Features of the Solar Active Cycles

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A Bayesian Analysis of the Correlations Among Sunspot Cycles

Sunspot numbers form a comprehensive, long-duration proxy of solar activity and have been used numerous times to empirically investigate the properties of the solar cycle. A number of correlations have been discovered over the 24 cycles for which observational records are available. Here we carry out a sophisticated statistical analysis of the sunspot record that reaffirms these correlations, and sets up an empirical predictive framework for future cycles. An advantage of our approach is that it allows for rigorous assessment of both the statistical significance of various cycle features and the uncertainty associated with predictions. We summarize the data into three sequential relations that estimate the amplitude, duration, and time of rise to maximum for any cycle, given the values from the previous cycle. We find that there is no indication of a persistence in predictive power beyond one cycle, and we conclude that the dynamo does not retain memory beyond one cycle. Based on sunspot records up to October 2011, we obtain, for Cycle 24, an estimated maximum smoothed monthly sunspot number of 97±15, to occur in January??C?February 2014 ± six months.     

What the Sunspot Record Tells Us About Space Climate

The records concerning the number, sizes, and positions of sunspots provide a direct means of characterizing solar activity over nearly 400 years. Sunspot numbers are strongly correlated with modern measures of solar activity including: 10.7-cm radio flux, total irradiance, X-ray flares, sunspot area, the baseline level of geomagnetic activity, and the flux of galactic cosmic rays. The Group Sunspot Number provides information on 27 sunspot cycles, far more than any of the modern measures of solar activity, and enough to provide important details about long-term variations in solar activity or “Space Climate.” The sunspot record shows: 1) sunspot cycles have periods of 131± 14 months with a normal distribution; 2) sunspot cycles are asymmetric with a fast rise and slow decline; 3) the rise time from minimum to maximum decreases with cycle amplitude; 4) large amplitude cycles are preceded by short period cycles; 5) large amplitude cycles are preceded by high minima; 6) although the two hemispheres remain linked in phase, there are significant asymmetries in the activity in each hemisphere; 7) the rate at which the active latitudes drift toward the equator is anti-correlated with the cycle period; 8) the rate at which the active latitudes drift toward the equator is positively correlated with the amplitude of the cycle after the next; 9) there has been a significant secular increase in the amplitudes of the sunspot cycles since the end of the Maunder Minimum (1715); and 10) there is weak evidence for a quasi-periodic variation in the sunspot cycle amplitudes with a period of about 90 years. These characteristics indicate that the next solar cycle should have a maximum smoothed sunspot number of about 145 ± 30 in 2010 while the following cycle should have a maximum of about 70 ± 30 in 2023.     

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.     

The Shape of Solar Cycle Described by a Modified Gaussian Function

The shape of each sunspot cycle is found to be well described by a modified Gaussian function with four parameters: peak size A, peak timing t _m, width B, and asymmetry α. The four-parameter function can be further reduced to a two-parameter function by assuming that B and α are quadratic functions of t _m, computed from the starting time (T ₀). It is found that the shape can be better fitted by the four-parameter function, while the remaining behavior of the cycle can be better predicted by the two-parameter function when using the data from a few (about two) months after the starting time defined by the smoothed monthly mean sunspot numbers. As a new solar cycle is ongoing, its remaining behavior can be constructed by the above four- or two-parameter function. A running test shows that the maximum amplitude of the cycle can be predicted to within 15% at about 25 months into the cycle based on the two-parameter function. A preliminary modeling to the first 24 months of data available for the current cycle indicates that the peak of cycle 24 may probably occur around June 2013±7 months with a size of 72±11. The above results are compared to those by quasi-Planck functions.     

Search for relationship between duration of the extended solar cycles and amplitude of sunspot cycle

Duration of the extended solar cycles is taken into the consideration. The beginning of cycles is counted from the moment of polarity reversal of large-scale magnetic field in high latitudes, occurring in the sunspot cycle n till the minimum of the cycle n + 2. The connection between cycle duration and its amplitude is established. Duration of the “latent” period of evolution of extended cycle between reversals and a minimum of the current sunspot cycle is entered. It is shown, that the latent period of cycles evolution is connected with the next sunspot cycle amplitude and can be used for the prognosis of a level and time of a sunspot maximum. The 24th activity cycle prognosis is made. The found dependences correspond to transport dynamo model of generation of solar cyclicity, it is possible with various speed of meridional circulation. Long-term behavior of extended cycle's lengths and connection with change of a climate of the Earth is considered. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar activity in the sixteenth and seventeenth centuries (a revision)

Maximum relative sunspot numbers for the 16th and 17th century were computed by means of the dependence of the maximum relative sunspot numbers on the solar cycle rise time and on the cycle asymmetry. In these dependencies four separate modes of relations, two for odd and two for even cycles, were identified. These modes are coupled two and two in even-odd cycle pairs. The rise times and the asymmetries of solar cycles in the 16th and 17th centuries were taken from cycle extreme estimates by Schove (1979), from auroral and telescopic sunspot observations during this period, but with some necessary corrections. Annual relative sunspot numbers and decade averages were estimated from the cycle maxima and the epochs of extremes. In addition, the efficiency of auroral records in latitudes lower than 55 deg was computed for the time interval 1500–1868. For this purpose the dependence of occurrence numbers of aurorae on the cycle and decade means of the relative sunspot numbers was derived.     

An Estimate for the Size of Cycle 23 Using Multivariate Relationships

A new technique based on multivariate analysis is described which allows for the prediction of the size or maximum amplitude of cycle 23. The technique uses the number of geomagnetic disturbances at selected times during the decline of cycle 22 (as precursors) to predict a maximum of about 152 (in terms of smoothed monthly mean sunspot numbers) for cycle 23. On the basis of this technique, hindcasts for cycles 17–22 are shown to agree with observed values within 5%.     

On the maximum rate of change in sunspot number growth and the size of the sunspot cycle

Statistically significant correlations exist between the size (maximum amplitude) of the sunspot cycle and, especially, the maximum value of the rate of rise during the ascending portion of the sunspot cycle, where the rate of rise is computed either as the difference in the month-to-month smoothed sunspot number values or as the average rate of growth in smoothed sunspot number from sunspot minimum. Based on the observed values of these quantities (equal to 10.6 and 4.63, respectively) as of early 1989, one infers that cycle 22's maximum amplitude will be about 175 ± 30 or 185 ± 10, respectively, where the error bars represent approximately twice the average error found during cycles 10–21 from the two fits.     

A Standard Law for the Equatorward Drift of the Sunspot Zones

The latitudinal location of the sunspot zones in each hemisphere is determined by calculating the centroid position of sunspot areas for each solar rotation from May 1874 to June 2011. When these centroid positions are plotted and analyzed as functions of time from each sunspot cycle maximum, there appear to be systematic differences in the positions and equatorward drift rates as a function of sunspot cycle amplitude. If, instead, these centroid positions are plotted and analyzed as functions of time from each sunspot cycle minimum, then most of the differences in the positions and equatorward drift rates disappear. The differences that remain disappear entirely if curve fitting is used to determine the starting times (which vary by as much as eight months from the times of minima). The sunspot zone latitudes and equatorward drift measured relative to this starting time follow a standard path for all cycles with no dependence upon cycle strength or hemispheric dominance. Although Cycle 23 was peculiar in its length and the strength of the polar fields it produced, it too shows no significant variation from this standard. This standard law, and the lack of variation with sunspot cycle characteristics, is consistent with dynamo wave mechanisms but not consistent with current flux transport dynamo models for the equatorward drift of the sunspot zones.     

Intermediate-term periodicities in solar activity

The presence of intermediate-term periodicities in solar activity, at approximately 323 and 540 days, has been claimed by different authors. In this paper, we have performed a search for them in the historical records of two main indices of solar activity, namely, the daily sunspot areas (cycles 12–21) and the daily Zürich sunspot number (cycles 6–21). Two different methods to compute power spectra have been used, one of them being especially appropriate to deal with gapped time series. The results obtained for the periodicity near 323 days indicate that it has only been present in cycle 21, while in previous cycles no significant evidence for it has been found. On the other hand, a significant periodicity at 350 days is found in sunspot areas and Zürich sunspot number during cycles 12–21 considered all together, also having been detected in some individual cycles. However, this last periodicity must be looked into with care due to the lack of confirmation for it coming from other features of solar activity. The periodicity around 540 days is found in cycles 12, 14, and 17 in sunspot areas, while during cycles 18 and 19 it is present, with a very high significance, in sunspot areas and Zürich sunspot number. It also appears at 528 days in sunspot areas during cycles 12–21. On the other hand, it is important to note the coincidence between the asymmetry, favouring the northern hemisphere, of sunspot areas and solar flares during cycle 19, and the fact that the periodicity at 540 days was only present, with high significance, in that hemisphere during that solar cycle.     

A new pattern for the north–south asymmetry of sunspots

We study the solar cycle evolution during the last 8 solar cycles using a vectorial sunspot area called the LA (longitudinal asymmetry) parameter. This is a useful measure of solar activity in which the stochastic, longitudinally evenly distributed sunspot activity is reduced and which therefore emphasizes the more systematic, longitudinally asymmetric sunspot activity. Interesting differences are found between the LA parameter and the more conventional sunspot activity indices like the (scalar) sunspot area and the sunspot number. E.g., cycle 19 is not the highest cycle according to LA. We have calculated the separate LA parameters for the northern and southern hemisphere and found a systematic dipolar-type oscillation in the dominating hemisphere during high solar activity times which is reproduced from cycle to cycle. We have analyzed this oscillation during cycles 16–22 by a superposed epoch method using the date of magnetic reversal in the southern hemisphere as the zero epoch time. According to our analysis, the oscillation starts by an excess of the northern LA value in the ascending phase of the solar cycle which lasts for about 2.3 years. Soon after the maximum northern dominance, the southern hemisphere starts dominating, reaching its minimum some 1.2–1.7 years later. The period of southern dominance lasts for about 1.6 years and ends, on an average, slightly before the end of magnetic reversal.     

On the level of skill in predicting maximum sunspot number: A comparative study of single variate and bivariate precursor techniques

Precursor prediction techniques have generally performed well in predicting the maximum amplitude of sunspot cycles, based on cycles 10–21. Single variate methods based on minimum sunspot amplitude have reliably predicted the size of the sunspot cycle 9 out of 12 times, where a reliable prediction is defined as one having an observed maximum amplitude within the prediction interval (determined from the average error). On the other hand, single variate methods based on the size of the geomagnetic minimum have reliably predicted the size of the sunspot cycle 8 of 10 times (geomagnetic data are only available since about cycle 12). Bivariate prediction methods have, thus far, performed flawlessly, giving reliable predictions 10 out of 10 times (bivariate methods are based on sunspot and geomagnetic data). For cycle 22, single variate methods (based on geomagnetic data) suggest a maximum amplitude of about 170 ± 25, while bivariate methods suggest a maximum amplitude of about 140 ± 15; thus, both techniques suggest that cycle 22 will be of smaller maximum amplitude than that observed during cycle 19, and possibly even smaller than that observed for cycle 21. Compared to the mean cycle, cycle 22 is presently behaving as if it is a + 2.6 cycle (maximum amplitude about 225). It appears then that either cycle 22 will be the first cycle not to be reliably predicted by the combined precursor techniques (i.e., cycle 22 is an anomaly, a statistical outlier) or the deviation of cycle 22 relative to the mean cycle will substantially decrease over the next 18 months. Because cycle 22 is a large amplitude cycle, maximum smoothed sunspot number is expected to occur early in 1990 (between December 1989 and May 1990).     

Detection of the Long-Term Microwave `Darkening' Before the 14 July 2000 Flare

The latitudinal distribution of sunspot groups over a solar cycle is investigated. Although individual sunspot groups of a solar cycle emerge randomly at any middle and low latitude, the whole latitudinal distribution of sunspot groups of the cycle is not stochastic and, in fact, can be represented by a probability density function of the distribution having maximum probability at about 15.5°. The maximum amplitude of a solar cycle is found to be positively correlated against the number of sunspot groups at high latitude (35°) over the cycle, as well as the mean latitude. Also, the relation between the asymmetry of sunspot groups and its latitude is investigated, and a pattern of the N-S asymmetry in solar activity is suggested.     

Group Sunspot Numbers: Sunspot Cycle Characteristics

We examine the `Group' sunspot numbers constructed by Hoyt and Schatten to determine their utility in characterizing the solar activity cycle. We compare smoothed monthly Group sunspot numbers to Zürich (International) sunspot numbers, 10.7-cm radio flux, and total sunspot area. We find that the Zürich numbers follow the 10.7-cm radio flux and total sunspot area measurements only slightly better than the Group numbers. We examine several significant characteristics of the sunspot cycle using both Group numbers and Zürich numbers. We find that the `Waldmeier Effect' – the anti-correlation between cycle amplitude and the elapsed time between minimum and maximum of a cycle – is much more apparent in the Zürich numbers. The `Amplitude–Period Effect' – the anti-correlation between cycle amplitude and the length of the previous cycle from minimum to minimum – is also much more apparent in the Zürich numbers. The `Amplitude–Minimum Effect' – the correlation between cycle amplitude and the activity level at the previous (onset) minimum is equally apparent in both the Zürich numbers and the Group numbers. The `Even–Odd Effect' – in which odd-numbered cycles are larger than their even-numbered precursors – is somewhat stronger in the Group numbers but with a tighter relationship in the Zürich numbers. The `Secular Trend' – the increase in cycle amplitudes since the Maunder Minimum – is much stronger in Group numbers. After removing this trend we find little evidence for multi-cycle periodicities like the 80-year Gleissberg cycle or the two- and three-cycle periodicities. We also find little evidence for a correlation between the amplitude of a cycle and its period or for a bimodal distribution of cycle periods. We conclude that the Group numbers are most useful for extending the sunspot cycle data further back in time and thereby adding more cycles and improving the statistics. However, the Zürich numbers are slightly more useful for characterizing the on-going levels of solar activity.     

The north–south asymmetry of filaments in solar cycles 16–21

In the present study, the north–south asymmetry of filaments in solar cycles 16–21 is investigated with the use of the solar filaments observed at the Observatoire de Paris, Section de Meudon from March 1919 to December 1989. Filament activity is found regularly dominated in each of cycles 16–21 in the same hemisphere as that inferred by sunspot activity, and it is found to run in a different asymmetrical behavior at different latitudinal bands, suggesting that the north–south asymmetry of filament activity should be a function of latitudes. The regularity on the north–south asymmetry of sunspot activity given by Li et al. (2002b) is demonstrated by filament activity. The periods in the north–south asymmetry of solar filament activity are 9.13, and 12.8 years without the solar cycle found.     

A nonlinear RLC solar cycle model

A simplified, monoparametric model, based on the Van der Pol nonlinear RLC electric oscillator, is found capable of describing the shape and related morphological properties (such as the Waldmeier effect) of the sunspot cycles. The model can also exhibit long periods of sunspot inactivity of the Maunder Minimum type. According to the model, the significant rise-to-fall time asymmetry of the most recent cycles suggests that it is unlikely that another cycle suppression will occur in the forthcoming decades. The complete sunspot record and the system's attractor are successfully emulated, given the sunspot number at cycle maxima.     

Sunspot groups at high latitude

Data of sunspot groups at high latitude (35°), from the year 1874 to the present (2000 January), are collected to show their evolutional behaviour and to investigate features of the yearly number of sunspot groups at high latitude. Subsequently, an evolutional pattern of sunspot group number at high latitude is given in this paper. Results obtained show that the number of sunspot groups of a solar cycle at high latitude rises to a maximum value about 1 yr earlier than the time of the maximum of sunspot relative numbers of the solar cycle, and then falls to zero more rapidly. The results also show that, at the moment, solar activity described by the sunspot relative numbers has not yet reached its minimum. In general, sunspot groups at high latitude have not appeared on the solar disc during the last 3 yr of a Wolf solar cycle. The asymmetry of the high latitude sunspot group number of a Wolf solar cycle can reflect the asymmetry of solar activity in the Wolf solar cycle, and it is suggested that one could further use the high latitude sunspot group number during the rising time of a Wolf solar cycle, maximum year included, to judge the asymmetry of solar activity over the whole solar cycle.     

Prediction of solar cycle 24 and beyond

In the previous study (Hiremath, Astron. Astrophys. 452:591, 2006a), the solar cycle is modeled as a forced and damped harmonic oscillator and from all the 22 cycles (1755–1996), long-term amplitudes, frequencies, phases and decay factor are obtained. Using these physical parameters of the previous 22 solar cycles and by an autoregressive model, we predict the amplitude and period of the present cycle 23 and future fifteen solar cycles. The period of present solar cycle 23 is estimated to be 11.73 years and it is expected that onset of next sunspot activity cycle 24 might starts during the period 2008.57±0.17 (i.e., around May–September 2008). The predicted period and amplitude of the present cycle 23 are almost similar to the period and amplitude of the observed cycle. With these encouraging results, we also predict the profiles of future 15 solar cycles. Important predictions are: (i) the period and amplitude of the cycle 24 are 9.34 years and 110 (±11), (ii) the period and amplitude of the cycle 25 are 12.49 years and 110 (±11), (iii) during the cycles 26 (2030–2042 AD), 27 (2042–2054 AD), 34 (2118–2127 AD), 37 (2152–2163 AD) and 38 (2163–2176 AD), the sun might experience a very high sunspot activity, (iv) the sun might also experience a very low (around 60) sunspot activity during cycle 31 (2089–2100 AD) and, (v) length of the solar cycles vary from 8.65 years for the cycle 33 to maximum of 13.07 years for the cycle 35.     

The Long-term Behavior of the North – South Asymmetry of Sunspot Activity

A new index, the cumulative difference of sunspot activity in the northern and southern hemispheres, respectively, is proposed to describe the long-term behavior of the North – South asymmetry of sunspot activity and to show the balance (or bias) of sunspot activity in the two solar hemispheres on a long-term scale. Sunspot groups and sunspot areas from June 1874 to January 2007 are used to show the advantage of the index. The index clearly shows a long-term characteristic time scale of about 12 cycles in the North – South asymmetry of sunspot activity. Sunspot activity is found to dominate in the southern hemisphere in cycle 23, and in cycle 24 it is predicted to dominate still in the southern hemisphere. A comparison of the new index with other similar indexes is also given.