The Schwabe and Gleissberg Periods in the Wolf Sunspot Numbers and the Group Sunspot Numbers

Three wavelet functions: the Morlet wavelet, the Paul wavelet, and the DOG wavelet have been respectively performed on both the monthly Wolf sunspot numbers (R_z) from January 1749 to May 2004 and the monthly group sunspot numbers (R_g) from June 1795 to December 1995 to study the evolution of the Gleissberg and Schwabe periods of solar activity. The main results obtained are (1) the two most obvious periods in both the R_z and R_g are the Schwabe and Gleissberg periods. The Schwabe period oscillated during the second half of the eighteenth century and was steady from the 1850s onward. No obvious drifting trend of the Schwabe period exists. (2) The Gleissberg period obviously drifts to longer periods the whole consideration time, and the drifting speed of the Gleissberg period is larger for R_z than for R_g. (3) Although the Schwabe-period values for R_z and R_g are about 10.7 years, the value for R_z seems slightly larger than that for R_g. The Schwabe period of R_z is highly significant after the 1820s, and the Schwabe period of R_g is highly significant over almost the whole consideration time except for about 20 years around the 1800s. The evolution of the Schwabe period for both R_z and R_g in time is similar to each other. (4) The Gleissberg period in R_z and R_g is highly significant during the whole consideration time, but this result is unreliable at the two ends of each of the time series of the data. The evolution of the Gleissberg period in R_z is similar to that in R_g.     

Recurrent magnetic activity,sunspot number and its rate of decline

Power spectral densities computed from low-latitude horizontal intensity of the Earth's magnetic field over two-year periods of declining phases of solar cycles 16 to 19 show a close relationship with the maximum relative sunspot number of the following solar cycles. The maximum sunspot number shows an exponential rise with the power density near 1/27 cd^?1; maximum R _z,however, increases linearly with power density near 1/14 cd^?1. It is also shown that the rate of decline of sunspot number in a solar cycle is almost exactly related, linearly, to power spectral density for the preceding solar cycle. Power densities near 1/27 and 1/14 cd^?1 in declining phase of solar cycle appear to be satisfactory indices for the maximum relative sunspot number of the following cycle and its rate of decline thereafter.     

On the long-term secular increase in sunspot number

Correlated with the maximum amplitude (R _max) of the sunspot cycle are the sum (R _sum) and the mean (R _mean) of sunspot number over the duration of the cycle, having a correlation coefficient r equal to 0.925 and 0.960, respectively. Runs tests of R _max, R _sum, and R _mean for cycles 0–21 have probabilities of randomness P equal to 6.3, 1.2, and 9.2%, respectively, indicating a tendency for these solar-cycle related parameters to be nonrandomly distributed. The past record of these parameters can be described using a simple two-parameter secular fit, one parameter being an 8-cycle modulation (the so-called Gleissberg cycle or long period) and the other being a long-term general (linear) increase lasting tens of cycles. For each of the solar-cycle related parameters, the secular fit has an r equal to about 0.7–0.8, implying that about 50–60% of the variation in R _max, R _sum, and R _mean can be accounted for by the variation in the secular fit.Extrapolation of the two-parameter secular fit of R _max to cycle 22 suggests that the present cycle will have an R _max = 74.5 ± 49.0, where the error bar equals ± 2 standard errors; hence, the maximum amplitude for cycle 22 should be lower than about 125 when sunspot number is expressed as an annual average or it should be lower than about 130 when sunspot number is expressed as a smoothed (13-month running mean) average. The long-term general increase in sunspot number appears to have begun about the time of the Maunder minimum, implying that the 314-yr periodicity found in ancient varve data may not be a dominant feature of present sunspot cycles.     

Comparison of three solar activity indices based on sunspot observations

The following sunspot formation indices are analyzed: the relative sunspot number R _z, the normalized sunspot group number R _g, and the total sunspot area A. Six empirical formulas are derived to describe the relations among these indices after 1908. The earlier data exhibit systematic deviations from these formulas, which can be attributed to systematic errors of the indices. The Greenwich data on the sunspot total area A and the sunspot group number in 1874–1880 are found to be doubtful. Erroneous data at the beginning of the Greenwich series must spoil the values of the index R _g in the XVII–XIX centuries. The Hoyt-Schatten series of R _g may be less reliable than the well-known Wolf number series R _z.     

Predicting the maximum amplitude for the sunspot cycle from the rate of rise in sunspot number

Examined are associational aspects as they relate the maximum amplitude R _M for the sunspot cycle to the rate of rise R _t during the ascending phase, where R _M is the smoothed sunspot number at cycle maximum and R _t is the sum of the monthly mean sunspot numbers for selected 6-month intervals (t) measured from cycle onset. One finds that, prior to about 2 yr into the cycle, the rate of rise is not a reliable predictor for maximum amplitude. Only during the latter half of the ascent do the fits display strong linearity, having a coefficient of correlation r 0.9 and a standard error S _yx 20. During the first four intervals, the expected R _M and the observed R _M were found to differ by no more than 20 units of smoothed sunspot number only 25, 42, 50, and 58 % of the time; during the latter four intervals, they differed by no more than 20 units 67, 83, 92, and 100% of the time.     

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 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.     

Periodicities in the occurrence rate of solar proton events

Power spectral analyses of the time series of solar proton events during the past three solar cycles reveal a periodicity around 154 days. This feature is prominent in all of the cycles combined, cycles 19 and 21 individually but is only weak in cycle 20. These results are consistent with the presence of similar periodicities between 152 and 155 days in the occurrence rate of major solar flares, the sunspot blocking function (P _s ), the 10.7 cm radio flux (F _10.7) and the sunspot number (R _z ). This suggests that the circa 154-days periodicity may be a fundamental characteristic of the Sun. Periods around 50–52 days are also found in the combined data set and in the three individual cycles in general agreement with the detection of this periodicity in major flares in cycle 19 and inP _s ,F _10.7, andR _z in cycle 21. The cause of the 155 day period remains unknown. The spectra contain lines (or show power at frequencies) consistent with a model in which the periodicity is caused by differential rotation of active zones and a model in which it is related to beat frequencies between solar oscillations, as proposed by Wolff.     

Wavelet Analysis of solar,solar wind and geomagnetic parameters

The sunspot number, solar wind plasma, interplanetary magnetic field, and geomagnetic activity index A _p have been analyzed using a wavelet technique to look for the presence of periods and the temporal evolution of these periods. The global wavelet spectra of these parameters, which provide information about the temporal average strength of quasi periods, exhibit the presence of a variety of prominent quasi periods around 16 years, 10.6 years, 9.6 years, 5.5 years, 1.3 years, 180 days, 154 days, 27 days, and 14 days. The wavelet spectra of sunspot number during 1873–2000, geomagnetic activity index A _p during 1932–2000, and solar wind velocity and interplanetary magnetic field during 1964–2000 indicate that their spectral power evolves with time. In general, the power of the oscillations with a period of less than one year evolves rapidly with the phase of the solar cycle with their peak values changing from one cycle to the next. The temporal evolution of wavelet power in R _z, v _sw, n, B _y, B _z, |B|, and A _p for each of the prominent quasi periods is studied in detail.     

Relative sunspot numbers in the first half of eighteenth century

We revised relative sunspot numbers in the time interval 1700–1748 for which Wolf derived their annual means. The frequency of daily observations, counting simultaneously the number of sunspots and the number of sunspot groups necessary for determinating Wolf's relative sunspot numbers, is in this time interval very low and covers, on average, 4.8% of the number of all days only. There also exist incomplete observations not convenient to determine relative sunspot numbers. To enlarge the number of daily relative sunspot numbers we used the nonlinear, two-step interpolation method derived earlier by Letfus (1996, 1999). After interpolation, the mean value increased to 13.8%. Waldmeier (1968) found that the scaling factor k can be derived directly from the observed number of spots f and from the number of sunspot groups g. From the observations made at Zürich (Wolf and his assistants, Wolfer), at Peckeloh, and at Moncalieri during the years 1861–1928, we derived a new, more correct empirical relation. The resulting annual relative sunspot numbers are given in Table II. However, only for 26 years (53.0%) from the total number of 49 years was it possible to derive annual relative sunspot numbers. The observations were missing for the other years. This corresponds with results of Wolf, which gives the annual relative sunspot numbers for all 49 years. For the years when the data were missing, he marked these values as interpolated or very uncertain ones. Most of the observations originate from two data series (Kirch, Plantade), for which Wolf derived a higher scaling factor (k=2.0) than followed from the newly derived relation (k=1.40). The investigated time interval covers four solar cycles. After our results, the height of the first cycle (No. –4), given by Wolf, should be lowered by about two-thirds, the following two cycles (Nos. –3 and –2) lowered by one-third, as given by Wolf, and only the height of the fourth one (No. –1) should be unchanged. The activity levels of the cycles, as represented by group sunspot numbers, are lower by about one-fourth and, in the case of the first one (No. –4) even by two-thirds of the levels derived by us. The group sunspot numbers, derived from a much greater number of observations, have also greater credibility than other estimates. The shapes of the cycles, as given by Wolf, can be considered only as their more or less idealized form.     

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.     

Structure of a simple sunspot from the inversion of IR spectral data

Analysis of spectral data of two neighboring infrared lines, Fe I 15648.5 Å (g = 3) and FeI 15652.9 Å (g_eff = 1.53) are carried out for a simple sunspot when it was near the solar disk center (μ = 0.92), to understand the basic structure of sunspot magnetic field. Inversions of Stokes profiles are carried out to derive different atmospheric parameters both as a function of location within the sunspot and height in the atmosphere. As a result of the inversion we have obtained maps of magnetic field strength, temperature, line‐of‐sight velocity, field inclination and azimuth for different optical depth layers between log(τ₅) = 0 and log(τ₅) = –2.0. In this paper we present few results from our inversion for a layer averaged between log(τ5) from 0.0 to –0.5.     

PREDICTIONS OF THE FEATURES FOR SUNSPOT CYCLE 23

Smoothed monthly mean Ap indices are decomposed into two components (Ap) _c and (Ap) _n. The former is directly correlated with the current sunspot numbers, while the latter is shown to achieve its maximum correlation with the sunspot numbers after some time lag. This latter property is used to develop a method for predicting the sunspot maximum based on the observed value of (Ap) _n maximum which occurs during the preceding cycle. The value of R _M for cycle 23 predicted by this method is 149.3 ± 19.9. A method to estimate the rise time (from solar minimum to maximum) has been developed (based on analyses of Hathaway, Wilson, and Reichmann, 1994) and yields a value of 4.2 years. Using an estimate that the minimum between cycles 22 and 23 occurred in May 1996, it is predicted that the sunspot maximum for cycle 23 will occur in July 2000.     

Predicting maximum sunspot number in solar cycle 24

A few prediction methods have been developed based on the precursor technique which is found to be successful for forecasting the solar activity. Considering the geomagnetic activity aa indices during the descending phase of the preceding solar cycle as the precursor, we predict the maximum amplitude of annual mean sunspot number in cycle 24 to be 111 ± 21. This suggests that the maximum amplitude of the upcoming cycle 24 will be less than cycles 21–22. Further, we have estimated the annual mean geomagnetic activity aa index for the solar maximum year in cycle 24 to be 20.6 ± 4.7 and the average of the annual mean sunspot number during the descending phase of cycle 24 is estimated to be 48 ± 16.8.     

Variations of the radiation signatures of noise storm-emitting sunspot groups during a solar cycle

The flux density of a noise storm continuum is known to depend on importance parameters of the associated sunspot group, e.g., its total area A. A study of the continua at 287, 234, 113, and 64 MHz, however, reveals in case of the two cycles Nos. 20 and 21 that the radiation signatures of sunspot groups, with a value of A kept fixed, vary systematically with time indicating regular changes of relevant parameters of the overlying loop systems with the phase of the solar cycle. A trend of intense continua at high frequencies (for definition, cf. Figures 1, 2(b)) to occur preferably during the first activity maximum of a solar cycle has been obtained in either case suggesting a decrease of the emissivity of sunspot groups with time. Vice versa, intense continua at lower frequencies (for definition, cf. Figures 1, 2(b)) were mainly observed during the later phase of both cycles. The latter effect is shown to be attributed rather to a long-term variation of the spectral characteristics of the type-I continua than to an enhanced number of intense type-III continua. From the result obtained it follows that non-potential loops extending to great heights into the corona or developing at least conditions favourable for the generation of an intense type-I continuum even at the frequencies < 100 MHz tend to occur more frequently above sunspot groups during the later phase of a solar cycle than above the comparable groups of its first activity maximum.Furthermore, characteristic periods have been found for both cycles during which the emissivity, especially of the very large sunspot groups, was significantly diminished with reference to the comparable groups of the adjacent time intervals.     

Wavelet Analysis of the Schwabe Cycle Properties in Solar Activity

Properties of the Schwabe cycles in solar activity are investigated by using wavelet transform. We study the main range of the Schwabe cycles of the solar activity recorded by relative sunspot numbers, and find that the main range of the Schwabe cycles is the periodic span from 8-year to 14-year. We make the comparison of 11-year‘s phase between relative sunspot numbers and sunspot group numbers. The results show that there is some difference between two phases for the interval from 1710 to 1810, while the two phases are almost the same for the interval from 1810 to 1990.     

Activity cycle of solar filaments

Long-term variation in the distribution of the solar filaments observed at the Observatorie de Paris, Section de Meudon from March 1919 to December 1989 is presented to compare with sunspot cycle and to study the periodicity in the filament activity, namely the periods of the coronal activity with the Morlet wavelet used. It is inferred that the activity cycle of solar filaments should have the same cycle length as sunspot cycle, but the cycle behavior of solar filaments is globally similar in profile with, but different in detail from, that of sunspot cycles. The amplitude of solar magnetic activity should not keep in phase with the complexity of solar magnetic activity. The possible periods in the filament activity are about 10.44 and 19.20 years. The wavelet local power spectrum of the period 10.44 years is statistically significant during the whole consideration time. The wavelet local power spectrum of the period 19.20 years is under the 95% confidence spectrum during the whole consideration time, but over the mean red-noise spectrum of α = 0.72 before approximate Carrington rotation number 1500, and after that the filament activity does not statistically show the period. Wavelet reconstruction indicates that the early data of the filament archive (in and before cycle 16) are more noiseful than the later (in and after cycle 17).     

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).     

The use of abnormal quiet days in Sq(H) for predicting the magnitude of sunspot maximum at the time of preceding sunspot minimum

The previously established connection between the occurence of AQDs (“abnormal quiet days” when the phase of the solar diurnal variation of horizontal magnetic field, Sq(H), at a mid-latitude northern hemisphere station is anomalous) at sunspot minimum and the magnitude of the following sunspot maximum is examined in the light of our recent improved understanding of the nature and cause of AQDs. A small contribution to the relationship is found to arise from variations from cycle to cycle in the additional northward field which is characteristic of AQDs and leads to a reduced Sq(H) amplitude at stations poleward of the Sq focus. However, the main factor which determines the connection is a variation from one sunspot minimum to another of the amplitude of the small southward bay-like field perturbations which constitute the AQD events, and evidence is presented which suggests that this parameter may be quantitatively related to the extent of southward swing of the B_z component of the interplanetary magnetic field which determines the energy transfer from the solar wind into the magnetospheric tail. It thus appears that the magnitude of southward swing in B_z might be another solar parameter which anticipates the size of a forthcoming sunspot cycle during its build-up over the declining phase of the previous cycle and at the minimum.