminimum at the end of cycle 19, the SN begins to progressfrom the maximum activity cycle, cycle 19, to a very low

activity cycle, seemingly indicating no phase lag. The phase lagprobably should be caused by the EMD analysis method, whichmakes Trend I too glossy due to the edge effect of the EMDmethod itself. Heristchi & Mouradian (2008) found that the solarrotation unfolding during a Schwable cycle is complex, withoutany systematic pattern. Here, it is inferred that the variation ofthe rotational cycle length may be related to the variation ofsunspot activity in the long run.

We thank the anonymous referees for their careful readingof the manuscript and constructive comments, which improvedthe original version of the manuscript. Data used here are alldownloaded from Web sites. The authors express their deepthanks to the staffs of these Web sites. The work is supportedby the NSFC under grants 10873032, 10921303, 11073010,and 40636031, the National Key Research Science Foundation(2011CB811406), and the Chinese Academy of Sciences.

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