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Solar cycle characteristics and their application in the prediction of cycle 25
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Solar cycle characteristics and their application in the prediction of cycle 25 F.Y. Lia,b,c,d,∗, D.F. Konga,c,d, J.L. Xiea,c,d, N.B. Xianga,c,d, J.C. Xua,b,c,d a
Yunnan Observatories, Chinese Academy of Sciences, Kunming, 650011, China University of Chinese Academy of Sciences, Beijing, 100049, China State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing, 100190, China d Key Laboratory of Solar Activity, National Astronomical Observatories, CAS, Beijing, 100012, China b c
A R T I C LE I N FO
A B S T R A C T
Keywords: sunspots Methods: statistical
The Solar Influences Data Analysis Center (SIDC) issued a new version (version 2) of the sunspot number data in July 2015. The 13-month smoothed monthly sunspot number from the new version is used for the first time to research the relations among the feature parameters of solar cycles under the bimodal distribution for the modern era cycles (10–23), and, their physical implications are discussed. These relations are utilized to predict the maximum amplitude of solar cycle 25. Cycle 25 is predicted to start in October 2020 and reach its maximum amplitude of 168.5 ± 16.3 in October 2024, thus, it should be stronger than cycle 24 but weaker than cycle 23.
1. Introduction The solar activity rising and falling within an 11-year cycle causes changes in space weather, which greatly influences space weather analysis, operations of low-Earth-orbiting satellites, and so on. A key indicator of solar activity is the sunspot number (Hathaway, 2015; Wilson, 1988). To understand solar cycles and predict solar activity, many characteristics of the solar cycles have been studied (Hathaway et al., 2002; Aparicio et al., 2012; Carrasco et al., 2016), based on the Wolf sunspot number, such as, the Waldemeier effect (Waldmeier, 1935) and the Even-Odd effect (Gnevyshev, 1948). However, the International Sunspot Number and Group Number do not match on various aspects, inducing confusions and contradictions. To resolve these problems, a new revised collection of sunspot group numbers was presented by Vaquero et al. (2016) and several series of the sunspot number have been presented (Clette and Lefèvre, 2016; Lockwood et al., 2016; Svalgaard and Schatten, 2016; Usoskin et al., 2016; Chatzistergos et al., 2017). In addition, a new revised version of the sunspot number (version 2), issued by the Solar Influences Data Analysis Center (SIDC) in 2015 (Clette et al., 2014; Ahluwalia and Ygbuhay, 2016), is chosen to be used in this study. Reviewing the solar cycle characteristics using this new version of data is necessary. Solar activity prediction is becoming increasingly important because solar activities influence the atmosphere (Pulkkinen, 2007), and thus affect the orbital lifetime of an increasing number of spacecraft, including manned space stations. Several methods and technologies have been proposed to predict solar cycles, such as precursor methods
∗
and statistical methods (Petrovay, 2010; Yoshida and Sayre, 2012; Ahluwalia and Jackiewicz, 2012). Precursor methods, which use the polar fields or geomagnetic indices, are more reliable than others using the sunspot number (Hathaway et al., 1999; Li et al., 2001; Pesnell, 2012) for cycles 21 and 22. However, some statistical methods have provided more reliable predictions for cycle 23 (Li et al., 2001; Wilson et al., 1998; Petrovay, 2010), whereas most of the precursor methods have overestimated the amplitude. Nevertheless, extrapolation methods and model-based methods occasionally show advantages (Pesnell, 2014). These former studies use the old version of sunspot number data (version 1) to make predictions for the solar cycle. In this study, the new version of sunspot number data is employed to predict the maximum amplitude of solar cycle 25 for the first time. The current observations suggest that cycle 24 should have already reached its maxima (Li et al., 2011) at 116.4 in April 2014. Therefore, now is the appropriate time to predict when cycle 25 will start and how strong it will be. In Section 2, characteristics of the modern era solar cycles and classical correlations among feature parameters are examined based on the new sunspot number index, and the re-examined relations are used to predict the maximum amplitude of cycle 25 and its timing. 2. Cycle characteristics and predictions 2.1. Data The 13-month smoothed monthly sunspot number of the new
Corresponding author. Yunnan Observatories, Chinese Academy of Sciences, Kunming, 650011, China. E-mail address: [email protected] (F.Y. Li).
https://doi.org/10.1016/j.jastp.2018.10.014 Received 21 November 2017; Received in revised form 15 October 2018; Accepted 19 October 2018 1364-6826/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Li, F.Y., Journal of Atmospheric and Solar-Terrestrial Physics, https://doi.org/10.1016/j.jastp.2018.10.014
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Table 1 Parameters of cycles 10–24 based on the new sunspot-number index: year of minimum, ascent duration, year of maximum, amplitude, descent duration, cycle length, and max-max cycle length. Cycle
Year of
Ascent
Year of
Number
Minimum
Duration
(Yrs) 1856.0 1867.2 1879.0 1890.2 1902.0 1913.6 1923.6 1933.7 1944.1 1954.3 1964.8 1976.2 1986.7 1996.9 2009.0
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Amplitude
Descent
Cycle
Max-Max
Maximum
Duration
Length
Cycle
(Yrs)
(Yrs)
(Yrs)
(Yrs)
Length (Yrs)
4.2 3.4 5.0 3.8 4.1 4.0 4.7 3.6 3.2 3.9 4.1 3.8 3.2 5.0 5.3
1860.1 1870.6 1884.0 1894.0 1906.1 1917.6 1928.3 1937.3 1947.4 1958.2 1968.9 1980.0 1989.9 2001.9 2014.3
7.1 8.3 6.2 8.0 7.5 6.0 5.4 6.8 6.9 6.6 7.3 6.7 7.0 7.1
11.2 11.8 11.2 11.8 11.5 10.0 10.1 10.4 10.2 10.5 11.4 10.5 10.2 12.0
10.5 13.3 10.1 12.1 11.5 10.7 9.0 10.1 10.8 10.7 11.1 9.9 12.0 12.4
186.2 234.0 124.4 146.5 107.1 175.7 130.2 198.6 218.7 285.0 156.6 232.9 212.5 180.3 116.4
sunspot number data (version 2) (Clette and Lefèvre, 2016) issued by the Solar Influences Data Analysis Center (SIDC) is used in this study. This version can be downloaded from the Sunspot Index and Long-term Solar Observations (SILSO) website (http://sidc.oma.be/silso/). The socalled modern era sunspot cycles refer to cycle 10 and thereafter, as the sunspots have been observed every day since the start of cycle 10 (Wilson, 1987a; Vaquero, 2007). Table 1 lists the parameters of modern era solar cycles 10–24 in columns 2–8: year of minimum, ascent duration, year of maximum, amplitude, descent duration, cycle length, and max-max cycle length, respectively. The ascent duration is the time the number of sunspots rise from minimum to maximum, whereas the descent duration is the time the number of sunspots decrease from maximum to minimum. The amplitude is the maximum smoothed monthly sunspot number (version 2) of a cycle. Therefore, the actual value of the amplitude of a cycle depends on the activity index used and the type of smoothing. The cycle length of a sunspot cycle is defined as the elapsed time from the minimum preceding its maximum to the minimum of the following cycle (Hathaway et al., 2002), and the max-max cycle length is the elapsed time from the maximum of a solar cycle to the maximum of the following cycle (Du, 2006).
Fig. 1. Period length of cycles 10–23, showing the ‘bimodality of the solar cycle’. 7 long-period cycles are shown as diamonds, whereas 7 short-period cycles as circles. The two horizontal lines represent the averages, respectively.
2.2. Bimodality of the solar cycle bimodal distribution should be considered an exact physical mechanism; that is, it only provides information about a pattern in the data. According to the Babcock-Leighton (BL) type flux transport dynamos (Babcock, 1961; Leighton, 1969), the cycle length increases if the meridional flow is weak, and the increase in cycle length results in two effects. On the one hand, the differential rotations could have more time to enhance the cycle strength by producing a toroidal magnetic field. On the other hand, the diffusing effect in the magnetic field has more time to weaken the cycle strength (Jiang et al., 2015). Because the diffusing effect in the magnetic field is more apparent than the productive process of the toroidal field, a relatively longer cycle should be weaker in strength. Thus, the random variation in the meridional flow leads to the variation in the strength of a solar cycle. Based on this, the bimodal distribution in Fig. 1 implies that there should be two clusters for the toroidal field.
Fig. 1 shows a bimodal distribution (Rabin et al., 1986) for the modern era cycles. As proposed by Wilson (1987b), according to the lengths of the solar cycles, the modern era cycles can be divided into two clusters (short-period cycles and long-period cycles) with an 8month gap (the smallest distance between the two clusters) separating these two clusters. This gap is called the Wilson Gap by Hathaway (2015) that is the so-called bimodality of solar cycles. Their cycle lengths may be clustered into two groups: 7 short-period cycles and 7 long-period cycles. The mean period length is 11.64 ± 0.40 years for long cycles and 10.21 ± 0.25 years for short cycles. This may not be a completely accurate distribution due to the limited sample data. However, compared with the normal and uniform distributions, Wilson (1987b) proposed that the bimodal distribution can better describe the observed distribution of cycle lengths. Then, from the results of Li et al. (2015) obtained with version 1 of the sunspot number data, there is no significant change in these relations between versions 1 and 2. It is useful to divide these cycles into the two groups to study their different statistical relations among some feature parameters of a solar cycle in the modern era. Nevertheless, there is no evidence to suggest that the
2.3. The solar memory We analyzed the different possibilities for the relationships of cycle 2
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previous and new versions of the sunspot number data and considering different sets of solar cycles, demonstrate that the method by Du (2006) is not suitable for prediction because it is based on a non-robust relationship. Therefore, the predictions should be treated with caution. The analysis in this study regarding sunspot number data can provide information about a pattern in the data but should not be considered as exact. All the relations between amplitude and cycle length in Figs. 2 and 3 are negative correlations. There may be a memory mechanism or a time delay in solar cycles, because the characteristics of these amplitudes and periods remain and appear after a time lag. This is the so-called memory of a solar magnetic cycle, which is built into flux transport dynamo models. Hathaway et al. (2003) found the negative correlation between the period of a cycle and the drift rate of the sunspot latitude bands as evidence that a deep meridional flow sets the sunspot cycle period. Stochastic fluctuations in the parameters, accounted by the inherent turbulent nature of the dynamo mechanism and the large-scale flows, can lead to cycle amplitude and period modulation (Wang et al., 2002). The meridional flow requires a finite time to carry down the surface poloidal field to the low-latitude tachocline, where the toroidal field of the next cycle is generated (Hathaway et al., 2003; Charbonneau and Dikpati, 2000). It is difficult to determine the length of the time lag as it is a very complicated modulation process. Additionally, it is difficult to explain the reasons for the difference between results for cycles 10–23 (the modern era cycles) and for cycles 1–23. One explanation could be the poor accuracy of the sunspot number data for the early cycles. Alternatively, the different results may be a result of the complicated modulation process.
Fig. 2. Relationship between current cycle (n) length and cycle (n+3) amplitude for the modern era cycles (solar cycles 10–23), with a correlation coefficient of −0.7587, significant at the 99% confidence level.
2.4. Ascent duration versus descent duration The linear relationships between the ascent duration ( AD ) and the descent duration (DD ) are separately shown in Fig. 4 for the longperiod cycles and short-period cycles in the modern era. For the longperiod cycles, the best linear fit is,
DD = −0.804⋅AD + 10.8
(1)
and the correlation coefficient is −0.8013, significant at the 97% confidence level. The standard error of this linear fitting is 0.38 years. For
Fig. 3. Relationship between current cycle length and following cycle amplitude for cycles 1–23, with a correlation coefficient of −0.6791, significant at the 99.9% confidence level.
lengths at different lags. For modern era Cycles 10–23, as shown in Fig. 2, the most significant correlation occurs between current cycle (n) length and latter cycle (n+3) amplitude with a correlation coefficient of −0.7587, significant at the 99% confidence level. In addition, this correlation is investigated in a larger sample size, as shown in Fig. 3. For cycles 1–23, the most statistically significant correlation occurs between current cycle (n) length and latter cycle (n+1) amplitude, obtained by Hathaway et al. (2002) with a correlation coefficient of − 0.6791, significant at the 99.9% confidence level, similar to Hathaway et al. (2002) obtained from the Zurich sunspot numbers and the Group sunspot numbers. According to the above mentioned research, the correlation between length and amplitude of the solar cycle depends on the data employed. Furthermore, other authors have found different results for this amplitude-length relationship. For example, Du and Du (2006) proposed a method to predict the maximum amplitude of solar cycles 24 and 25 by employing the relationship between the maximum amplitude and the cycle length of two earlier solar cycles considering solar cycles 9–23. However, Carrasco et al. (2017a, b), applying both the
Fig. 4. The linear relationships between the ascent duration and descent duration separately for the long-period cycles (diamonds, with a correlation coefficient of −0.8013, significant at the 97% confidence level) and shortperiod cycles (circles, with a correlation coefficient of −0.8736, significant at the 98% confidence level) for modern era cycles 10–23. The thin lines are their linear regression lines. 3
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Fig. 5. Monthly sunspot number (the blue thin line) for cycle 24 from December 2008 (the minimum time of cycle 24) to August 2018 and its smoothed value (the black dashed line) to February 2018 (http://sidc.oma. be/silso/).
Fig. 6. Relationship between the descent duration (DD) and the max-max cycle length (DA) for modern era cycles 10–23 and its linear fitting regression line with a correlation coefficient of 0.825, significant at the 99.9% confidence level.
the short-period cycles, the best linear fit is,
DD = −0.916⋅AD + 9.897
DA(n) = DD(n)+AD(n+1), the correlation between the descent duration (DD (n) ) of a cycle and the ascent duration ( AD (n + 1) ) of the following cycle can be inferred from Equation (3) as
(2)
and the correlation coefficient is −0.8736, significant at the 98% confidence level. The standard error of linear fitting is 0.25 years. The negative correlation between the ascent duration and the descent duration implies that if a cycle raise quickly, it will decline slowly, and vice versa. Therefore, changes in solar cycle lengths should not be too large. The length of a solar cycle is leaded by the meridional flow in the Babcock-Leighton(BL) type flux transport dynamos (Babcock, 1961; Hathaway et al., 2003; Leighton, 1969). The meridional flow rate is always approximately 10ms−1 (Featherstone and Miesch, 2015); thus, the length of a solar cycle is stabilized approximately 11 years. This is in agreement with the negative correlation between the ascent duration and the descent duration. Fig. 5 shows the monthly sunspot number from December 2008 (the start time of cycle 24) to August 2018 and its smoothed value to February 2018. More than eight years have elapsed since the start of cycle 24, thus, it is most likely that cycle 24 has already reached its maximum (Li et al., 2011). That is, the maximum amplitude of cycle 24 is 116.4 in April 2014, and the ascent duration is 5.33 years. Based on the linear relationships between the ascent duration (AD) and the descent duration (DD), the descent duration of cycle 24 should be 6.51 ± 0.38 years if it is a long-period cycle, or 5.01 ± 0.25 years if it is a short-period cycle. Therefore, cycle 25 should begin in October 2020 (± 0.38 years) if cycle 24 is a long-period cycle, or in April 2019 (± 0.25 years) if it is a shortperiod cycle.
AD (n + 1) = 0.186⋅DD (n) + 2.826
According to the correlation in Equation (3), DA (24) should be 10.55 ± 0.69 years if DD (24) = 6.51 years, or 8.77 ± 0.69 years if DD (24) = 5.01 years. Thus, the ascent duration of cycle 25 ( AD (25) ) is calculated to be 4.04 ± 0.69 years if cycle 24 is a long-period cycle, or 3.76 ± 0.69 years if it is a short-period cycle. Therefore, cycle 25 will reach its maxima in October 2024 if cycle 24 is a long-period cycle, or in January 2023 if cycle 24 is a short-period cycle. 2.6. The modified Waldmeier Effect There is a linear relationship between the amplitude ( Am ) and rising rate (r) for cycles 10–24 (including solar cycle 24, unlike with previous correlations), which is shown in Fig. 7. This is the so-called modified Waldmeier Effect (Waldmeier, 1935; Carrasco et al., 2016; Li et al., 2017):
Am = 2.676⋅r + 56.89, r = Am / AD
(5)
where r is the ratio of the amplitude ( Am ) to ascent duration ( AD ). The correlation coefficient is 0.9432, significant at the 99.9% confidence level, and the standard error of this regression is 16.3. The rate of the rise of the sunspot number to maximum is positively correlated to the cycle amplitude. That is, the more quickly the sunspot number of a cycle rises, the larger the amplitude of the cycle is. According to this relationship, the amplitude of cycle 25 is predicted to be 168.5 ± 16.3 if cycle 24 is a long-period cycle ( AD (25) = 4.04 ) or 197.3 ± 16.3 if cycle 24 is a short-period cycle ( AD (25) = 3.76). Past studies, such as the prediction by the NASA's Marshall Space Flight Center (http:// solarscience.msfc.nasa.gov/predict.shtml) and the NOAA's Space Weather Prediction Center (http://www.swpc.noaa.gov/ products/ solar-cycle-progression), suggest that cycle 24 is a long-period cycle (Li et al., 2011; Ahluwalia, 2016), that will be completed at the end of 2019 (Hathaway and Upton, 2016) or at the beginning of 2020 (Carrasco et al., 2017a). Thus, this study makes the following prediction: cycle 24 is a long-period cycle, that will be completed in October 2020 (± 0.38 years) and cycle 25 will reach its peak at 168.5 ± 16.3 in October 2024. This peak is greater than the amplitude of cycle 24
2.5. Max-max cycle length versus decline time The max-max cycle length (DA (n) ) is the period between two successive maxima, equal to the sum of the descent duration (DD (n) ) of a cycle and the ascent duration ( AD (n + 1) ) of its next cycle (Du and Du, 2006). It is found to be correlated with the descent duration (DD (n) ) (Li et al., 2005, 2015) as follows:
DA (n) = 1.186⋅DD (n) + 2.826
(4)
(3)
which is similar to the results Du (2006) obtained using the previews version of the sunspot number data. This correlation is shown in Fig. 6, with a correlation coefficient of 0.825, significant at the 99.9% confidence level. The standard error of this regression is 0.69 years. As 4
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(Gnevyshev, 1948), but weaker than cycle 23 (Kakad et al., 2017). This prediction matches the speculation by Ahluwalia and Ygbuhay (2012) that cycle 24 may herald the onset of a Dalton-like minimum in the 21st century, and is in agreement with the inference that cycle 24 is located at the bottom of the Gleissberg cycle (Hathaway, 2015). Acknowledgments The authors are grateful to Dr. Li, K. J. for his constructive proposals and helpful suggestions. This work is supported by the National Natural Science Foundation of China (11573065, 11633008, and 11703085), the Specialized Research Fund for State Key Laboratories, and the Chinese Academy of Sciences. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jastp.2018.10.014.
Fig. 7. Correlation between the rising rate (the ratio of the amplitude to ascent duration) and the sunspot number amplitude, which is the so-called modified Waldmeier effect for modern era cycles 10–24. The solid line is the linear regression line with a correlation coefficient of 0.9351, significant at the 99.9% confidence level.
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(Steinhilber and Beer, 2013; Zharkova et al., 2015), but less than that of cycle 23, as the amplitude of cycle 23 (Am(23)) is 180.3, and the amplitude of cycle 24 (Am(24)) is 116.4. As a test, we used this method to predict solar cycle 24. The method predicts that solar cycle 24 would have reached its peak at 123.6 ± 16.3 in August 2012, which was fulfilled while the actual amplitude is 116.4. However, the maximum for Solar cycle 24 is predicted to have occurred in August 2012, almost two years before the actual occurrence in April 2014. Since these statistical relations among the characteristic parameters is based on regarding the solar cycle as a unimodal profile, this method is limited in predicting the date of the maximum amplitude, especially for bimodal cycles such as cycle 24. 3. Summary In this paper, the 13-month smoothed monthly sunspot number of the new sunspot number data (version 2) is used for the first time to study the solar cycle characteristics for modern era cycles 10–24 (Li et al., 2005). The modern era cycles can be divided into short-period cycles and long-period cycles (Rabin et al., 1986; Wilson, 1987a), showing a bimodal distribution. There is a significant negative correlation with time lag between cycle length and amplitude. We speculate that there is a memory mechanism in solar cycle (Hathaway et al., 2002). Linear relations between the ascent duration (AD) and descent duration (DD) are found for short-period cycles and long-period cycles, and these correlations result in a stable duration of a solar cycle at approximately 11 years. The max-max cycle length DA (n) (Du, 2006) is found to be correlated with the descent duration DD (n) . The correlation between the amplitude (Am) and rising rate (r = Am / AD) for cycles 10–24 is found to be positive. According to these statistical relations among the characteristic parameters of sunspot cycle profiles, cycle 25 is predicted to start in October 2020 (± 0.38 years) and reach its maximum amplitude of 168.5 ± 16.3 in October 2024 if cycle 24 is a longperiod cycle or to begin in April 2019 (± 0.25 years) and reach its maximum amplitude of 197.3 ± 16.3 in January 2023 if cycle 24 is a short-period cycle. However, past studies suggest that cycle 24 is a longperiod cycle (Li et al., 2011; Ahluwalia, 2016), therefore, we expect the former. In this paper, it is predicted that cycle 25 is more likely to be stronger than cycle 24 (Cameron et al., 2016; Hathaway and Upton, 2016), which is in agreement with the Gnevyshev-Ohl (even-odd) rule 5
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Solar cycle characteristics and their application in the prediction of cycle 25 F.Y. Lia,b,c,d,∗, D.F. Konga,c,d, J.L. Xiea,c,d, N.B. Xianga,c,d, J.C. Xua,b,c,d a
Yunnan Observatories, Chinese Academy of Sciences, Kunming, 650011, China University of Chinese Academy of Sciences, Beijing, 100049, China State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing, 100190, China d Key Laboratory of Solar Activity, National Astronomical Observatories, CAS, Beijing, 100012, China b c
A R T I C LE I N FO
A B S T R A C T
Keywords: sunspots Methods: statistical
The Solar Influences Data Analysis Center (SIDC) issued a new version (version 2) of the sunspot number data in July 2015. The 13-month smoothed monthly sunspot number from the new version is used for the first time to research the relations among the feature parameters of solar cycles under the bimodal distribution for the modern era cycles (10–23), and, their physical implications are discussed. These relations are utilized to predict the maximum amplitude of solar cycle 25. Cycle 25 is predicted to start in October 2020 and reach its maximum amplitude of 168.5 ± 16.3 in October 2024, thus, it should be stronger than cycle 24 but weaker than cycle 23.
1. Introduction The solar activity rising and falling within an 11-year cycle causes changes in space weather, which greatly influences space weather analysis, operations of low-Earth-orbiting satellites, and so on. A key indicator of solar activity is the sunspot number (Hathaway, 2015; Wilson, 1988). To understand solar cycles and predict solar activity, many characteristics of the solar cycles have been studied (Hathaway et al., 2002; Aparicio et al., 2012; Carrasco et al., 2016), based on the Wolf sunspot number, such as, the Waldemeier effect (Waldmeier, 1935) and the Even-Odd effect (Gnevyshev, 1948). However, the International Sunspot Number and Group Number do not match on various aspects, inducing confusions and contradictions. To resolve these problems, a new revised collection of sunspot group numbers was presented by Vaquero et al. (2016) and several series of the sunspot number have been presented (Clette and Lefèvre, 2016; Lockwood et al., 2016; Svalgaard and Schatten, 2016; Usoskin et al., 2016; Chatzistergos et al., 2017). In addition, a new revised version of the sunspot number (version 2), issued by the Solar Influences Data Analysis Center (SIDC) in 2015 (Clette et al., 2014; Ahluwalia and Ygbuhay, 2016), is chosen to be used in this study. Reviewing the solar cycle characteristics using this new version of data is necessary. Solar activity prediction is becoming increasingly important because solar activities influence the atmosphere (Pulkkinen, 2007), and thus affect the orbital lifetime of an increasing number of spacecraft, including manned space stations. Several methods and technologies have been proposed to predict solar cycles, such as precursor methods
∗
and statistical methods (Petrovay, 2010; Yoshida and Sayre, 2012; Ahluwalia and Jackiewicz, 2012). Precursor methods, which use the polar fields or geomagnetic indices, are more reliable than others using the sunspot number (Hathaway et al., 1999; Li et al., 2001; Pesnell, 2012) for cycles 21 and 22. However, some statistical methods have provided more reliable predictions for cycle 23 (Li et al., 2001; Wilson et al., 1998; Petrovay, 2010), whereas most of the precursor methods have overestimated the amplitude. Nevertheless, extrapolation methods and model-based methods occasionally show advantages (Pesnell, 2014). These former studies use the old version of sunspot number data (version 1) to make predictions for the solar cycle. In this study, the new version of sunspot number data is employed to predict the maximum amplitude of solar cycle 25 for the first time. The current observations suggest that cycle 24 should have already reached its maxima (Li et al., 2011) at 116.4 in April 2014. Therefore, now is the appropriate time to predict when cycle 25 will start and how strong it will be. In Section 2, characteristics of the modern era solar cycles and classical correlations among feature parameters are examined based on the new sunspot number index, and the re-examined relations are used to predict the maximum amplitude of cycle 25 and its timing. 2. Cycle characteristics and predictions 2.1. Data The 13-month smoothed monthly sunspot number of the new
Corresponding author. Yunnan Observatories, Chinese Academy of Sciences, Kunming, 650011, China. E-mail address: [email protected] (F.Y. Li).
https://doi.org/10.1016/j.jastp.2018.10.014 Received 21 November 2017; Received in revised form 15 October 2018; Accepted 19 October 2018 1364-6826/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Li, F.Y., Journal of Atmospheric and Solar-Terrestrial Physics, https://doi.org/10.1016/j.jastp.2018.10.014
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Table 1 Parameters of cycles 10–24 based on the new sunspot-number index: year of minimum, ascent duration, year of maximum, amplitude, descent duration, cycle length, and max-max cycle length. Cycle
Year of
Ascent
Year of
Number
Minimum
Duration
(Yrs) 1856.0 1867.2 1879.0 1890.2 1902.0 1913.6 1923.6 1933.7 1944.1 1954.3 1964.8 1976.2 1986.7 1996.9 2009.0
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Amplitude
Descent
Cycle
Max-Max
Maximum
Duration
Length
Cycle
(Yrs)
(Yrs)
(Yrs)
(Yrs)
Length (Yrs)
4.2 3.4 5.0 3.8 4.1 4.0 4.7 3.6 3.2 3.9 4.1 3.8 3.2 5.0 5.3
1860.1 1870.6 1884.0 1894.0 1906.1 1917.6 1928.3 1937.3 1947.4 1958.2 1968.9 1980.0 1989.9 2001.9 2014.3
7.1 8.3 6.2 8.0 7.5 6.0 5.4 6.8 6.9 6.6 7.3 6.7 7.0 7.1
11.2 11.8 11.2 11.8 11.5 10.0 10.1 10.4 10.2 10.5 11.4 10.5 10.2 12.0
10.5 13.3 10.1 12.1 11.5 10.7 9.0 10.1 10.8 10.7 11.1 9.9 12.0 12.4
186.2 234.0 124.4 146.5 107.1 175.7 130.2 198.6 218.7 285.0 156.6 232.9 212.5 180.3 116.4
sunspot number data (version 2) (Clette and Lefèvre, 2016) issued by the Solar Influences Data Analysis Center (SIDC) is used in this study. This version can be downloaded from the Sunspot Index and Long-term Solar Observations (SILSO) website (http://sidc.oma.be/silso/). The socalled modern era sunspot cycles refer to cycle 10 and thereafter, as the sunspots have been observed every day since the start of cycle 10 (Wilson, 1987a; Vaquero, 2007). Table 1 lists the parameters of modern era solar cycles 10–24 in columns 2–8: year of minimum, ascent duration, year of maximum, amplitude, descent duration, cycle length, and max-max cycle length, respectively. The ascent duration is the time the number of sunspots rise from minimum to maximum, whereas the descent duration is the time the number of sunspots decrease from maximum to minimum. The amplitude is the maximum smoothed monthly sunspot number (version 2) of a cycle. Therefore, the actual value of the amplitude of a cycle depends on the activity index used and the type of smoothing. The cycle length of a sunspot cycle is defined as the elapsed time from the minimum preceding its maximum to the minimum of the following cycle (Hathaway et al., 2002), and the max-max cycle length is the elapsed time from the maximum of a solar cycle to the maximum of the following cycle (Du, 2006).
Fig. 1. Period length of cycles 10–23, showing the ‘bimodality of the solar cycle’. 7 long-period cycles are shown as diamonds, whereas 7 short-period cycles as circles. The two horizontal lines represent the averages, respectively.
2.2. Bimodality of the solar cycle bimodal distribution should be considered an exact physical mechanism; that is, it only provides information about a pattern in the data. According to the Babcock-Leighton (BL) type flux transport dynamos (Babcock, 1961; Leighton, 1969), the cycle length increases if the meridional flow is weak, and the increase in cycle length results in two effects. On the one hand, the differential rotations could have more time to enhance the cycle strength by producing a toroidal magnetic field. On the other hand, the diffusing effect in the magnetic field has more time to weaken the cycle strength (Jiang et al., 2015). Because the diffusing effect in the magnetic field is more apparent than the productive process of the toroidal field, a relatively longer cycle should be weaker in strength. Thus, the random variation in the meridional flow leads to the variation in the strength of a solar cycle. Based on this, the bimodal distribution in Fig. 1 implies that there should be two clusters for the toroidal field.
Fig. 1 shows a bimodal distribution (Rabin et al., 1986) for the modern era cycles. As proposed by Wilson (1987b), according to the lengths of the solar cycles, the modern era cycles can be divided into two clusters (short-period cycles and long-period cycles) with an 8month gap (the smallest distance between the two clusters) separating these two clusters. This gap is called the Wilson Gap by Hathaway (2015) that is the so-called bimodality of solar cycles. Their cycle lengths may be clustered into two groups: 7 short-period cycles and 7 long-period cycles. The mean period length is 11.64 ± 0.40 years for long cycles and 10.21 ± 0.25 years for short cycles. This may not be a completely accurate distribution due to the limited sample data. However, compared with the normal and uniform distributions, Wilson (1987b) proposed that the bimodal distribution can better describe the observed distribution of cycle lengths. Then, from the results of Li et al. (2015) obtained with version 1 of the sunspot number data, there is no significant change in these relations between versions 1 and 2. It is useful to divide these cycles into the two groups to study their different statistical relations among some feature parameters of a solar cycle in the modern era. Nevertheless, there is no evidence to suggest that the
2.3. The solar memory We analyzed the different possibilities for the relationships of cycle 2
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previous and new versions of the sunspot number data and considering different sets of solar cycles, demonstrate that the method by Du (2006) is not suitable for prediction because it is based on a non-robust relationship. Therefore, the predictions should be treated with caution. The analysis in this study regarding sunspot number data can provide information about a pattern in the data but should not be considered as exact. All the relations between amplitude and cycle length in Figs. 2 and 3 are negative correlations. There may be a memory mechanism or a time delay in solar cycles, because the characteristics of these amplitudes and periods remain and appear after a time lag. This is the so-called memory of a solar magnetic cycle, which is built into flux transport dynamo models. Hathaway et al. (2003) found the negative correlation between the period of a cycle and the drift rate of the sunspot latitude bands as evidence that a deep meridional flow sets the sunspot cycle period. Stochastic fluctuations in the parameters, accounted by the inherent turbulent nature of the dynamo mechanism and the large-scale flows, can lead to cycle amplitude and period modulation (Wang et al., 2002). The meridional flow requires a finite time to carry down the surface poloidal field to the low-latitude tachocline, where the toroidal field of the next cycle is generated (Hathaway et al., 2003; Charbonneau and Dikpati, 2000). It is difficult to determine the length of the time lag as it is a very complicated modulation process. Additionally, it is difficult to explain the reasons for the difference between results for cycles 10–23 (the modern era cycles) and for cycles 1–23. One explanation could be the poor accuracy of the sunspot number data for the early cycles. Alternatively, the different results may be a result of the complicated modulation process.
Fig. 2. Relationship between current cycle (n) length and cycle (n+3) amplitude for the modern era cycles (solar cycles 10–23), with a correlation coefficient of −0.7587, significant at the 99% confidence level.
2.4. Ascent duration versus descent duration The linear relationships between the ascent duration ( AD ) and the descent duration (DD ) are separately shown in Fig. 4 for the longperiod cycles and short-period cycles in the modern era. For the longperiod cycles, the best linear fit is,
DD = −0.804⋅AD + 10.8
(1)
and the correlation coefficient is −0.8013, significant at the 97% confidence level. The standard error of this linear fitting is 0.38 years. For
Fig. 3. Relationship between current cycle length and following cycle amplitude for cycles 1–23, with a correlation coefficient of −0.6791, significant at the 99.9% confidence level.
lengths at different lags. For modern era Cycles 10–23, as shown in Fig. 2, the most significant correlation occurs between current cycle (n) length and latter cycle (n+3) amplitude with a correlation coefficient of −0.7587, significant at the 99% confidence level. In addition, this correlation is investigated in a larger sample size, as shown in Fig. 3. For cycles 1–23, the most statistically significant correlation occurs between current cycle (n) length and latter cycle (n+1) amplitude, obtained by Hathaway et al. (2002) with a correlation coefficient of − 0.6791, significant at the 99.9% confidence level, similar to Hathaway et al. (2002) obtained from the Zurich sunspot numbers and the Group sunspot numbers. According to the above mentioned research, the correlation between length and amplitude of the solar cycle depends on the data employed. Furthermore, other authors have found different results for this amplitude-length relationship. For example, Du and Du (2006) proposed a method to predict the maximum amplitude of solar cycles 24 and 25 by employing the relationship between the maximum amplitude and the cycle length of two earlier solar cycles considering solar cycles 9–23. However, Carrasco et al. (2017a, b), applying both the
Fig. 4. The linear relationships between the ascent duration and descent duration separately for the long-period cycles (diamonds, with a correlation coefficient of −0.8013, significant at the 97% confidence level) and shortperiod cycles (circles, with a correlation coefficient of −0.8736, significant at the 98% confidence level) for modern era cycles 10–23. The thin lines are their linear regression lines. 3
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Fig. 5. Monthly sunspot number (the blue thin line) for cycle 24 from December 2008 (the minimum time of cycle 24) to August 2018 and its smoothed value (the black dashed line) to February 2018 (http://sidc.oma. be/silso/).
Fig. 6. Relationship between the descent duration (DD) and the max-max cycle length (DA) for modern era cycles 10–23 and its linear fitting regression line with a correlation coefficient of 0.825, significant at the 99.9% confidence level.
the short-period cycles, the best linear fit is,
DD = −0.916⋅AD + 9.897
DA(n) = DD(n)+AD(n+1), the correlation between the descent duration (DD (n) ) of a cycle and the ascent duration ( AD (n + 1) ) of the following cycle can be inferred from Equation (3) as
(2)
and the correlation coefficient is −0.8736, significant at the 98% confidence level. The standard error of linear fitting is 0.25 years. The negative correlation between the ascent duration and the descent duration implies that if a cycle raise quickly, it will decline slowly, and vice versa. Therefore, changes in solar cycle lengths should not be too large. The length of a solar cycle is leaded by the meridional flow in the Babcock-Leighton(BL) type flux transport dynamos (Babcock, 1961; Hathaway et al., 2003; Leighton, 1969). The meridional flow rate is always approximately 10ms−1 (Featherstone and Miesch, 2015); thus, the length of a solar cycle is stabilized approximately 11 years. This is in agreement with the negative correlation between the ascent duration and the descent duration. Fig. 5 shows the monthly sunspot number from December 2008 (the start time of cycle 24) to August 2018 and its smoothed value to February 2018. More than eight years have elapsed since the start of cycle 24, thus, it is most likely that cycle 24 has already reached its maximum (Li et al., 2011). That is, the maximum amplitude of cycle 24 is 116.4 in April 2014, and the ascent duration is 5.33 years. Based on the linear relationships between the ascent duration (AD) and the descent duration (DD), the descent duration of cycle 24 should be 6.51 ± 0.38 years if it is a long-period cycle, or 5.01 ± 0.25 years if it is a short-period cycle. Therefore, cycle 25 should begin in October 2020 (± 0.38 years) if cycle 24 is a long-period cycle, or in April 2019 (± 0.25 years) if it is a shortperiod cycle.
AD (n + 1) = 0.186⋅DD (n) + 2.826
According to the correlation in Equation (3), DA (24) should be 10.55 ± 0.69 years if DD (24) = 6.51 years, or 8.77 ± 0.69 years if DD (24) = 5.01 years. Thus, the ascent duration of cycle 25 ( AD (25) ) is calculated to be 4.04 ± 0.69 years if cycle 24 is a long-period cycle, or 3.76 ± 0.69 years if it is a short-period cycle. Therefore, cycle 25 will reach its maxima in October 2024 if cycle 24 is a long-period cycle, or in January 2023 if cycle 24 is a short-period cycle. 2.6. The modified Waldmeier Effect There is a linear relationship between the amplitude ( Am ) and rising rate (r) for cycles 10–24 (including solar cycle 24, unlike with previous correlations), which is shown in Fig. 7. This is the so-called modified Waldmeier Effect (Waldmeier, 1935; Carrasco et al., 2016; Li et al., 2017):
Am = 2.676⋅r + 56.89, r = Am / AD
(5)
where r is the ratio of the amplitude ( Am ) to ascent duration ( AD ). The correlation coefficient is 0.9432, significant at the 99.9% confidence level, and the standard error of this regression is 16.3. The rate of the rise of the sunspot number to maximum is positively correlated to the cycle amplitude. That is, the more quickly the sunspot number of a cycle rises, the larger the amplitude of the cycle is. According to this relationship, the amplitude of cycle 25 is predicted to be 168.5 ± 16.3 if cycle 24 is a long-period cycle ( AD (25) = 4.04 ) or 197.3 ± 16.3 if cycle 24 is a short-period cycle ( AD (25) = 3.76). Past studies, such as the prediction by the NASA's Marshall Space Flight Center (http:// solarscience.msfc.nasa.gov/predict.shtml) and the NOAA's Space Weather Prediction Center (http://www.swpc.noaa.gov/ products/ solar-cycle-progression), suggest that cycle 24 is a long-period cycle (Li et al., 2011; Ahluwalia, 2016), that will be completed at the end of 2019 (Hathaway and Upton, 2016) or at the beginning of 2020 (Carrasco et al., 2017a). Thus, this study makes the following prediction: cycle 24 is a long-period cycle, that will be completed in October 2020 (± 0.38 years) and cycle 25 will reach its peak at 168.5 ± 16.3 in October 2024. This peak is greater than the amplitude of cycle 24
2.5. Max-max cycle length versus decline time The max-max cycle length (DA (n) ) is the period between two successive maxima, equal to the sum of the descent duration (DD (n) ) of a cycle and the ascent duration ( AD (n + 1) ) of its next cycle (Du and Du, 2006). It is found to be correlated with the descent duration (DD (n) ) (Li et al., 2005, 2015) as follows:
DA (n) = 1.186⋅DD (n) + 2.826
(4)
(3)
which is similar to the results Du (2006) obtained using the previews version of the sunspot number data. This correlation is shown in Fig. 6, with a correlation coefficient of 0.825, significant at the 99.9% confidence level. The standard error of this regression is 0.69 years. As 4
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(Gnevyshev, 1948), but weaker than cycle 23 (Kakad et al., 2017). This prediction matches the speculation by Ahluwalia and Ygbuhay (2012) that cycle 24 may herald the onset of a Dalton-like minimum in the 21st century, and is in agreement with the inference that cycle 24 is located at the bottom of the Gleissberg cycle (Hathaway, 2015). Acknowledgments The authors are grateful to Dr. Li, K. J. for his constructive proposals and helpful suggestions. This work is supported by the National Natural Science Foundation of China (11573065, 11633008, and 11703085), the Specialized Research Fund for State Key Laboratories, and the Chinese Academy of Sciences. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jastp.2018.10.014.
Fig. 7. Correlation between the rising rate (the ratio of the amplitude to ascent duration) and the sunspot number amplitude, which is the so-called modified Waldmeier effect for modern era cycles 10–24. The solid line is the linear regression line with a correlation coefficient of 0.9351, significant at the 99.9% confidence level.
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(Steinhilber and Beer, 2013; Zharkova et al., 2015), but less than that of cycle 23, as the amplitude of cycle 23 (Am(23)) is 180.3, and the amplitude of cycle 24 (Am(24)) is 116.4. As a test, we used this method to predict solar cycle 24. The method predicts that solar cycle 24 would have reached its peak at 123.6 ± 16.3 in August 2012, which was fulfilled while the actual amplitude is 116.4. However, the maximum for Solar cycle 24 is predicted to have occurred in August 2012, almost two years before the actual occurrence in April 2014. Since these statistical relations among the characteristic parameters is based on regarding the solar cycle as a unimodal profile, this method is limited in predicting the date of the maximum amplitude, especially for bimodal cycles such as cycle 24. 3. Summary In this paper, the 13-month smoothed monthly sunspot number of the new sunspot number data (version 2) is used for the first time to study the solar cycle characteristics for modern era cycles 10–24 (Li et al., 2005). The modern era cycles can be divided into short-period cycles and long-period cycles (Rabin et al., 1986; Wilson, 1987a), showing a bimodal distribution. There is a significant negative correlation with time lag between cycle length and amplitude. We speculate that there is a memory mechanism in solar cycle (Hathaway et al., 2002). Linear relations between the ascent duration (AD) and descent duration (DD) are found for short-period cycles and long-period cycles, and these correlations result in a stable duration of a solar cycle at approximately 11 years. The max-max cycle length DA (n) (Du, 2006) is found to be correlated with the descent duration DD (n) . The correlation between the amplitude (Am) and rising rate (r = Am / AD) for cycles 10–24 is found to be positive. According to these statistical relations among the characteristic parameters of sunspot cycle profiles, cycle 25 is predicted to start in October 2020 (± 0.38 years) and reach its maximum amplitude of 168.5 ± 16.3 in October 2024 if cycle 24 is a longperiod cycle or to begin in April 2019 (± 0.25 years) and reach its maximum amplitude of 197.3 ± 16.3 in January 2023 if cycle 24 is a short-period cycle. However, past studies suggest that cycle 24 is a longperiod cycle (Li et al., 2011; Ahluwalia, 2016), therefore, we expect the former. In this paper, it is predicted that cycle 25 is more likely to be stronger than cycle 24 (Cameron et al., 2016; Hathaway and Upton, 2016), which is in agreement with the Gnevyshev-Ohl (even-odd) rule 5
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