A Preliminary Estimate of the Size of the Coming Solar Cycle 24, based on Ohl’s Precursor Method

For many purposes (e.g., satellite drag, operation of power grids on Earth, and satellite communication systems), predictions of the strength of a solar cycle are needed. Predictions are made by using different methods, depending upon the characteristics of sunspot cycles. However, the method most successful seems to be the precursor method by Ohl and his group, in which the geomagnetic activity in the declining phase of a sunspot cycle is found to be well correlated with the sunspot maximum of the next cycle. In the present communication, the method is illustrated by plotting the 12-month running means aa(min ) of the geomagnetic disturbance index aa near sunspot minimum versus the 12-month running means of the sunspot number Rz near sunspot maximum [aa(min ) versus Rz(max )], using data for sunspot cycles 9 – 18 to predict the Rz(max ) of cycle 19, using data for cycles 9 – 19 to predict Rz(max ) of cycle 20, and so on, and finally using data for cycles 9 – 23 to predict Rz(max ) of cycle 24, which is expected to occur in 2011 – 2012. The correlations were good (∼+0.90) and our preliminary predicted Rz(max ) for cycle 24 is 142±24, though this can be regarded as an upper limit, since there are indications that solar minimum may occur as late as March 2008. (Some workers have reported that the aa values before 1957 would have an error of 3 nT; if true, the revised estimate would be 124±26.) This result of the precursor method is compared with several other predictions of cycle 24, which are in a very wide range (50 – 200), so that whatever may be the final observed value, some method or other will be discredited, as happened in the case of cycle 23.     

An Estimate for the Size of Sunspot Cycle 24

For the sunspot cycles in the modern era (cycle?10 to the present), the ratio of R _Z(max)/R _Z(36th month) equals 1.26±0.22, where R _Z(max) is the maximum amplitude of the sunspot cycle?using smoothed monthly mean sunspot number and R _Z(36th month) is the smoothed monthly mean sunspot number 36 months after cycle?minimum. For the current sunspot cycle?24, the 36th month following the cycle?minimum occurred in November 2011, measuring?61.1. Hence, cycle?24 likely will have a maximum amplitude of about 77.0±13.4 (the one-sigma prediction interval), a value well below the average R _Z(max) for the modern era sunspot cycles (about 119.7±39.5).     

Solar Cycle Predictions based on Solar Activity at Different Solar Latitudes

Many methods of predictions of sunspot maximum number use data before or at the preceding sunspot minimum to correlate with the following sunspot maximum of the same cycle, which occurs a few years later. Kane and Trivedi (Solar Phys. 68, 135, 1980) found that correlations of R _z(max) (the maximum in the 12-month running means of sunspot number R _z) with R _z(min) (the minimum in the 12-month running means of sunspot number R _z) in the solar latitude belt 20° – 40°, particularly in the southern hemisphere, exceeded 0.6 and was still higher (0.86) for the narrower belt > 30° S. Recently, Javaraiah (Mon. Not. Roy. Astron. Soc. 377, L34, 2007) studied the relationship of sunspot areas at different solar latitudes and reported correlations 0.95 – 0.97 between minima and maxima of sunspot areas at low latitudes and sunspot maxima of the next cycle, and predictions could be made with an antecedence of more than 11 years. For the present study, we selected another parameter, namely, SGN, the sunspot group number (irrespective of their areas) and found that SGN(min) during a sunspot minimum year at latitudes > 30° S had a correlation +0.78±0.11 with the sunspot number R _z(max) of the same cycle. Also, the SGN during a sunspot minimum year in the latitude belt (10° – 30° N) had a correlation +0.87±0.07 with the sunspot number R _z(max) of the next cycle. We obtain an appropriate regression equation, from which our prediction for the coming cycle 24 is R _z(max )=129.7±16.3.     

Short-Term Periodicities in Sunspot Activity and Flare Index Data during Solar Cycle 23

The short-term periodicities in sunspot numbers, sunspot areas, and flare index data are investigated in detail using the Date Compensated Discrete Fourier Transform (DCDFT) for the full disk of the Sun separately over the rising, the maximum, and the declining portions of solar cycle 23 (1996 – 2006). While sunspot numbers and areas show several significant periodicities in a wide range between 23.1 and 36.4 days, the flare index data do not exhibit any significant periodicity. The earlier conclusion of Pap, Tobiska, and Bouwer (1990, Solar Phys. 129, 165) and Kane (2003, J. Atmos. Solar-Terr. Phys. 65, 1169), that the 27-day periodicity is more pronounced in the declining portion of a solar cycle than in the rising and maximum ones, seems to be true for sunspot numbers and sunspot area data analyzed here during solar cycle 23.     

Dependence of SSNM on SSNm – a Reconsideration for Predicting the Amplitude of a Sunspot Cycle

An improved correlation between maximum sunspot number (SSN_M) and the preceding minimum (SSN_m) is reported when the monthly mean sunspot numbers are smoothed with a 13-month running window. This relation allows prediction of the amplitude of a sunspot cycle by making use of the sunspot data alone. The estimated smoothed maximum sunspot number (126±26) and time of maximum epoch (second half of 2000) of cycle 23 are in good agreement with the predictions made by some of the precursor methods.     

Predicting Solar Cycle 24 Using a Geomagnetic Precursor Pair

We describe using Ap and F_10.7 as a geomagnetic-precursor pair to predict the amplitude of Solar Cycle 24. The precursor is created by using F_10.7 to remove the direct solar-activity component of Ap. Four peaks are seen in the precursor function during the decline of Solar Cycle 23. A recurrence index that is generated by a local correlation of Ap is then used to determine which peak is the correct precursor. The earliest peak is the most prominent but coincides with high levels of non-recurrent solar activity associated with the intense solar activity of October and November 2003. The second and third peaks coincide with some recurrent activity on the Sun and show that a weak cycle precursor closely following a period of strong solar activity may be difficult to resolve. A fourth peak, which appears in early 2008 and has recurrent activity similar to precursors of earlier solar cycles, appears to be the “true” precursor peak for Solar Cycle 24 and predicts the smallest amplitude for Solar Cycle 24. To determine the timing of peak activity it is noted that the average time between the precursor peak and the following maximum is ≈?6.4 years. Hence, Solar Cycle 24 would peak during 2014. Several effects contribute to the smaller prediction when compared with other geomagnetic-precursor predictions. During Solar Cycle 23 the correlation between sunspot number and F_10.7 shows that F_10.7 is higher than the equivalent sunspot number over most of the cycle, implying that the sunspot number underestimates the solar-activity component described by F_10.7. During 2003 the correlation between aa and Ap shows that aa is 10 % higher than the value predicted from Ap, leading to an overestimate of the aa precursor for that year. However, the most important difference is the lack of recurrent activity in the first three peaks and the presence of significant recurrent activity in the fourth. While the prediction is for an amplitude of Solar Cycle 24 of 65±20 in smoothed sunspot number, a below-average amplitude for Solar Cycle 24, with maximum at 2014.5±0.5, we conclude that Solar Cycle 24 will be no stronger than average and could be much weaker than average.     

Prediction of the Maximum Amplitude and Timing of Sunspot Cycle 24

Precursor techniques, in particular those using geomagnetic indices, often are used in the prediction of the maximum amplitude for a sunspot cycle. Here, the year 2008 is taken as being the sunspot minimum year for cycle 24. Based on the average aa index value for the year of the sunspot minimum and the preceding four years, we estimate the expected annual maximum amplitude for cycle 24 to be about 92.8±19.6 (1-sigma accuracy), indicating a somewhat weaker cycle 24 as compared to cycles 21 – 23. Presuming a smoothed monthly mean sunspot number minimum in August 2008, a smoothed monthly mean sunspot number maximum is expected about October 2012±4 months (1-sigma accuracy).     

Bimodality and the Hale cycle

Because of the bimodal distribution of sunspot cycle periods, the Hale cycle (or double sunspot cycle) should show evidence of modulation between 20 and 24 yr, with the Hale cycle having an average length of about 22 yr. Indeed, such a modulation is observed. Comparison of consecutive pairs of cycles strongly suggests that even-numbered cycles are preferentially paired with odd-numbered following cycles. Systematic variations are hinted in both the Hale cycle period and R _sum (the sum of monthly mean sunspot numbers over consecutively paired sunspot cycles). The preferred even-odd cycle pairing suggests that cycles 22 and 23 form a new Hale cycle pair (Hale cycle 12), that cycle 23 will be larger than cycle 22 (in terms of R _M, the maximum smoothed sunspot number, and of the individual cycle value of R _sum), and that the length of Hale cycle 12 will be longer than 22 yr. Because of the strong correlation (r = 0.95) between individual sunspot cycle values of R _sum and R _M, having a good estimate of R _Mfor the present sunspot cycle (22) allows one to predict its R _sum, which further allows an estimation of both R _Mand R _sum for cycle 23 and an estimation of R _sum for Hale cycle 12. Based on Wilson's bivariate fit (r = 0.98), sunspot cycle 22 should have an R _Mequal to 144.4 ± 27.3 (at the 3- level), implying that its R _sum should be about 8600 ± 2200; such values imply that sunspot cycle 23 should have an R _sum of about 10500 ± 2000 and an R _Mof about 175 ± 40, and that Hale cycle 12 should have an R _sum of about 19100 ± 3000.     

A Prediction of Solar Cycle 24 Using a Modified Precursor Method

Based on cycles 17 – 23, linear correlations are obtained between 12-month moving averages of the number of disturbed days when Ap is greater than or equal to 25, called the Disturbance Index (DI), at thirteen selected times (called variate blocks 1, 2,… , each of six-month duration) during the declining portion of the ongoing sunspot cycle and the maximum amplitude of the following sunspot cycle. In particular, variate block 9, which occurs just prior to subsequent cycle minimum, gives the best correlation (0.94) with a minimum standard error of estimation of ± 13, and hindcasting shows agreement between predicted and observed maximum amplitudes to about 10%. As applied to cycle 24, the modified precursor technique yields maximum amplitude of about 124±23 occurring about 45±4 months after its minimum amplitude occurrence, probably in mid to late 2011.     

Prediction of maximum sunspot number in solar cycle 23

Instantaneous predictions of the maximum monthly smoothed sunspot number in solar cycle 23 have been made with a linear regressive model, which gives the predicted maximum value as a function of the smoothed sunspot numbers corresponding to a given month from the minimum in all preceding cycles. These predictions indicate that the intensity of solar activity in the current cycle will be at an average level.     

Prediction of the Maximum Amplitude of Solar Cycles Using the Ascending Inflexion Point

To predict solar cycle maximum in terms of smooth sunspot numbers, a method based on the slope at the inflexion point observed during the ascending phase of the cycle is proposed. Application to cycle 23 (beginning in May 1996) gives a predicted value of 103±20 (r.m.s.) for the sunspot number maximum. A comparison with predictions using other methods is given.     

Prediction of the height of solar cycle 23

Presuming a bimodal behaviour of even-odd solar cycle pairs (i.e., four modes designated asA, B, C, andD), we predict the amplitude of solar cycle 23. The bimodal properties include the dependence of maximum relative sunspot number (RM) on cycle rise time (TR) separately for odd-following and even cycles (both in two split modes), and the dependencies of odd-following on even cycles separately for cycle rise times and maximum relative sunspot numbers (each also split into two mode pairs). The procedure was first to identify the proper mode for cycle 22 (modeA), which then explicitly defines the mode for cycle 23 (modeC). The presumed mode-inherent relations were then used to estimate the rise time for cycle 23 (3.7 0.5 yr) and its maximum amplitude (195.1 17.1). A second estimate of maximum amplitude, based directly on a presumed mode-inherent relation between maximum amplitudes for even and odd cycle pairs, yields a somewhat lower value (181.3 44.3). Thus, the results of this analysis supports previous findings that cycle 23 may be one of the largest amplitude cycles ever observed.     

Can the Solar Cycle Amplitude Be Predicted Using the Preceding Solar Cycle Length?

In this work, the evolution of the relationship between Solar Cycle Length of solar cycle n (SCL _n ) and Solar Cycle Amplitude of the solar cycle n+1 (SCA _n+1) is studied by using the R _Z and R _G sunspot numbers. We conclude that this relationship is only strongly significant in a statistical sense during the first half of the historical record of R _Z sunspot number whereas it is considerably less significant for the R _G sunspot number. In this sense we assert that these simple lagged relationships should be avoided as a valid method to predict the following solar activity amplitude.     

Amplitude and period of the dynamo wave and prediction of the solar cycle

The relation of the solar cycle period and its amplitude is a complex problem as there is no direct correlation between these two quantities. Nevertheless, the period of the cycle is of important influence to the Earth's climate, which has been noted by many authors. The present authors make an attempt to analyse the solar indices data taking into account recent developments of the asymptotic theory of the solar dynamo. The use of the WKB method enables us to estimate the amplitude and the period of the cycle versus dynamo wave parameters in the framework of the nonlinear development of the one-dimensional Parker migratory dynamo. These estimates link the period T and the amplitude a with dynamo number D and thickness of the generation layer of the solar convective zone h. As previous authors, we have not revealed any considerable correlation between the above quantities calculated in the usual way. However, we have found some similar dependences with good confidence using running cycle periods. We have noticed statistically significant dependences between the Wolf numbers and the running period of the magnetic cycle, as well as between maximum sunspot number and duration of the phase of growth of each sunspot cycle. The latter one supports asymptotic estimates of the nonlinear dynamo wave suggested earlier. These dependences may be useful for understanding the mechanism of the solar dynamo wave and prediction of the average maximum amplitude of solar cycles. Besides that, we have noted that the maximum amplitude of the cycle and the temporal derivative of the monthly Wolf numbers at the very beginning of the phase of growth of the cycle have high correlation coefficient of order 0.95. The link between Wolf number data and their derivative taken with a time shift enabled us to predict the dynamics of the sunspot activity. For the current cycle 23 this yields Wolf numbers of order 107±7.     

The Relationship between Prediction Accuracy and Correlation Coefficient

The correlation coefficient (r) between the maximum amplitude (R _m) of a sunspot cycle and the preceding minimum aa geomagnetic index (aa _min), in terms of geomagnetic cycle, can be fitted by a sinusoidal function with a four-cycle periodicity superimposed on a declining trend. The prediction index (χ) of the prediction error relative to its estimated uncertainty based on a geomagnetic precursor method can be fitted by a sinusoidal function with a four-and-half-cycle periodicity. A revised prediction relationship is found between the two quantities: χ<1.2 if r varies in a rising trend, and χ>1.2 if r varies in a declining trend. The prediction accuracy of R _m depends on the long-term variation in the correlation. These results indicate that the prediction for the next cycle inferred from this method, R _m(24)=87±23 regarding the 75% level of confidence (1.2-σ), is likely to fail. When using another predictor of sunspot area instead of the geomagnetic index, similar results can be also obtained. Dynamo models will have better predictive powers when having considered the long-term periodicities.     

Did predictions of the maximum sunspot number for solar cycle No. 22 come true?

Solar cycle No. 22 which started in 1986 seems to have already passed through a maximum. The maximum annual mean sunspot number was 157 for 1989. The maximum twelve-month running average was 159, centered on July 1989. For cycle 21, the similar value was 165 centered at December 1979. Thus, cycle 22 is slightly weaker than cycle 21. Schatten and Sofia (1987) had predicted a stronger cycle 22 (170 ± 25) as compared to cycle 21 (140 ± 20). Predictions based on single variable analysis, viz., R _z (max) versus aa(min) were 165 and came true. Predictions based on a bivariate analysis, viz., R _z (max) versus aa(min) and R _z (min) were 130 and proved to be underestimates. Other techniques gave over- or underestimates.     

Empirical estimate of p-mode frequency shift for solar cycle 23

We have obtained empirical relations between the p-mode frequency shift and the change in solar activity indices. The empirical relations are determined on the basis of frequencies obtained from BBSO and GONG stations during solar cycle 22. These relations are applied to estimate the change in mean frequency for the cycle 21 and 23. A remarkable agreement between the calculated and observed frequency shifts for the ascending phase of cycle 23, indicates that the derived relations are independent of epoch and do not change significantly from cycle to cycle. We propose that these relations could be used to estimate the shift in p-mode frequencies for past, present and future solar activity cycles, if the solar activity index is known. The maximum frequency shift for cycle 23 is estimated to be 265±90 nHz, corresponding to a predicted maximum smoothed sunspot number 118.1±35.     

Long-Term Sunspot Number Prediction based on EMD Analysis and AR Model

The Empirical Mode Decomposition (EMD) and Auto-Regressive model (AR) are applied to a long-term prediction of sunspot numbers. With the sample data of sunspot numbers from 1848 to 1992, the method is evaluated by examining the measured data of the solar cycle 23 with the prediction: different time scale components are obtained by the EMD method and multi-step predicted values are combined to reconstruct the sunspot number time series. The result is remarkably good in comparison to the predictions made by the solar dynamo and precursor approaches for cycle 23. Sunspot numbers of the coming solar cycle 24 are obtained with the data from 1848 to 2007, the maximum amplitude of the next solar cycle is predicted to be about 112 in 2011-2012.     

Prediction of Solar Cycle 24 Using Sunspot Number near the Cycle Minimum

Correlations between monthly smoothed sunspot numbers at the solar-cycle maximum [R _max] and duration of the ascending phase of the cycle [T _rise], on the one hand, and sunspot-number parameters (values, differences and sums) near the cycle minimum, on the other hand, are studied. It is found that sunspot numbers two?–?three years around minimum correlate with R _max or T _rise better than those exactly at the minimum. The strongest correlation (Pearson’s r=0.93 with P<0.001 and Spearman’s rank correlation coefficient r _S=0.95 with P=9×10^?12) proved to be between R _max and the sum of the increase of activity over 30 months after the cycle minimum and the drop of activity over 30 or 36 months before the minimum. Several predictions of maximal amplitude and duration of the ascending phase for Solar Cycle 24 are given using sunspot-number parameters as precursors. All of the predictions indicate that Solar Cycle 24 is expected to reach a maximal smoothed monthly sunspot number (SSN) of 70?–?100. The prediction based on the best correlation yields the maximal amplitude of 90±12. The maximum of Solar Cycle 24 is expected to be in December 2013?–?January 2014. The rising and declining phases of Solar Cycle 24 are estimated to be about 5.0 and 6.3 years, respectively. The minimum epoch between Solar Cycles 24 and 25 is predicted to be at 2020.3 with minimal SSN of 5.1?–?5.4. We predict also that Solar Cycle 25 will be slightly stronger than Solar Cycle 24; its maximal SSN will be of 105?–?110.     

Prediction of Solar Cycle 24 Using Geomagnetic Precursors: Validation and Update

In the previous study (Dabas et al. in Solar Phys. 250, 171, 2008), to predict the maximum sunspot number of the current solar cycle 24 based on the geomagnetic activity of the preceding sunspot minimum, the Ap index was used which is available from the last six to seven solar cycles. Since a longer series of the aa index is available for more than the last 10 – 12 cycles, the present study utilizes aa to validate the earlier prediction. Based on the same methodology, the disturbance index (DI), which is the 12-month moving average of the number of disturbed days (aa≥50), is computed at thirteen selected times (called variate blocks 1,2,…,13; each of them in six-month duration) during the declining portion of the ongoing sunspot cycle. Then its correlation with the maximum sunspot number of the following cycle is evaluated. As in the case of Ap, variate block 9, which occurs exactly 48 months after the current cycle maximum, gives the best correlation (R=0.96) with a minimum standard error of estimation (SEE) of ± 9. As applied to cycle 24, the aa index as precursor yields the maximum sunspot number of about 120±16 (the 90% prediction interval), which is within the 90% prediction interval of the earlier prediction (124±23 using Ap). Furthermore, the same method is applied to an expanded range of cycles 11 – 23, and once again variate block 9 gives the best correlation (R=0.95) with a minimum SEE of ± 13. The relation yields the modified maximum amplitude for cycle 24 of about 131±20, which is also close to our earlier prediction and is likely to occur at about 43±4 months after its minimum (December 2008), probably in July 2012 (± 4 months).