# Magnetic Helicity as a Predictor of the Solar Cycle

## Abstract

It is well known that the polar magnetic field is at its maximum during solar minima, and that the behaviour during this time acts as a strong predictor of the strength of the following solar cycle. This relationship relies on the action of differential rotation (the Omega effect) on the poloidal field, which generates the toroidal flux observed in sunspots and active regions. We measure the helicity flux into both the northern and the southern hemispheres using a model that takes account of the Omega effect, which we apply to data sets covering a total of 60 years. We find that the helicity flux offers a strong prediction of solar activity up to five years in advance of the next solar cycle. We also hazard an early guess as to the strength of Solar Cycle 25, which we believe will be of similar amplitude and strength to Cycle 24.

## Keywords

Solar cycle, observations Helicity, magnetic Sunspots, statistics## 1 Introduction

In this article we investigate the suitability of the magnetic-helicity flux as a predictor of solar activity.

Solar activity and its associated phenomena and drivers are known to have wide-ranging effect on the heliosphere, including (for example) how cosmic rays pass through said regions (Ferreira and Potgieter, 2004). The Sun is the only star close enough for us to routinely observe magnetic activity regions with high spatial and temporal resolution. The magnetic field produced by the solar dynamo is utterly fundamental to furthering this understanding (Charbonneau, 2005; Cameron and Schüssler, 2015). Solar dynamos are in a rare class where we see a self-sustained and reinforced magnetic field (Moffatt, 1978). This and other conditions on the Sun (for example, sustained nuclear fusion) are not currently reproducible on the Earth, making it an excellent laboratory for studying more exotic and extreme physical phenomena.

There have been many attempts to make predictions of the solar-activity cycle, which itself tends to be quantified by either sunspot/active-region number or area. Prediction methodologies can be split into three subsets: extrapolation methods, precursor methods, and model-based predictions (Munoz-Jaramillo, Balmaceda, and Deluca, 2013). An extrapolation method would take advantage of, for example, a sunspot data series, and any mathematical relations that can be derived from them, whilst a precursor method takes advantage of other observables such as poloidal field strength during solar minima. Finally, a model-based prediction (arguably a combination of the two methods) takes a range of data sets in an attempt to model the solar cycle, which are then passed through evolution equations. A model-based prediction could include physical effects such as the \(\alpha \Omega \) dynamo. Two recent reviews are Hathaway (2009) and Petrovay (2010).

One notable example is the work of Choudhuri, Chatterjee, and Jiang (2007), who use a mean-field dynamo model. In that article, they gave a prediction of the strength of Cycle 24, using a dynamo model-based prediction system, which has now been revealed to be accurate in at least an amplitude sense (a large reduction from Cycle 23). An earlier prediction for the same cycle, made by Dikpati, de Toma, and Gilman (2006), using a surface-flux-transport model, which had been successful in re-predicting previous solar cycles, predicted that Cycle 24 would in fact exceed Cycle 23 by a similar percentage to the decrease predicted by Choudhuri, Chatterjee, and Jiang (2007). This variability shows the difficulties in predicting solar cycles, even when using similar methods.

The work described in the current article fits within the definition of a precursor method, taking advantage of the magnetic-helicity flux during the preceding solar minima. We will attempt to display a relationship between this quantity and the strength of the following solar maxima, which is quantified here by sunspot number. We hope such a relationship would give us the ability to predict the strength of the cycle approximately five years in advance.

*i.e.*\(\boldsymbol{B}\cdot \hat{\boldsymbol{n}} = 0\)) is given by

The Omega effect is typically attributed to the polar magnetic-field lines, which move slower than those at the Equator due to differential rotation. As a consequence, they wrap themselves around the solar axis, generating toroidal field. Thus, when we measure the strength of the polar field at solar minimum, we should get an estimate of how much toroidal flux is being stored for the production of sunspots in the next solar cycle. However, polar field (in this article and similarly in others) is defined as a field average over a 15-degree polar cap, whilst our helicity flux takes account of all of the field lines in a hemisphere. We believe that this makes helicity flux a better measure of the Omega effect than polar field.

*etc.*(van Driel-Gesztelyi, Démoulin, and Mandrini, 2003).

There have been studies into the effectiveness of measuring the build-up of magnetic helicity to measure the likelihood of a solar eruption (see Pariat *et al.*, 2017). This is admittedly restricted to singular events, rather than a study of helicity flow and build-up over the whole Sun. However, one advantage of this type of study is that one can take account of the scale of the active region, rather than assuming uniformity.

Sunspots will typically appear in pairs: regions of intense, oppositely signed magnetic toroidal flux. These are often referred to as bipolar magnetic regions (BMRs). This magnetic flux rises up from the convection zone via a process known as magnetic buoyancy (Parker, 1955). Sunspots are generated in their highest number around an equatorial band – where the toroidal field is assumed to be strongest (a consequence of meridional flow) (Mordvinov, Grigoryev, and Peshcherov, 2012). Sunspots will typically appear as dark spots on the photosphere, with temperatures between 3000 – 4500 K, in contrast with the surrounding material at a temperature of \({\approx}\, 5700~\mbox{K}\). This is due to an increase in magnetic pressure from the intense toroidal flux penetrating the surface, which limits convection. This effect can be observed in the values of the plasma \(\beta \). Given their relation to the toroidal field (Balogh, Hudson, and Kristof, 2016), sunspots are an ideal measure of the Sun’s activity.

The Sun’s activity minimum is defined as the period in which we see very few sunspots, in between long periods of notable activity. At this time the Sun’s poloidal field is observed to be maximal. The action of differential rotation on the poloidal field (the Omega effect) then causes a maximal helicity flux in line with maximum poloidal field. The poloidal field that has been “wrapped up” by differential rotation goes on to be the toroidal flux rising out sunspots (the Babcock–Leighton mechanism).

Observations of the Sun have been performed for hundreds of years. Sunspot records are semi-reliably available back to 1610 (Hoyt and Schatten, 1998), although there is evidence of observations being made as early as the year 939 (Vaquero and Gallego, 2002). Other activity indices include the solar radio flux index (F_{10.7}), interplanetary magnetic field (IMF), flare index, polar faculae (Sheeley, 2008), and coronal index (Fe xiv emission) (Usoskin, 2005).

There are multiple repositories containing some subset of these indices over a variety of time periods. The data used in this work are provided by the Wilcox Solar Observatory. Equally comprehensive is the Space Weather Prediction Centre (National Oceanic and Atmospheric Administration – NOAA), also offering up-to-date measurements alongside historical archives. Finally, the Kislovodsk Mountain Astronomical Station offers a large collection of data over a large time period – some of which is employed in this work.

In this paper, we analyse the data provided by the Wilcox Solar Observatory using a variety of techniques described within. This includes hemispherical splitting, and a direct comparison with the prediction capabilities of the polar field, which is currently the most commonly used precursor indicator of the solar cycle. We also test the strength of the inverse of our hypothesis: that sunspot number can predict a magnetic-helicity flux cycle. We also analyse some reconstructed magnetic-field data (Makarov and Tlatov, 2000), to which we apply the same techniques described in section one.

## 2 Data and Analysis Techniques

The data show a resemblance between the helicity cycles during solar decline/minima and the following solar-activity cycle. If we take account of the relationship between helicity flux and magnetic flux (\({\sim}\, \Phi ^{2}\)) *versus* that between sunspot number and magnetic flux (\({\sim}\, \Phi \)), the differences in amplitude are approximately accounted for. The similarities seem to be much weaker for the weaker cycle (the third pair of peaks (Cycle 24), 2000 onwards). This cycle is also distinctive in that the helicity maximum lies at around \(20\%\) of that of the previous cycle. The cause of this anomaly could be the recent extended solar minimum (Frohlich, 2013). Regardless, both cycles are anomalously low within their own set, which is an inherent similarity.

The conclusion drawn is that a larger set of data is required. Magnetic-field data, however, are largely unavailable before the dates already graphed. We perform an analysis on the available data, with this restriction in mind.

### 2.1 Dynamic Linear Modelling and Kalman Smoothing

Both data sets of Figure 1 have high frequency noise, making it difficult to identify similar overall trends. In an attempt to smooth the data, we employ two data-analysis tools: dynamic linear modelling (DLM) and Kalman smoothing (KS). Smoothed data are commonly used when working with sunspot number, for the purposes of prediction (Petrovay, 2010).

*most likely*to generate the data set that we observe, with associated distributions.

This two-peak structure is particularly notable in the final sunspot cycle, due to the disparity between the height of the two peaks, and the structures of the cycles.

### 2.2 Pearson Correlation Coefficient

One standardized method for testing how well two data sets are correlated is the Pearson correlation coefficient (which we denote by \(P\)).

The correlation between the two data sets will be maximised at a given phase shift. We find that there is a clear difference between the optimum lag times, with the first (larger) peaks having a phase shift of 60 Carrington Rotations (4.5 years), whilst the second peaks are shifted by 92 Carrington rotations (6.9 years). These shifts give \(P\)-values for their relevant cycle pairs of 0.88 and 0.84 respectively, both of which indicate strong positive correlation. In all cases, unless otherwise stated, we use all data points that define the cycle to calculate correlation (between each successive minima).

Note that the third pair of peaks is not very well correlated in either figure, nor do we attempt to perform an optimisation on those data sets. Figure 1 shows how distinct they are. We therefore do not perform an analysis on this peak for sunspot number over the entire disc.

### 2.3 Integration

Values of integrated helicity flow and sunspot number with helicity shifted forwards.

Peak pair | Helicity data integrated | Sunspot number integrated | Ratio of integrations |
---|---|---|---|

1 | 59.50 CR | 55.15 CR | 94% |

2 | 42.39 CR | 52.40 CR | 81% |

3 | 17.5 CR | 26.50 CR | 66% |

Integrating helicity flux [\(\mathrm{d}H / \mathrm{d}t\)] over time clearly gives the total helicity [\(H\)] that passed through the corona in a given time period. For sunspot number, a temporal integral is less physically meaningful. However, we do note the relation between sunspot size, activity, and its period of existence (Henwood, Chapman, and Willis, 2010). For recurring sunspots (those that traverse the entire solar disc and re-appear in a following Carrington rotation), the temporal integral should take account of their increased size and activity.

The ratio column of Table 1 (and all subsequent tables) is obtained by dividing the smaller quantity by the larger, regardless of association. We see that, for the second peak in particular, the areas under the curves are very similar. The weakest comparison comes from the third pair of peaks. All three results match with a 66\(\%\) threshold, but with an average of 80\(\%\), indicating good area matching.

## 3 Hemispherical Helicity and Sunspots

Ratios of the values of integrated helicity flow and sunspot number separated by hemisphere.

Peak pair | Integration ratio (North) | Integration ratio (South) | Ratios of sums |
---|---|---|---|

2 | 73.5% | 98.7% | 94.9% |

3 | 93.2% | 78.0% | 83.1% |

Maximising correlation for the first pair of peaks for each hemisphere gives \(P = 0.76\) with a lag of 77 CR in the North, and \(P = 0.72\) and a 88 CR lag in the South. We encounter a problem when attempting to map the second helicity cycle onto its sunspot counterpart, as the shifted helicity has a length exceeding the sunspot data range. Correlating the minima gave a lag of 94 CR for the northern hemisphere and 98 for the southern. This fits in with our pattern of varying lag time. The two values of \(P\) obtained indicate strong positive correlation between the cycles.

Ratios of the values of integrated polar field and sunspot number separated by hemisphere.

Peak pair | Integration ratio (North) | Integration ratio (South) |
---|---|---|

2 | 54.8% | 90.4% |

3 | 41.1% | 55% |

### 3.1 Summation of Hemispheres

Integrated helicity flux and polar field when using hemisphere summation.

Peak pair | Helicity integration ratio | Polar field integration ratio |
---|---|---|

1 | 90.0% | 67.9% |

2 | 75.4% | 72.3% |

3 | 58.4% | 45.2% |

### 3.2 Sunspot Area

The limited extent of the hemispherical sunspot number data can be partially overcome with publicly available hemispherical sunspot area, which is recorded dating back to 1874, available from NASA ( solarscience.msfc.nasa.gov/greenwch.shtml ).

## 4 Interpolated Sunspot Data

Temmer *et al.* (2006) have generated a catalogue of sunspot data, split by hemisphere, for the years 1945 – 2004. This was performed using drawings taken from two separate observatories, the results from which were normalised using the international sunspot number. More details can be found in their article.

Calculating the integration ratios for the North gives a value of \(74\%\) for the North and \(82\%\) for the South. These values have a median of \(78\%\), which is considerably lower than that obtained when one uses sunspot number over the entire solar disk (\(90\%\)). This could be due to the reconstruction, given the improvements observed in the ratios when using the WDC data (especially in the case of the final cycle).

## 5 Sunspots Predicting Helicity Flow

In order to assert statistically that it is indeed helicity predicting the behaviour of the sunspot cycle, and not the converse, we must test the strength of said converse. Therefore, in this section we look at the strength of the theory that sunspot number predicts helicity flow. We must note that by including an additional sunspot cycle, the normalisation of the sunspots is changed slightly.

We notice that the minima in particular are very well matched, as opposed to the changing phase shift of the helicity → sunspot figures. In particular, the minima of the final pair of peaks, where we saw an elongated helicity cycle, are well correlated.

Integrated helicity flux and sunspot number with sunspot number shifted forwards.

Peak pair | Helicity data integrated | Sunspot number integrated | Ratio of integrations |
---|---|---|---|

1 | 59.5 CR | 53.50 CR | 90.0% |

2 | 42.39 CR | 48.90 CR | 87.2% |

3 | 17.5 CR | 46.79 CR | 37.4% |

We find that, excluding the final pair of peaks, a value of \(P = 0.804\) is achieved.

One explanation for these results is that cycle influence works not only in one direction, but both. We appear to see the cycle amplitude/strength (where strength is indicated by the integrated area) of a sunspot cycle being dictated by helicity flow, whilst the length of a helicity flux cycle is strongly correlated with that of the previous activity cycle.

## 6 Comparisons with Polar Field

The strength of the polar field during solar minima has often been used to predict the strength of the following solar maxima (Jiang, Chatterjee, and Choudhuri, 2007). The dipole moment has also been used (such as in the predictions made in Choudhuri, Chatterjee, and Jiang, 2007), back to 1978 (Schatten *et al.*, 1978). However, given that the polar field is more directly related to the Omega effect being described by our helicity flux, as well as a good measure of the dipole moment, we choose to compare the effectiveness of magnetic-helicity predictions with this benchmark. Helicity flux, as stated earlier, is a closer measure of the Omega effect than the polar field, making it likely to be more strongly correlated with sunspot number.

Of note is the amplitude differences between the polar field and magnetic helicity. Given that the expression used to calculate the magnetic-helicity flux is directly dependent upon the poloidal field, we would expect a closer relation in amplitude. These differences are likely due to the distribution of flux being more concentrated at mid to low latitudes.

Integrated polar field and helicity flux ratios with integrated sunspot number.

Peak pair | Polar field integration ratio | Helicity flux integration ratio |
---|---|---|

1 | 68% | 94% |

2 | 77% | 81% |

3 | 52% | 66% |

The ratios are consistently below those of Table 1, which have been included in the rightmost column. This indicates a larger disparity between the strength and structure of the polar-field cycles, *versus* that of sunspots. The structural comments made earlier continue to apply, which is expected, given the presence of \(B_{n}\) in the expression for the magnetic-helicity flux.

## 7 Reconstructed Magnetic Field Harmonics

There have been attempts to “recreate” measurements of the large-scale solar magnetic field using a variety of proxies. One example of this is described by Makarov and Tlatov (2000), who used H\(\alpha \) maps to calculate a spherical-harmonic decomposition of the said field, up to degree \(\ell = 10\). This technique is broadly described by Makarov and Sivaraman (1989). The authors have generously provided their decomposition data over the period 1958 – 2015, covering Carrington rotations 1400 – 2161. This period contains two additional solar cycles that are not available in the Wilcox data.

We must, however, take care when using reconstruction data due to possible inaccuracies, and we therefore suggest that any conclusions drawn from this work are taken as secondary to that in the previous sections.

Markarov and Tlatov have also provided data covering Rotations 800 – 1400 (1913 – 1958), although they have made it clear that this second data set is more likely to be inaccurate than the more recent reconstructions. We therefore split our analysis into two sections, each dealing with one of the two periods.

### 7.1 1958 – 2015

#### 7.1.1 Polar Field

Integrated polar-field and helicity-flux integration ratios.

Peak pair | Helicity flux ratio (smooth) | Polar field ratio (smooth) |
---|---|---|

1 | 39.60% (51.89%) | 96.20% (90.20%) |

2 | 59.57% (80.69%) | 97.69% (83.40%) |

3 | 63.44% (83.57%) | 81.98% (83.66%) |

4 | 46.30% (60.23%) | 99.40% (94.86%) |

The advantages of smoothing are shown in the bracketed values, making the most notable difference for the helicity-flux ratio. For the two central cycles, we note that smoothing brings the ratios of the polar field and helicity flux to almost equal values.

#### 7.1.2 Hemispherical Sunspot Number

Northern integrated sunspot number and helicity-flux integration ratios using hemispherical splitting.

Peak pair | Helicity flux (smooth) | Sunspot number (smooth) | Integration ratio (smooth) |
---|---|---|---|

1 | 18.32 (29.99) | 41.90 (59.17) | 43.7% (50.6%) |

2 | 32.80 (54.21) | 42.29 (61.66) | 77.6% (88.0%) |

3 | 30.91 (51.29) | 38.10 (54.53) | 81.1% (94.1%) |

4 | 21.55 (35.40) | 31.56 (45.28) | 68.3% (78.2%) |

#### 7.1.3 Comparisons with Wilcox Data

In this subsection, we compare the outputs of the harmonics in the reconstructed data with the outputs produced by the Wilcox data sets. This will give a measure of the accuracy of the reconstructed data, indicating how firmly we can make conclusions from the results that it provides.

There is a strong correlation between the two sets of helicity flux, indicating a good level of accuracy in the reconstructed data. Notable differences include the amplitude of the final observed cycle, and the “smoothness” of the cycles, particularly notable during 1980 – 1990. These differences are mostly removed using the KS process described in earlier sections. These differences in structure are what likely caused the integration procedure of the previous section to be skewed, and they are strongest during the aforementioned cycle.

Again, the final cycle is the most troublesome, giving the largest difference between the two data sets. The length of the cycles is the same for the two sets, but the amplitude is notably (approximately two times) larger in the reconstructed data. The cycle 1980 – 1990 has the largest amplitude of both sets of data, meaning that they share a common normalisation point. This means that we cannot assume that the amplitude difference is due to normalisation issues. The amplitude differences could instead be due to either an issue with the H\(\alpha \) maps used to determine the magnetic harmonics, or some underlying physical mechanism linked to the extended solar minima experienced during this period. More likely is that the reconstruction technique is simply not as accurate as real-time data taken by the Wilcox Solar Observatory, and they should thus be considered as less meaningful.

The reconstructed data also has a different relationship with its polar field than that of the Wilcox data. For Wilcox, we see an increasing cycle-amplitude difference between the two quantities with decreasing cycle strength. In the reconstructed data, however, the difference between helicity flux and polar field is fairly constant from cycle to cycle.

The exact reasons for the differences described are not known, but they are worthy of further investigation, being of particular importance for the helicity cycles of 1980 – 1990 and 2000 – 2012.

### 7.2 1913 – 2015

## 8 Predicting Solar Cycle 25

We are unfortunately not at a point where we can be sure that the helicity flux has reached its peak, making the accuracy of any prediction lower than we would like.

Over the course of this article we have demonstrated statistically that the helicity-flux cycle is a good predictor of the following activity cycle. With this in mind, based on the assumption that Cycle 25’s preceding helicity-flux cycle has reached its maxima, we predict that the amplitude of Cycle 25 will be only slightly higher than that of Cycle 24.

Using our three existing cycles, we estimate that Cycle 25 will have an amplitude of 117 (given as the maximum value of sunspots per Carrington rotation over a cycle), \(50 \%\) of that of Cycle 22, calculated by regression analysis on the amplitudes. This is quite similar to that of Cycle 24’s amplitude of 104. With only three data points, the standard statistical error was found to be quite un-realistic, and we therefore do not include it here.

Previous sections have also indicated that the helicity flux as a prediction mechanism is best when we compare the areas under curves. At the time of writing, the current helicity-flux cycle has an area of 5.63 CR. If we again assume that the helicity-flux cycle has reached its halfway point, this would give us a total area of \({\approx}\, 11.26\) CR.

Performing regression analysis on the areas, as an indication of the overall strength of the cycle (as in Table 1) gives a predicted normalised area of 24.48 CR for Cycle 25. This is again approximately equivalent to the area of Cycle 24 (26.50 CR). Errors have once again been neglected.

Gopalswamy *et al.* (2018) similarly found that Cycle 25 will be akin to Cycle 24.

## 9 Conclusion

We have completed a fairly comprehensive statistical analysis of the hypothesis that helicity flux during solar minima can be used to predict the strength of the following solar maxima. We found a strong indication of causation between the two sets of data. This was most noticeable when we performed our analysis on a hemispherical basis, where we saw ratios consistently outperforming those calculated over the entire solar disc. In some cases, this ratio was more than doubled (\(41.1\%\) to \(93.2\%\)) when using helicity flux as opposed to polar field. Notably, the strength of the polar field as a precursor seemed to decrease when we looked at individual hemispheres. This result indicates that future attempts to predict the strength of solar cycles should use a hemispherical model. In particular, the relationship appears stronger than that offered by the polar field, which is currently the most popular precursor indicator of solar activity. The only advantage that we found when using polar field was that it seemed to occasionally excel in terms of a more-exact amplitude prediction. However, this only occurred when the analysis was performed over the entire disc, rather than with respect to hemispheres. The helicity flux outperforming the polar field indicates that magnetic-field activity in regions beyond the polar cap is important for the progression of the solar dynamo into its next maxima.

In an attempt to strengthen our result, we obtained reconstructed data, but this was found to imply that the polar field was the stronger indicator, even if helicity flux still offered a high correlation. However, these data were found to be inaccurate in some places, and highly chaotic. Some underlying structure of magnetic cycles appeared to be absent. The reconstruction then, with the necessary amount of trepidation, does also indicate a causal relationship between helicity flux and sunspot number.

In conclusion, we believe that we have demonstrated the statistical link between these two physical quantities, using two different data sets over a period covering approximately 50 years.

Additionally, we made a speculative guess as to the amplitude of the forthcoming solar cycle, Cycle 25, which we believe would be approximately the same (perhaps slightly greater) amplitude and strength as Cycle 24.

## Notes

### Acknowledgements

The Wilcox Solar Observatory data used in this study to calculate magnetic-helicity flux were obtained via the web site wso.stanford.edu , courtesy of J.T. Hoeksema, with thanks. Sunspot Number Source: WDC-SILSO, Royal Observatory of Belgium, Brussels. The reconstructed field data were generously provided by Andrey Tlatov, from Makarov and Tlatov (2000). G. Hawkes would like to thank the STFC for their funding under grant ST/N504063/1. The authors would also like to thank Tim Jupp for useful discussions, and the anonymous reviewers for helping to improve this article.

### Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

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