Predicting the maximum amplitude of solar cycle 25 and its timing

Journal of Atmospheric and Solar-Terrestrial Physics 135 (2015) 72–76

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Predicting the maximum amplitude of solar cycle 25 and its timing K.J. Li a,b, W. Feng c,d,n, F.Y. Li a,e a

Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China Key Laboratory of Solar Activity, National Astronomical Observatories, CAS, Beijing 100012, China Research Center of Analysis and Measurement, Kunming University of Science and Technology, Kunming 650093, China d Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China e University of Chinese Academy of Sciences, Beijing 100049, China b c

art ic l e i nf o

a b s t r a c t

Article history: Received 28 May 2015 Received in revised form 7 September 2015 Accepted 16 September 2015

Some classical statistical relations among feature parameters of solar cycle profiles are utilized to predict the maximum amplitude of the next upcoming solar cycle and its timing. Resultantly, cycle 25 should start in November 2019 and have its maximum amplitude of 109.1 in October 2023, being a moderate solar activity cycle. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Sun: activity Sun: general Methods: data analysis

1. Introduction Solar activity prediction is not only important for the understanding of solar activity itself, but also of importance for practical application, due to that it may provide some information on possible variations of the solar-terrestrial environment ahead of time (Hathaway, 2010; Petrovay, 2010; Pesnell, 2012). Large numbers of methods and technologies have been proposed to give predictions of solar activity to the recent 4 solar cycles. For cycles 21 (Brown, 1984; Brown and Simon, 1986) and 22 (Kunches, 1993; Li et al., 2001), precursor methods gave better predictions than others (Brown and Simon, 1986; Li et al., 2001). However, for cycle 23 (Kane, 2001; Li et al., 2001), most precursor methods overrated the maximum sunspot number, meanwhile some statistical methods need be paid more attentions (Li et al., 2001; Petrovay, 2010). For cycle 24, Pesnell (2012) collected 75 predictions. The present status of measuring sunspot number suggests that cycle 24 should already have reached its maximum, 81.8 in April 2014. Among the collections, the method of solar polar magnetic field as precursor gave better predictions than others (Svalgaard et al., 2005; Schatten, 2005). Hathaway (2010) and Petrovay (2010) gave the general summing-up for solar activity prediction. In this study, some classical relations among feature parameters of solar cycle

n Corresponding author at: Research Center of Analysis and Measurement, Kunming University of Science and Technology, Kunming 650093, China. E-mail address: [email protected] (W. Feng).

http://dx.doi.org/10.1016/j.jastp.2015.09.010 1364-6826/& 2015 Elsevier Ltd. All rights reserved.

profiles are used to forecast the maximum sunspot number of solar cycle 25 and its timing.

2. Estimating the maximum amplitude of solar cycle 25 The so-called modern era sunspot cycles are generally referred to cycle 10 and thereafter, and they differ from prior cycles in that sunspots have been observed every day since cycle 10 (Wilson, 1987; Li et al., 2005). Feature parameters of a solar cycle profile just for the modern era sunspot cycles, such as ascent duration, descent duration, cycle length, minimum and maximum amplitudes, and so on are used to display relations among them at followings, and further the relations are then used to forecast the maximum amplitude of solar cycle 25. A noticeable feature for the modern era cycles is that their cycle lengths may be clustered into two groups: a short-period-cycle cluster and a long-period-cycle cluster, and this is the so-called “bimodality of the solar cycle” (Rabin et al., 1986; Wilson, 1987; Wilson et al., 1996). Fig. 1 illustrates the bimodality for cycles 10– 23. UP to now, 7 cycles belong to the short-period-cycle cluster, averaging about 123 months in length and ranging from 121 to 126 months, and 7 cycles to the long-period-cycle cluster, averaging about 139 months and ranging from 134 to 145 months. Here cycle 23 is regarded to begin in October 1996 (Wilson et al., 1996; Li et al., 2005) and ends in December 2008, having cycle length of 145 months. As pointed out by Wilson (1987), an 8-month gap (the smallest distance between the two clusters)

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Fig. 1. Period length of the modern era cycles (cycles 10–23). Period lengths cluster near 139 and 123 months, showing the “bimodality of the solar cycle”. 7 shortperiod cycles are illustrated by triangles, while 7 long-period cycles by circles.

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Fig. 2. Two different linear relations about the descent duration of a cycle versus the ascent duration of the cycle for the two clusters of the modern era cycles (cycles 10–23). The short-period-cycle cluster is marked by triangles, while the long-period-cycle cluster is denoted by circles.

separates the two clusters, and this gap is called “Wilson Gap” by Hathaway (2010). Furthermore, the two clusters statistically show two different linear relations about the descent duration DD(n) of a cycle (for example, solar cycle n) and the ascent duration AD(n) of the cycle (Wilson et al., 1996; Li et al., 2005). Fig. 2 displays the scatter plot of DD(n) varying with respect to AD(n) for cycles 10– 23. For the short-period-cycle cluster, the linear relation is that DD(n) = 122.027 − 0.978 × AD(n), the correlation coefficient between practical observations and the theoretical fitting line is
0.970, which is of significance at the 99.9% confidence level (the number of data points is 7, and the tabulated value is 0.951 at this confidence level), and the standard error of the fitting is 1.8 months (Li et al., 2005). For the long-period-cycle cluster, the linear relation is that DD(n) = 164.460 − 1.521 × AD(n), the correlation coefficient is
0.973, which is also statistically significant at the 99.9% confidence level, and the standard error is 2.0 months. The present cycle 24 begins in December 2008 and progresses

Fig. 3. Monthly mean sunspot number (SSN, the solid line) from the minimum time (December 2008) of cycle 24 to April 2015 and its smoothed value (the dashed line).

into cycle over 6 years up to now (April 2015). Fig. 3 shows monthly mean sunspot number (SSN) from the minimum time of cycle 24 to April 2015 and its smoothed value, which are downloaded from the Solar Influences Data Analysis Center (SIDC).1 The smoothed SSN reaches its second peak in April 2014, and the peak SSN is 81.8. It is known that the modern era cycles generally have one or at most two local peaks and their ADs are generally less than 6 years. If the second peak were not the maximum SSN of cycle 24, then the present observations of SSN shown in Fig. 3 implies that the maximum SSN would form the third peak later in 2015 or even later, with the AD being about 7 years or even longer. So the present status of SSN suggests that cycle 24 should already have reached its maximum in all probability, and its AD is 64 months, spanning from December 2008 to April 2014. Based on the two fitting lines, the descent duration of cycle 24 should be DD(24) = 59.3 ± 1.8 months, if cycle 24 is a short-period cycle, or DD(24) = 67.1 ± 2.0 if cycle 24 belongs to the long-period-cycle cluster. Therefore cycle 25 should begin in March 2019 (71.8 months), if cycle 24 is a short-period cycle, or in November 2019 (7 2.0 months) if cycle 24 belongs to the long-period-cycle cluster. A statistical or physical correlation of two consecutive solar cycles probably provides a method, through which one could forecast or estimate characteristic values of the following solar cycle using the known feature values of the former cycle (Thompson, 1993; Wang et al., 2002b; Li et al., 2005). DD(n) of cycle n is found correlated with DD(n) plus AD(n + 1), namely the interval (DA(n)) between the maximum time of cycles n and n þ1 (Thompson, 1993; Wang et al., 2002b; Li et al., 2005). This correlation is illustrated in Fig. 4 for the modern era cycles, which may be expressed well by a regressive line: DA(n) = 2.726 + 1.1804 × DD(n). The correlation coefficient between observations and the regressive line is 0.839, which is of significance at the 99% confidence level, and the standard error of the regression is about 0.677 years. Therefore, the ascent duration AD(25) of cycle 25 is 3.613 70.677 (years) when DD(24) = 59 months, or 3.730 70.677 (years) when DD(24) = 67 months. This is to say, cycle 25 should peak in October 2022 if cycle 24 is a short-period cycle, or in August 2023 if cycle 24 is a long-period cycle. 1

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Fig. 4. Relation of the descent duration DD(n) of a cycle (for example, solar cycle n) versus DD(n) plus the ascent duration AD(n + 1) of its following cycle, namely the interval (DA(n)) between the maximum time of cycles n and n þ1 for the modern era cycles. The solid line is the linear fitting to the relation which is marked by ▵ s.

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Fig. 6. Statistical relation of the maximum sunspot number versus the ascent duration of a solar cycle for the modern era cycles. The dashed line is the linear fitting to the relation, while cycle 21, which is marked by the triangle, is excluded.

error is much lower. In the following, just the results given by the latter are utilized. The maximum smoothed monthly sunspot number of a cycle is found statistically related with the ascent duration of the cycle, this is the so-called Waldmeier effect (Waldmeier, 1935, 1939; Hathaway, 2010; Petrovay, 2010). Here, the Waldmeier effect for the modern era cycles is illustrated in Fig. 6. Except the strongest cycle, cycle 21, a linear fitting may give a good description to the effect:

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Fig. 5. Relation of DD(n) versus DA(n) for the similar cycles of cycle 24. The solid line is the linear fitting to the relation which is marked by ▵ s.

The relation of DA(n) and DD(n) should be closer for the similar cycles of a certain solar cycle (Wang, 1992; Wang et al., 2002a). For a certain solar cycle, its similar cycles are those whose feature parameters of cycle profiles, such as the maximum amplitude Amax, AD, and DD are close to the certain solar cycle (Wang, 1992; Wang et al., 2002a; Li et al., 2005). For cycle 24, cycles 1, 7, 12, 13, 14, and 16 are chosen to be its similar cycles. For these cycles, DA(n) = 0.8369 + 1.5503 × DD(n), which are shown in Fig. 5. The correlation coefficient between observations and the regressive line is 0.979, which is of significance at the 99.9% confidence level, and the standard error of the regression is about 0.287 years. Therefore, the ascent duration AD(25) of cycle 25 is 3.543 70.287 (years) when DD(24) = 59 months, or 3.910 70.287 (years) when DD(24) = 67 months. This is to say, cycle 25 should peak in October 2022 if cycle 24 is a short-period cycle, or in October 2023 if cycle 24 is a long-period cycle. The relation of DA(n) versus DD(n) gives almost the same peak time of cycle 25 for the modern era cycle as that for the similar cycles of cycle 24, but for the latter standard

¯ )2 1 (AD − AD , + n ¯ )2 n ∑i = 1 (ADi − AD

where the standard error s is 6.7883 × 10−4 , n is the number of ¯ = ∑n AD /n, the mean ascent data used to fit, being 14, AD i i=1 duration, and t(α /2)(n − 2) is the tabulated Students t-test value at the α probability level (Li et al., 2002). The correlation coefficient being 0.9713, which is significant at the 99.99% confidence level. Therefore, the maximum smoothed monthly sunspot number for cycle 25 is predicted to be 109.1 when the ascent duration of the cycle is taken 3.910 years, and meanwhile it is located at the interval of 96.1–125.9 with the 95% probability. Or, it is forecasted to be 129.0 when the ascent duration of the cycle is taken 3.542 years, and meanwhile it is located at the interval of 111.1–153.7 with the 95% probability. Fig. 7 gives the monthly SSN averaged over the similar cycles of cycle 24. Shown also in the figure is the monthly SSN predicted respectively by the NASA's Marshall Space Flight Center2 and the NOAA's Space Weather Prediction Center.3 These three lines of SSN match well and all suggest that cycle 24 should end at the end of 2019. Thus, this study prefers to give such prediction: cycle 24 should be a long-period cycle, ending in November 2019 ( 72.0 months), and cycle 25 should peak to be 109.1 in October 2023, and the maximum SSN is located at the interval of 96.1–125.9 with the 95% probability. 2 3

http://solarscience.msfc.nasa.gov/predict.shtml http://www.swpc.noaa.gov/products/solar-cycle-progression

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Fig. 7. Monthly sunspot number (SSN) averaged over the similar cycles of cycle 24 (the thin solid line) and SSN predicted respectively by the NASA's Marshall Space Flight Center (the dashed line) and the NOAA's Space Weather Prediction Center (the dotted line). The thick solid line is the smoothed SSN.

Table 1 Predictions of solar cycle 25. Chistyakov (1983) Kontor et al. (1983) Du (2006) Du and Du (2006) Quassim et al. (2007) Hiremath (2008) Pishkalo (2008) Rigozo et al. (2011) Abdel-Fattah et al. (2013) Hamid and Galal (2013)

121 117 102.6 7 22.4 111.6 7 17.4 116 1107 11 112.3 7 33.4 132.1 90.7 78 118

2028.5 2024

2020 2023 2023.4 7 0.7 2023.3 2020 2022–2023

75

Many researches have already given predictions of the maximum SSN to cycle 25 up to now. A high predicted value of maximum SSN, 159 725 was given by Solanki et al. (2002), and 144.3 727.6 by Du et al. (2006). On the other hand, some much lower forecast values were proposed, for example, 50 710 by Javaraiah (2014), 50 715 by Abdusmatov (2007), and 707 30 by Hathaway and Wilson (2004). Meanwhile, most predictions show a moderate solar activity strength for cycle 25, which are in agreement with this study, and they are listed in Table 1. Cycle 24 is located at the bottom of the Gleissberg cycle (see Fig. 34 of Hathaway, 2010), and cycle 25 should start to rebound from the bottom, suggesting that cycle 25 should be possibly stronger than cycle 24. The Gnevyshev–Ohl (even–odd) rule shows that an odd cycle is stronger than its former neighboring even cycle (Gnevyshev and Ohl, 1948), implying that cycle 25 should be probably stronger than cycle 24. All in all, this study infers a moderate solar activity cycle for cycle 25. Time will display. In this study, the prediction of solar activity for cycle 25 is basically given by the statistical relations among the characteristic parameters of sunspot cycle profiles. More relations of characteristic parameters (Kakad, 2011; Hazra et al., 2015; Otkidychev and Popova, 2015) and more proxies of solar activity, such as x-ray fluxes, 10.7 cm flux, polar field strengths, etc. should (Munoz-Jaramillo et al., 2013) provide more robust predictions. This is an opening issue. Finally, one issue should be pointed out. On July 1, 2015, the Solar Influences Data Analysis Center produced a milestone event in the history of sunspot number: a new version of sunspot number is announced (Clette et al., 2014). The characteristic parameters of solar cycles used in this study come from the National Geophysical Data Center of NOAA, which are determined mainly on the original version of sunspot number. The influence of the new version on the determination of characteristic parameters needs to be investigated, and further based on the new version, a new frame of predictions of solar activity needs to be built up in the future.

3. Conclusions

Acknowledgments

For the modern era solar cycles, the ascent duration (AD) of a cycle is statistically related to the descent duration (DD) of the cycle. The present cycle 24 is inferred to already pass its maximum time at present (August 2015), thus its descent duration may be suggested through the statistical relation. The descent duration of a solar cycle, for example, cycle n, DD(n) is statistically correlated with DD(n) plus AD(n + 1), namely the interval (DA(n)) of the maximum time of cycles n nad n þ1. Accordingly, the ascent duration of cycle 25 could be suggested. The maximum smoothed monthly sunspot number (SSN) of a cycle is statistically correlated with the ascent duration of the cycle, therefore we could forecast the maximum SSN of cycle 25. Resultantly, cycle 25 is forecasted to start in November 2019, reaching its maximum, 109.1 in October 2023, and the maximum SSN is located at the interval of 96.1– 125.9 with the 95% probability. In this study, the modern era sunspot cycles are split into short and long cycles, as done by Rabin et al. (1986), Wilson (1987), and Wilson et al. (1996). If the cycles in each classification are exactly the same length then ADþDD should be constant: 122 months for the short cycles and 140 months for the long cycles. Thus for the short cycles we would expect DD = 122 − AD , which is very close to that found through regression. For the long cycles we would expect DD = 140 − AD , but the slope of such the linear relation is
1.5, somewhat far away from 1.0. Therefore, each classification with the same cycle length is an approximation, but DD should highly linearly correlated with AD.

We thank the scientific edit and two anonymous referees for their careful reading of the paper and constructive comments which improved the original version of the paper. Monthly sunspot number and its smoothed value come from the WDC-SILSO, Royal Observatory of Belgium, Brussels, and are freely downloaded form its SIDC website. Our many thanks are given to the staff of the WDC-SILSO. This work is supported by the National Natural Science Foundation of China (11573065, 11221063 and 11273057), the 973 programs 2012CB957801 and 2011CB811406, and the Chinese Academy of Sciences.

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