# A New Formula for Predicting Solar Cycles

###### Abstract

A new formula for predicting solar cycles based on the current theoretical understanding of the solar cycle from flux transport dynamo is presented. Two important processes – Babcock-Leighton mechanism and variation in meridional circulation, which are believed to be responsible for irregularities of the solar cycle, are constrained using observational data. We take the polar field near minima of the cycle as a measure of the randomness in the Babcock-Leighton process, and the decay rate near the minima as a consequence of the change in meridional circulation. We couple these two observationally derived quantities into a single formula to predict the amplitude of the future solar cycle. Whether this formula is reasonable to predict future solar cycle is also discussed using simulation from the flux transport dynamo model.

## 1 Introduction

The sunspot cycle with the approximate period of 11 years is one of the most intriguing natural cycles known to mankind. Solar disturbances, which become more frequent during the peak of this cycle, control the space environment of the Earth and affect our lives in various ways. Developing a method for predicting the strength of a solar cycle in advance is of utmost societal importance (Pesnell, 2008; Petrovay, 2010; Choudhuri, 2018). The aim of this paper is to propose a formula for predicting solar cycles. To apply this formula, we need values of certain quantities which become available towards the end of the previous cycle. Once these values are known, it will be possible to use this formula to predict the forthcoming cycle. We discuss how we arrive at this formula by analyzing the data of the last few solar cycles. We also look at the question whether this formula can be justified on the basis of the flux transport dynamo model, the most successful theoretical model for explaining different aspects of the solar cycle.

It has been known that there is a correlation between the polar field of the Sun during the solar minimum and the strength of the next cycle, allowing us to use this polar field as a predictor (Svalgaard et al., 2005; Schatten, 2005). The theoretical explanation of this correlation on the basis of the flux transport dynamo model was provided by Jiang et al. (2007). The poloidal field is generated in the flux transport dynamo model by the Babcock–Leighton (BL) mechanism from the decay of tilted bipolar sunspots. Since there is a scatter in the tilt angles of bipolar sunspots around the average given by Joy’s law (Longcope & Choudhuri, 2002; Wang et al., 2015), the BL mechanism has an inherent randomness, leading to the unequal production of the poloidal field in different cycles (Karak & Miesch, 2017). Since this poloidal field is brought to the polar region by the meridional circulation to produce the polar field at the end of the cycle and also diffuses to the bottom of the convection zone to act as the seed of the next cycle, we have this correlation between the polar field at the end of a cycle and the strength of the next cycle. Choudhuri et al. (2007) developed a methodology of incorporating the randomness of the BL mechanism in the theoretical flux transport dynamo model and predicted cycle 24 before its onset. Their prediction turned out to be the first successful prediction of a solar cycle from a theoretical dynamo model.

When the works mentioned in the last paragraph were being done, it was not yet realized that there can be another important source of irregularities in the solar cycle—fluctuations in the meridional circulation (MC). We know that there is a periodic variation of MC with the solar cycle, presumably due to the Lorentz force of the dynamo-generated magnetic field (Hazra & Choudhuri, 2017). There is indirect evidence that there have been random fluctuations in MC in the past with longer time scales (Karak & Choudhuri, 2011). It is the time scale of MC which sets the period of the flux transport dynamo. A slower MC makes cycles longer. If the diffusion time scale is shorter than the MC time scale—which is the case if we assume a value of diffusion based on simple mixing length arguments—then longer cycles become weaker due to a more prolonged action of diffusion. In other words, there would be an anti-correlation between the duration of the cycle and the cycle strength. This anti-correlation helps in explaining some features of observational data such as the Waldmeier effect (Karak & Choudhuri, 2011). It seems that there is a time delay in the effect of MC on the cycle strength. As a result, the peak of a cycle depends not on the value of MC at that time, but a few years earlier (Hazra et al., 2015). If the MC was weaker a few years earlier, that would make the decay rate of the previous cycle smaller. We actually find a correlation between the strength of a cycle and the decay rate of the previous cycle (Hazra et al., 2015).

We conclude that the irregularities of solar cycle are caused by the combined effect of two factors: (i) randomness in BL mechanism for poloidal field generation, and (ii) fluctuations in MC. Choudhuri & Karak (2012) developed a theoretical model of the grand minima of solar cycles by including both of these in their dynamo model. Now our aim is to develop a method for predicting a future cycle by taking both these factors into consideration. The polar field at the sunspot minimum captures the effect of randomness in the BL mechanism in the previous cycle. On the other hand, the decay rate of the previous cycle provides the information about MC during the phase which is important for determining the strength of the next cycle. So we expect the peak strength of the cycle to be given by a formula of the type

(1) |

Our job now is to check whether the strengths of the past cycles can be matched by some suitable choices of the indices and . We also look at the question whether we find any support for such a formula from the simulations of the flux transport dynamo.

The next Section is devoted to a discussion of the possibility of a formula of type given in Eq. 1 based on observational data. Then Sec. 3 discusses the support for it in theoretical dynamo simulations. Whether such a formula can help us in predicting the upcoming cycle is discussed in Sec. 4. Finally our conclusions are summarized in the last Section.

## 2 Observational Study

We have reasonably trustworthy data of sunspot number from at least the beginning of the 20th century. We can easily obtain reliable values of the peak sunspot number and the decay rate for several previous solar cycles from these data. To check whether a formula like in Eq. 1 worked for the past cycles, we need the values of the polar field during several sunspot minima. We have actual regular measurements of the polar field only from the mid-1970s. However, there are several proxies which indicate how the polar field might have evolved at earlier times. The important proxies we may consider are: (i) polar flux obtained from polar faculae number as presented by Muñoz-Jaramillo et al. (2012); (ii) the parameter obtained by Makarov et al. (2001) from the position of neutral lines (indicated by filaments) on the solar surface; and (iii) the Polar Network Index (PNI) developed by Priyal et al. (2014) from chromospheric networks seen in Kodaikanal Ca K spectroheliograms. The bottom panel of Fig. 1 plots polar fields inferred from these three different proxies, below the sunspot number shown in the upper panel. For all these three proxies for the polar field, we plot their normalized values by putting their maximum values in the range equal to 1. We see that during much of the time the three proxies give very similar values of the polar magnetic field. However, during a few solar minima (such as the minimum before cycle 16), we find that some of the proxies diverge widely. We do not know the reason behind this. Since the polar faculae number happens to be the most widely studied proxy for the polar field (Sheeley, 1991; Muñoz-Jaramillo et al., 2012), we use the polar field obtained from this proxy in our analysis. We should keep in mind that the value of obtained from this proxy may not be very reliable during the times when the different proxies give very different results.

For the sunspot number, we have used two datasets. One, the group sunspot number
from the Greenwich Observatory
^{1}^{1}1https://solarscience.msfc.nasa.gov/greenwch/spot_num.txt;
and another one, the calibrated monthly total sunspot number (Clette et al., 2014) from WDC-SILSO, Royal Observatory
of Belgium, Brussels
^{2}^{2}2http://www.sidc.be/silso/datafiles.
Both of the datasets are smoothed by
using Gaussian filter with FWHM = 2 years and are shown in the upper panel of Fig. 1. As we mentioned earlier, the decay rate during the late phase of the
cycle is a good precursor to predict the next cycle. We now calculate the decay rate during the late phase
of the cycle following the procedures as explained in Hazra et al. (2015). We first present the results obtained by
using the Greenwich sunspot number. Then we shall present results based on SIDC data.

Fig. 2 presents the results we get by using the Greenwich sunspot number. Fig. 2a shows the correlation between the decay rate calculated during the late phase of the cycle and the peak amplitude of the next cycle. The correlation coefficient is less than what we get () by considering all 23 cycles with same Gaussian smoothing with FWHM = 2 years (see table 1 of Hazra et al. (2015)). In all of our calculations, we have considered the data from cycle 15 up to the present time, as the polar faculae data are not available before cycle 15. The different campaigns of the polar faculae data are calibrated with the direct measurement of the polar fields from Wilcox solar observatory from the mid-1970s. We use the calibrated long-term polar flux dataset for the polar fields presented by Muñoz-Jaramillo et al. (2012). We determine the solar minima from the sunspot data of Greenwich observatory first and then calculate the polar field near the solar minima by averaging over a span of one year before the minimum and one year after the minimum. The correlation between polar field near minima and the next cycle amplitude is shown in Figure 2b. For the polar field , we have used the polar flux (in Mx) which is obtained by multiplying the normalized polar field shown in Fig. 1 by a factor of following the calibration given by Muñoz-Jaramillo et al. (2012). We see that The correlation () is rather poor—mainly due to the reason that cycles 16 and 21 show large scatters. Finally, to check whether and can be used as good precursors for predicting the next cycle, Figure 2c and 2d show the correlations of the peak of the next cycle with them. We find that among all possible values of and appearing in (1), the combination of [case (i)] and [case (ii)] give the highest correlations with the amplitude of the next cycle. The correlation coefficients for case (i) is 0.70 (null hypothesis rejected with probability 96.3) and for case (ii) is 0.73 (null hypothesis rejected with probability 97.3). These correlation coefficients are significantly higher than the correlation coefficient in the case of polar fields alone and slightly higher than the correlation coefficient in the case of decay rates alone.

It should be clear from Fig. 2 that the points corresponding to cycles 16 and 21 in all the plots make correlation coefficients lower than what they would otherwise be. These points are indicated in brown color. Interestingly, we see in Fig. 1 that the various proxies of the polar field preceding these cycles did not match each other well. We have no explanation for this. However, this raises the question whether polar faculae counts at these times were good indicators for the polar field. In particular, the cycle 16 was the weakest cycle in the century. Since, according to the flux transport dynamo model with reasonably diffusivity (Jiang et al., 2007), the polar field during the minima preceding the cycles are expected to be correlated with the strengths of the cycles, we expect the polar field before the cycle 16 would be weak. But in the Polar faculae count and PNI, the polar field is comparable to the polar field preceding the strongest cycle 19. However, the polar field from index of Makarov et al. (2001) before cycle 16 is weak as expected from theory. We also point out that, during the minima preceding cycles 20 and 21, there was a dearth of spectroheliogram plates and some error may be introduced in the PNI count during these times (Priyal et al., 2014). Had we used as the proxy of the polar field rather than the polar faculae count, then the points corresponding to cycles 16 and 21 in Fig. 2 would have considerably less scatter. Since we are unsure of the polar field during the minima preceding cycles 16 and 21, we have also calculated the correlation coefficients for our newly proposed precursors without the points 16 and 21. The correlation coefficient for case (i) with square root dependence on polar fields and decay rate is 0.97 with null hypothesis rejected with probability 99.9. For case (ii), the correlation coefficient without cycle 16 and 21 is 0.98 (99.9). Having these large values of correlation coefficients, we argue that the newly proposed precursor formulae hold a promise to predict the future solar cycle based on the polar field measurement preceding the cycle and decay rate at late phase of the previous cycle. These two quantities represent the two physical causes behind irregularities in the solar cycles, namely fluctuations in the B-L mechanism and fluctuations in the meridional circulation respectively. We also point out that, without the cycles 16 and 21, the correlation coefficient for polar field alone with the next cycle amplitude is around 0.90 (99.5). This is significantly higher than what we get when these cycles are included, but less than the the correlation coefficients obtained with the newly suggested precursors. Therefore, the precursors we are suggesting would be very helpful in predicting a future solar cycle with better accuracy.

Next, we repeat the same exercise with the newly calibrated international sunspot numbers from SIDC, Royal Observatory of Belgium. The results are presented in Fig. 3. Each panel in Fig. 3 is similar to the Fig. 2. It is clear from figure 3 that the newly proposed precursors are more highly correlated with the peak amplitude of the cycle compared to either the decay rate of the previous cycle alone or polar field during the preceding minima alone. We find that the correlation of decay rate with the next cycle amplitude is 0.62 (94.5) and the same for polar fields near minima with the next cycle is also 0.62 (94.5). On the other hand, the newly proposed precursors (Fig. 3(c) and (d)) have correlation coefficients of 0.69 (97.4) for case(i) and 0.70 (97.6) for case(ii), which are higher than the correlation coefficients found from the polar field alone (Fig. 3 (a)) and the decay rate alone (Fig. 3 (b)). The correlation coefficients do get improved if we exclude points 16 and 21. However, when we use SIDC data, we find that the correlation coefficient obtained from the polar field alone becomes quite high and the correlation coefficients obtained with the new precursor formulae (for each case (i) and case (ii)) become slightly less. This shows the hazard of doing statistical analysis with very few data points. Although our analysis is severely restricted by the very limited number of data points and the values of the polar field are uncertain in some cases even among these few data points, we still find tantalizing hints that the new precursor formulae we are suggesting are better for predicting a future cycle than the polar field at the minimum alone or the decay rate of the previous cycle alone. We have to wait for a few more cycles (at least for half a century) before we can draw firmer conclusions.

## 3 Theoretical Interpretation

As explained in the previous Section 2, our observational study motivated us to introduce new precursors for predicting the future solar cycle. In this section, we discuss whether we can provide any justification for these new precursors from a theoretical flux transport dynamo model. Presently, the flux transport dynamo model is the most promising model to explain the various features of the solar cycle and its irregularities. The cyclic oscillation between the poloidal field and the toroidal field produces the solar cycle in this model (Choudhuri et al., 1995; Durney, 1995; Choudhuri & Dikpati, 1999; Chatterjee et al., 2004). The differential rotation stretches the poloidal field to generate the toroidal field. Then the toroidal field rises up to the photosphere due to magnetic buoyancy and creates bipolar magnetic regions which appear with tilts produced by the Coriolis force (D’Silva & Choudhuri, 1993). The decay of tilted bipolar magnetic regions due to turbulent diffusion produces the poloidal field via the Babcock-Leighton mechanism (Babcock, 1961; Leighton, 1969). For a detailed explanation of the model, please see the reviews by Choudhuri (2011), Charbonneau (2014) and Karak et al. (2014). The irregularities in the solar cycle mainly arise because of the inherent randomness in the Babcock-Leighton mechanism, as first pointed out by Choudhuri (1992) and then analyzed by many authors (Charbonneau & Dikpati, 2000; Karak & Choudhuri, 2011; Hazra et al., 2015) who successfully reproduced many observed irregularities in the solar cycle. But some of the irregular properties (e.g., the Waldmeier Effect, the correlation between decay rate and the amplitude of the next cycle) are not reproduced using only the fluctuations in the Babcock-Leighton mechanism. Karak (2010) pointed out that the fluctuations in the meridional circulation can be another source of irregularities in the solar cycle. By including fluctuations in the meridional circulation, Karak & Choudhuri (2011) successfully reproduced the Waldmeier effect and Hazra et al. (2015) reproduced the correlation of the decay rate with the next cycle amplitude.

We now present some theoretical results obtained by introducing fluctuations in both the Babcock-Leighton process and the meridional circulation in our dynamo model. The parameters for the flux transport models are chosen the same as in section 4.3 of Hazra et al. (2015). A 100 fluctuation is introduced in the Babcock-Leighton with coherence time of one month, whereas a 30 fluctuation with 30 years of coherence time is introduced in the meridional flow. This gives us reasonably irregular solar cycles. The decay rate during the late phase of the cycle is calculated from the output of the theoretical simulation and its correlation with the amplitude of the next cycle is shown in Fig. 4(a). This is the same as the results presented in Hazra et al. (2015). We have calculated the peak polar field near the minima of the cycles. The correlation between this polar field near minima and the next cycle amplitude is shown in Fig. 4(b). The two bottom panels of Fig. 4 show how well the new precursors introduced by us on the basis of the observational data are correlated with the amplitude of the next cycle. It may be noted that the theoretical correlation between the polar field at the minima and the peak of the next cycle, as shown in Fig. 4(b), is already very high—considerably higher than what is seen in the observational data (see Figs. 2(b) and 3(b)). In such a situation, it is somewhat difficult to ascertain whether the precursors give even better correlations. We made several independent runs of our code and found that the correlations computed in different runs are often slightly different, although they have the same statistical nature. In Fig. 4 we have presented results from one run in which the both the precursors (case (i)) and (case (ii)) give better correlations with the next cycle amplitude than , but about the same as . In fact, the correlation coefficients for the precursors are marginally lower than that for . Since alone gives such a good correlation in our theoretical model, precursors which combine it with the less strongly correlated quantity tend to have slightly less correlations.

To verify theoretically whether our proposed precursors are really better for predicting a future cycle than either or , we need a theoretical dynamo model which faithfully reproduces all the different features of the solar cycle. The model presented in section 4.3 of Hazra et al. (2015) reproduced most of the features of the solar cycle. However, we now realize that it produces a tighter correlation between the polar field at the minima and the next cycle amplitude compared to what is observed. As pointed out by Jiang et al. (2007), this correlation becomes better on increasing the value of the turbulent diffusion. So we need to do calculations with a model having a lower value of diffusion in order to increase the scatter in the correlation between and the next cycle. On the other hand, Chatterjee et al. (2004) showed that the dynamo solution tends to become quadrupolar (contrary to what is observed) on decreasing the value of diffusion. It is, therefore, needed to construct a theoretical model which prefers dipolar parity but gives more scatter in the correlation between and the next cycle. On the basis of several trial runs using different combinations of parameters, so far we have not been able to come up with a theoretical model which has this property. So we are right now unable to give very strong arguments on the basis of a theoretical dynamo model that the precursors suggested by us are better in predicting a future cycle than and . Based on the theoretical calculations we have presented, we can certainly say that our precursors are very good for predicting the next cycle, although the correlation of with the next cycle is already so strong in the theoretical model that it is not clear whether the precursors correspond to significant improvements.

## 4 Future Cycle Prediction

As we believe that the two precursors we have suggested (case(i) and case(ii)) are particularly well suited to predict future cycles, we now write down appropriate formulae based on these precursors which can be readily used for predicting future cycles. Since the sunspot database from SIDC is the best calibrated and most trustworthy sunspot number database, we consider the best straight line fits of the data points for the two cases of the SIDC database only (fig. 3(c) and 3(d)). We perform the least square fitting for both the cases, i.e. case (i) with (fig. 3(c)) and case (ii) with (fig: 3(d)), to arrive at formulae from which the amplitude of the next solar cycle can be calculated, after knowing the values of the precursors and around the minimum before the cycle. The formula representing the case (i) () is

(2) |

whereas for case (ii) (with ) the formula is

(3) |

Note that the best fit straight lines in fig. 3(c) and 3(d) do not pass through the origin. In other words, a future cycle is never predicted to have zero strength for any combination of positive values of and . It is thus clear that the formulae we have arrived at cannot handle the situation of a grand minimum. Presumably, these formulae would give good results when the various parameters lie within a reasonable range of values. When we know the appropriate values of the polar field and decay rate (of the previous cycle) at the time of a solar minimum, we can use these formulae for predicting the next cycle.

Finally, we calculate the peak sunspot number of cycle 25 based on our newly obtained precursor formulae given in Eq. 2 and 3. As these formulae need and values, we calculate them individually. We calculate the polar field near the minimum preceding cycle 25 by using polar field data from Wilcox Solar Observatory. Although throughout the paper we have used the calibrated long-term polar flux dataset from Mount Wilson Observatory(Muñoz-Jaramillo et al., 2013), we use Wilcox Solar Observatory polar field data here because it is the most reliable directly measured available polar field dataset. The long-term data of polar flux, which we had used for obtaining the new precursor formulas (Eq. 2 and 3), has three cycles overlapping with the WSO direct polar field data. Therefore, we have calibrated the MWO polar flux data with respect to the WSO polar field data. After implementing the calibration factor, the polar flux near the minimum preceding solar cycle 25 corresponding to the WSO data turns out to be () Mx. We use this value of for predicting the solar cycle 25. Next, we need to calculate the decay rate just before the minimum preceding cycle 25. We have calculated it by following the same procedure as in Sec. 2. From the SIDC sunspot number database, the decay rate just before the minimum preceding cycle 25 is found to be 23.51. We plug these values of and into the new precursor formulae (2 and 3) to get an estimate of the peak amplitude of cycle 25.

## 5 Conclusion

Predicting a solar cycle before its onset is a challenge which interests both solar physicists and the general public. Even a decade ago, it was a rather uncertain art. Pesnell (2008) combined all the predictions that were made for cycle 24 in Fig. 1 of his paper. It was clear that the various predictions covered virtually the entire range of all possible values of the peak sunspot number. During the intervening years, our understanding of the physical basis for the solar cycle prediction has deepened considerably. The aim of the present paper has been to come up with a simple formula for predicting the forthcoming cycle on the basis on this new understanding.

We now believe that the irregularities of the solar cycle are produced primarily by two factors: fluctuations in the BL mechanism and fluctuations in MC. So, in order to predict a cycle, we need to include contributions from both these factors. Since the polar field at the beginning of the cycle provides the relevant information for fluctuations in the BL mechanism and the decay rate at the end of the previous cycle provides the relevant information for fluctuations in MC, we have looked for formulae combining and which have good correlation with the peak of the next cycle. The formulae have to be calibrated by using the data of the past cycles. The main bottleneck in this process is the lack of polar field data before the 1970s. We have pointed out three proxies for the polar field. During much of the time, these three proxies give very similar values for the polar field. However, there have been intervals during which some of the proxies diverged and we do not have a reliable information about the polar field in those intervals. We saw that some of the data points in our correlation plots with largest scatters corresponded to these intervals. Probably our formulae will get properly calibrated only after about half a century when solar astronomers have additional data for about 4–5 more cycles. It is possible that improved calibration will lead to formulae which can handle even the grand minima (which our present formulae cannot handle). In this paper, we merely try to do as good a calibration as possible at the present time. Based on the formulae we have arrived at, we have presented our prediction for the upcoming cycle 25.

Unfortunately our theoretical flux transport dynamo model could not provide too much help in coming up with the right formulae. In our group, we have developed a theoretical model of the solar cycle by using a combination of parameters in the flux transport dynamo model. This model has explained many aspects of the solar cycle extremely well. However, we now realize that this model gives a much a better correlation between the polar field at the end of a cycle and the strength of the next cycle than what is actually observed. This model is, therefore, not particularly suited for addressing the questions we are interested in. The only thing we can say is that the theoretical model corroborates that the formulae we have proposed should be good for predicting future cycles.

In summary, we have to say that the formulae proposed in this paper are of somewhat provisional nature at the present time, since we have very limited amount of past data to calibrate these formulae. However, we believe that these formulae show the right way for predicting future solar cycles. We expect that these formulae will get improved as solar astronomers get more data to calibrate them in future and will eventually prove very powerful tools for predicting solar cycles before their onset.

This work is partially supported by the JC Bose Fellowship awarded to A.R.C. by the Department of Science and Technology, Government of India. G.H. is supported by the Zhuoyue Postdoctoral Fellowship, Beihang University, China.

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