Temporal evolution of statistical features of the sunspot cycles

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Advances in Space Research 50 (2012) 669–675 www.elsevier.com/locate/asr

Temporal evolution of statistical features of the sunspot cycles Maxim Ogurtsov a,b,⇑, Hogne Jungner c a

Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194 021, Polytechnicheskaya 26, Laboratory of Cosmic rays, St. Petersburg, Russia b Central Astronomical Observatory at Pulkovo, 196140, Pulkovskoje ave, 65/2, Laboratory of Space weather problems, St. Petersburg, Russia c The University of Helsinki, Dating Laboratory, P.O. Box 64, Gustaf Ha¨llstro¨minkatu 2, FI-00014, Helsinki, Finland Received 11 January 2011; received in revised form 25 March 2011; accepted 29 March 2011 Available online 2 April 2011

Abstract Analysis of the general statistical features of the sunspot cycles in the period 1700–1996 AD, including the Gnevyshev–Ohl rule, Waldmeier rule and an amplitude–period effect, was performed for both Wolf numbers and group sunspot numbers. It was shown that for both solar indices all the statistical effects are weaker over the time interval 1700–1855 AD than over the time interval 1856–1996 AD. Possible causes of this difference are discussed. Ó 2011 Published by Elsevier Ltd. on behalf of COSPAR. Keywords: Solar variability; Solar magnetic field

1. Introduction Solar magnetic activity has important consequences for life on earth. Presence of a link between Sun’s activity and processes in the Earth’s atmosphere and the geospace has been reliably established. Thus increase of our knowledge about solar variability is not only of theoretical interest but is also of importance for daily life. Sunspots are the outstanding visible proofs of the magnetic activity of the Sun. The 11-year cyclicity is the most prominent feature of sunspot activity. It is known, that sunspot cycles exhibit several significant statistical features, including an even– odd or Gnevyshev–Ohl rule, the so called Waldmeier effect and an amplitude–period effect. Analysis of these statistical effects still attracts attention of many researchers (see e.g. Nagovitsyn et al., 2009; Hathaway, 2010; Dikpati et al., 2010). Hathaway et al. (2002) showed that the statistical features, stated above, are confirmed throughout 1755– 1996 (cycles 1–22 in Zurich numbering) in both Wolf ⇑ Corresponding author at: Ioffe Physico-Technical Institute of Russian Academy of Sciences, St. Petersburg, Russia. Tel.: +7 812 292 79 62; fax: +7 812 297 10 17. E-mail address: [email protected]ffe.ru (M. Ogurtsov).

0273-1177/$36.00 Ó 2011 Published by Elsevier Ltd. on behalf of COSPAR. doi:10.1016/j.asr.2011.03.035

(Zurich) sunspot number RZ and group sunspot number RG , introduced by Hoyt and Schatten (1998). However, some differences in statistical characteristics of cycles in the solar indices RZ and RG were obtained and some characteristics were found to be weakly confirmed (see Hathaway et al., 2002; Hathaway, 2010). It should be noted that sunspot series are inhomogeneous and the reliability of different periods of both Wolf and Hoyt and Schatten series is rather different. Eddy (1976) graded the reliability of RZ into the following epochs: it is poor for 1700–1748, questionable for 1749–1817, good for 1818–1847 and reliable after 1848. Systematic uncertainties of the RG values are assessed to be about 10% before 1640, less than 5% from 1640–1728 and from 1800–1849, 15%–20% from 1728–1799, and about 1% since 1849 (Hoyt and Schatten, 1998). It is evident that in both solar series the data are much more reliable and precise after 1848 than over the preceding period 1700–1848 (cycles 4 to 9). It is therefore important to estimate how uncertainty of our knowledge about sunspot number influences the reliability of the statistical rules stated above. For this purpose it is reasonable to analyze intervals with good data and less reliable data separately. Usoskin et al. (2001) performed such analysis for the Gnevyshev–Ohl effect in the cycle intensity.

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In present work we analyzed by the same way the other five statistical effects. We selected the time intervals 1700–1855 (less reliable data, cycles 4 to 9) and 1856–2008 (good data, cycles 10–23) and examined the statistical rules over the two periods separately. The both time intervals include 14 cycles. 2. General statistical features of sunspot cycles The Gnevyshev–Ohl (even–odd) rule (Gnevyshev and Ohl, 1948, Vitinsky et al., 1986) has a few statements. The most known are the following: (a) The amplitude (maximum value over the cycle) of an odd-numbered cycle 2N + 1 is larger than the amplitude of the preceding even-numbered cycle 2N. Hereafter we call this the GO(1) rule. (b) The amplitude of an odd-numbered solar cycle 2N + 1 correlates with the amplitude of the even cycle 2N (Hathaway et al., 2002). We call this the GO(2) rule. (c) The total sum of the number of sunspots during a cycle 2N (intensity of the cycle 2N) has a good correlation with the intensity of the odd-numbered cycle 2N + 1 while correlation between corresponding intensities of the cycles 2N  1 and 2N is weak. This is the original formulation of Gnevyshev and Ohl (1948). We call this the GO(3) rule. The Waldmeier relation (Waldmeier, 1935) links the length of the ascending phase of a cycle s to maximum value of sunspot number. For 23 cycles in the Wolf number record the Waldmeier rule leads to the expression: 1=2 s ¼ ð45  12ÞRZ;max ;

When we evaluate the dependence of the accuracy of the above stated empirical relations on the precision of sunspot data, it is reasonable to use the amplitudes RZ,max and RG,max as arguments in the functions describing the Waldmeier and the amplitude–period relations. Thus, in present work we considered functions s = f(Rmax), and Dt = f(Rmax) following the approach of Veselovski and Tarsina (2002). The main parameters, needed for our analysis, including amplitudes of cycles (Rmax), intensities of cycles Rcycle R, durations of cycles (Dt), durations of ascending phases (s), are summarized in Tables 1A and 2A in Appendix. The majority of Wolf number parameters (cycles 4 to 23) are taken from ftp://ftp.ngdc.noaa.gov/ STP/SOLAR_DATA/SUNSPOT_NUMBERS/docs/maxmin.new. The parameters of group sunspot numbers were obtained using 13 point average for monthly data for cycles 0–22, and using annual data for the cycles 4 to 1. The length of group sunspot number cycles, determined by us, is close to the corresponding values, determined by Usoskin and Mursula (2003) for the cycles 6–22 (standard deviation between the two data sets is only 0.3 year). For the cycles 4 to 5 the standard deviation is higher (1.3 year). For some of the individual early cycles (cycle 5, cycle 1) the difference can reach 2 years.

ð1Þ

according to (Veselovski and Tarsina, 2002), where RZ,max is the maximum Wolf number during a cycle (cycle amplitude) and s the duration of the ascending phase in years (period from minimum to reach the maximum). The amplitude–period relation is characterized by: (a) A negative correlation between the amplitude of the solar cycle Rmax and its length Dt. This relation was mentioned in the works of Dicke (1978) and Friis-Christensen and Lassen (1991). Hereafter we call this effect AP(1). The Dt value is defined as the time interval between subsequent solar cycle minima. (b) A negative correlation between the amplitude of the solar cycle N and the length of the previous cycle N  1. This link has been reported by (Chernosky, 1954; Wilson et al, 1998). We call this effect AP(2).

Fig. 1. A – Wolf sunspot number, B – group sunspot number after Hoyt and Schatten (1998). Violation of Gnevyshev–Ohl rule are marked with arrows.

Table 1 Statistical features of solar cycles through 1755–1996 AD. Coefficient of correlation between X and Y over the time interval AD 1755–1996 (cycles 1–22) Wolf number RZ

Group sunspot number RG

Effect

X

Y

Present work

Hathaway et al. (2002)

Present work

Hathaway et al. (2002)

GO(2) Waldmeier AP(1) AP(2)

R2Nþ1 max sN DtN DtN1

R2N max RNmax RNmax RNmax

0.45 0.82 0.37 0.67

0.47 0.73 0.27 0.69

0.62 0.32 0.30 0.30

0.61 0.34 0.27 0.49

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Fig. 2. Different correlations: A, B – Gnevyshev–Ohl (2); C, D – Gnevyshev–Ohl (3); E, F – Waldmeier; G, H – amplitude–period (1); I, J – amplitudeperiod (2). Empty circles correspond to the time interval AD 1700–1855 (cycles 4 to 9). Filled squares correspond to the time interval AD 1856–2008 (cycles 10–23) for Wolf number and AD 1856–1996 (cycles 10–22) for group sunspot number. Left panel describes statistical effects in Zurich Wolf number. Right panel the same effects in group sunspot number. A standard deviation of 1.0 (1r corridor) is shown with dotted lines.

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3. Comparison of Wolf and group sunspot cycle characteristics

are 37.5 during 1700–855 and 30.1 during 1856–2008 (25.6 during 1856–1996) for Wolf number, and 23.6 during 1700–1855 and 13.8 during 1856–996 for group sunspot number. We calculated values e during the time interval 1856–1996 for Wolf numbers in order to compare them with the corresponding values, calculated for group sunspot numbers. In Fig. 2C and D the dependence of the sum of Wolf number and group sunspot number over the cycle 2N + 1 on the corresponding sum over the previous cycle 2N are shown. The corresponding linear relations are: X X R2N RZ2N þ1 ¼ ð307:93  162:52Þ þ 0:49 Z : X X ð3Þ 2N þ1 RG ¼ ð130:91  157:03Þ þ 0:79 R2N G ;

We first calculated coefficients of linear correlation for four effects over the time interval AD 1755–1996 (cycles 1–22) and compared the obtained values with the results of Hathaway et al. (2002). The correlation coefficients are shown in Table 1. As can be seen from Table 1 the coefficients of correlation calculated in the present work are close to that calculated by Hathaway et al. (2002). The small differences are most likely a result of different timing of maxima and minima for each cycle. It is important to note, that in spite of not fully identical ways of estimating the cycle parameters the results are rather similar. We then tested the relationship for the time intervals 1700–1855 and 1856–2008 separately. First we examined the GO(1) effect. In Fig. 1 yearly average of both solar indices RZ and RG are plotted and violations of GO(1) rule (cycle 2N is more intense than cycle 2N + 1) are shown with arrows. As can be seen from Fig. 1A in Wolf numbers GO(1) rule is violated for three pairs of cycles for 1700–1855 (cycles 4 to 9) while there is one violation for 1856–2008 (cycles 10–23). Only the unusual cycle 23 violates the GO(1) rule during the last 150 years (see e.g. Nagovitsyn et al., 2009). In group sunspot numbers (Fig. 1B) GO(1) rule is not valid for 4 pairs of cycles of 7 for 1700–1855 (cycles 4 to 9) and there are no violations for 18561996 (cycles 10– 22). That means that GO(1) rule works almost perfectly throughout 18562008 but does not work properly during the preceding epoch. In Fig. 2A and B the dependence of the maximum Wolf number and group sunspot number during cycle 2N + 1 on the corresponding maximum during the previous cycle 2N are shown. The dependence could be described by a linear relationship over the entire time interval 1700–2008 (28 cycles) for Wolf numbers and 1700–1996 (27 cycles) for group sunspot numbers. The corresponding expressions, with including 1r uncertainty, are:

The correlation coefficient of the non-linear correlation is 0.87 throughout all the 28 cycles. But for cycles 1023 (1856–2008) the dependence (4) is more clear than for cycles 4 to 9 (1700–1855). The corresponding standard deviations are 0.44 (0.43 for 1856–1996) and 0.80. In the dataset of Hoyt and Schatten the link between the length of ascending phase and cycle maximum is less evident. In that case we can use a linear expression for the dependence:

RZ2N þ1 ¼ ð58:54  37:51Þ þ 0:54R2N Z ;

s ¼ ð5:84  1:24Þ  0:015RG;max :

RG2N þ1

¼ ð24:10  30:61Þ þ

0:86R2N G :

ð2Þ

Coefficients of linear correlation throughout the entire time interval are 0.47 for Wolf number and 0.71 for group sunspot number. Standard deviations (e) from linear fits

Coefficients of linear correlation throughout the entire time interval are 0.51 for Wolf number and 0.71 for group sunspot number. Standard deviations from linear fits are 170.3 during 1700–1855 and 125.9 during 1856–2008 (130.8 during 1856–1996) for Wolf number, and 165.4 during 1700–1855 and 70.6 during 1856–1996 for group sunspot number. Waldmeier rule for solar indices RZ and RG is illustrated in Fig. 2E and F. Fig. 2E shows that in Wolf number the length of ascending phase depends on cycle amplitude in a nonlinear way: 1=2

s ¼ ð57:08  7:02ÞRZ;max  1:23:

ð4Þ

ð5Þ

The correlation coefficient for the linear correlation between rise time and cycle amplitude is 0.44 for group sunspot numbers. The standard deviations are 1.66 (1700–1855) and 0.83 (1856–1996) respectively.

Table 2 General statistical features of cycles in solar indices RG and RZ after 1700 AD. Standard deviation from the obtained fit RZ

Gnevyshev–Ohl Rule (2) Gnevyshev–Ohl Rule (3) Waldmeier rule Amplitude–period effect (1) Amplitude–period effect (2)

RG

Cycles 4 to 9 (1700–1855)

Cycles 10–22 (1856–1996)

Cycles 10–23 (1856–2008)

Cycles 4 to 9 (1700–1855)

Cycles 10–22 (1856–1996)

37.5 170.3 0.80 1.40 1.05

25.6 130.8 0.43 0.87 0.91

30.1 125.9 0.44 0.93 0.93

23.6 165.4 1.66 1.89 1.88

13.8 70.6 0.83 0.78 0.72

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Table 1A Main characteristics of 11-year solar cycles based on the data on Wolf number RZ. Cycle number

Time of minimum

Time of maximum

Rmax Z

sRZ

DtRZ

P

4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

1698.0 1712.0 1723.5 1734.0 1745.0 1755.2 1766.5 1775.5 1784.7 1798.3 1810.6 1823.3 1833.9 1843.5 1856.0 1867.2 1878.9 1889.6 1901.7 1913.6 1923.6 1933.8 1944.2 1954.3 1964.9 1976.5 1986.8 1996.7

1705.5 1718.2 1727.5 1738.7 1750.3 1761.5 1769.7 1778.4 1788.1 1805.2 1816.4 1829.9 1837.2 1848.1 1860.1 1870.6 1883.9 1894.1 1907.0 1917.6 1928.4 1937.4 1947.5 1957.9 1968.9 1979.9 1989.6 2000.3

58.0 63.0 122.0 111.0 92.6 86.5 115.8 158.5 141.2 49.2 48.7 71.7 146.9 131.6 97.9 140.5 74.6 87.9 64.2 105.4 78.1 119.2 151.8 201.3 110.6 165.5 158.5 119.8

7.5 6.2 4.0 4.7 5.3 6.3 3.2 2.9 3.4 6.9 5.8 6.6 3.3 4.6 4.1 3.4 5.0 4.5 5.3 4.0 4.8 3.6 3.3 3.6 4.0 3.4 2.8 3.6

14.0 11.5 10.5 11.0 10.2 11.3 9.0 9.2 13.6 12.3 12.7 10.6 9.6 12.5 11.2 11.7 10.7 12.1 11.9 10.0 10.2 10.4 10.1 10.6 11.6 10.3 9.9 12.5

219 336 535 567 445.3 468 535 617 840.2 281.9 236.8 396.9 655 691.9 548.5 618.4 383.3 461.8 372.4 445.9 410.3 609.1 751.5 970.7 692.8 830.1 780.4 676.4

Table 2A Main characteristics of 11–year solar cycles based on data on group sunspot number RG.

cycleRZ

Cycle number

Time of minimum

Time of maximum

Rmax G

DtRG

sRG

P

4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

1698.0 1712.0 1723.0 1733.0 1745.2 1755.5 1766.3 1775.5 1784.2 1799.9 1810.1 1823.3 1834.2 1843.6 1856.0 1867.4 1878.8 1889.8 1902.1 1913.6 1923.7 1934.0 1944.3 1954.3 1964.5 1976.5 1986.3

1705.0 1719.0 1728.0 1736.0 1750.3 1761.6 1769. 7 1778.6 1788.5 1801.6 1816.7 1829.8 1837.1 1848.9 1860.7 1870.9 1884.2 1894.1 1906.1 1917.6 1928.6 1937.3 1947.6 1958.3 1970.5 1979.6 1991.2

5.5 33.9 64.2 48.6 64.1 65.6 103.5 71.8 89.1 51.1 32.6 64.7 116.0 91.0 85.3 101.2 67.2 96.6 64.6 109.7 81.2 123.6 143.5 186.1 108.9 154.7 156.2

14.0 11.0 10.0 12.2 10.3 10.8 9.2 8.7 15.7 10.2 13.2 10.9 9.4 12.4 11.4 11.4 11 12.3 11.5 10.1 10.3 10.3 10.0 10.2 12.0 9.8 9.8

7.0 7.0 5.0 3.0 5.1 6.1 3.4 3.1 4.3 1.7 6.6 6.5 2.9 5.3 4.7 3.5 5.4 4.3 4.0 4.0 4.9 3.3 3.3 4.0 6.0 3.1 4.9

27.1 134.5 277 251.8 282.3 359.3 510.7 390.1 599.6 186.1 159.5 341.4 510.1 536.3 462.1 476.9 334.7 484.1 382.7 495.8 474.5 643.7 716.2 873.7 704.3 831.5 764.1

cycleRG

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The effect AP(1) is illustrated in Fig. 2G and H. It is described by the expressions: DtN ¼ ð12:54  1:15Þ  0:013  RNZ;max ; DtN ¼ ð12:36  1:34Þ  0:015  RNG;max :

ð6Þ

It can be seen from Fig. 2G and H that in both RZ and RG the relationship between the cycle amplitude and its length is not strong. Coefficient of linear correlation for Wolf numbers (1700–2008) is 0.41 and for group sunspot number is 0.40 (1700–1996). The standard deviation for a linear fit in Wolf number is 1.40 during 1700–1855 and 0.93 during 1856–2008 (0.87 during 1856–1996). In group sunspot number the corresponding values e are 1.88 (1700– 1855) and 0.72 (1856–1996). The relationship between the cycle amplitude and the period of the previous cycle is shown in Fig. 2I and J. It is expressed as: DtN 1 ¼ ð13:35  0:97Þ  0:02  RNZ;max ; DtN 1 ¼ ð12:35  1:46Þ  0:014  RNG;max :

ð7Þ

The AP(2) effect is more pronounced in Wolf number as has been obtained also by Hathaway et al. (2002). The correlation coefficient between the amplitude and the period of the previous cycle is 0.63 for RZ values (1700–2008) and 0.34 for RG values (1700–1996). Standard deviation from linear fit in Wolf number is 1.05 for 1700–1855 and 0.93 for 1856–2008 (0.91 for 1856–1996). In group sunspot number the standard deviation is 1.89 (1700–1855) and 0.78 (1856–1996). Significance of the correlation coefficients r, calculated above for all the 27 cycles, was estimated: (a) by means of a number of statistical experiments including permutation of the initial data sets and (b) by theoreticalq evaluation ffiffiffiffiffiffiffi n2 using Student t-distribution of the value t ¼ r 1r 2 with n2 degrees of freedom. In all cases the significance was found to be more than 0.90 (>0.999 for Waldmeier effect in Wolf numbers). Table 2 summarizes the results of the performed analysis. One can see from Fig. 2 and Table 2 that for every case fits to the data show less scatter for the period 1856–2008 (or 1856–1996) than for 1700–1855. Throughout 1856– 1996 all the statistical relations, besides the Waldmeier rule, are manifested in Group sunspot number better than in Wolf number. During the previous epoch 1700–1855 the situation is opposite – the analyzed statistical features in RG cycles are less observable. 4. Discussion and conclusion We examined six statistical features in 27 cycles of Wolf number and group sunspot number records. We found that all the effects, including the even–odd, amplitude–period and Waldmeier relations, are significant at more than 0.90 confidence level. We obtained that over 1856–1996 all the statistical relationships are manifested

appreciably stronger than over 1700–1855 both in RZ an RG. Two possible reasons for the difference of the statistical effects through the two periods can be mentioned. The first one is uncertainty and less reliability of RZ an RG data during the earlier epoch. In that case the fact that the characteristics of solar cycles are more pronounced over the later time interval 1857–1996 can be a result of increased precision of the sunspot data. Another possibility is connected with possible temporal evolution of the solar statistical features. According to Nagovitsyn (1997) the Gnevyshev–Ohl effect is governed by the 900 year cycle. It changes with time and was weaker during 1670–1840 than during 1800–1990. In that case time variation of statistical characteristics of the solar cycles could carry important information about the corresponding changes in the solar dynamo. E.g. Karak and Choudhuri (2011) showed that the Waldmeier effect can be explained on the basis of flux transport dynamo models for sunspot cycles in which it arises from fluctuations in the meridional circulation. That means that weakening of the Waldmeier effect through 1700– 1855 might be a result of decrease in meridional circulation during this epoch. It should be noted that Usoskin et al. (2001,2009) proposed that the longest cycle 4 actually consisted of two cycles. The existence of an additional short cycle AD 1793–1800 can not be excluded since the information about sunspot activity through 1790–1794 is very uncertain. Including the new cycle changes the statistical features of the sunspot cycles. This cycle modifies the GO(3) rule, making it consistently valid before the Dalton minimum (Usoskin et al., 2001; Nagovitsyn et al., 2009). However the assumption about the lost solar cycle violates the Waldmeier relation (Yakovchuk and Veselovski, 2006).The hypothesis of Usoskin et al. (2001) has also been disputed by Krivova et al. (2002), and Zolotova and Ponyavin (2007). Thus, further research is necessary for more decisive conclusions. But in any case, the uncertainty in solar cycle data from older periods should be kept in mind when long term activity of the sun is discussed.

Acknowledgements M.G. Ogurtsov expresses his thanks to the exchange program between the Russian and Finnish Academies (project No. 16), program “Origin, Structure, and Evolution of Objects of the Universe” of the Presidium of Russian Academy of Sciences, program of the SaintPetersburg Scientific Center of RAS for 2011, RFBR, Grants No. 07-02-00379, 09-02-00083, 10-05-00129, 1102-00755 for financial support. H. Jungner is a member of the Millennium project (Contract No 017008 GOCE). Authors are grateful to the anonymous referees and to editor P.A. Shea for constructive comments on an earlier version of the paper, which greatly helped to improve it.

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