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ISSN 0016 7932, Geomagnetism and Aeronomy, 2012, Vol. 52, No. 7, pp. 857860. © Pleiades Publishing, Ltd., 2012.

Special Points in 11 Year Variations in the Latitudinal Characteristics of Sunspot Activity
E. V. Miletsky and Yu. A. Nagovitsyn
Central Astronomical (Pulkovo) Observatory, Russian Academy of Sciences, Pulkovskoe sh. 65, St. Petersburg, 196140 Russia e mail: eugenm@gao.spb.ru
Received March 5, 2012

Abstract--It has been indicated that special moments (turning points), when certain characteristics of the latitudinal (equatorward) drift of the sunspot drift zone suddenly change, exist in each 11 year solar cycle. The moment when a sunspot formation low latitude boundary minimum (T2), coordinated in time with the end of a polar magnetic field polarity reversal, exists has a special place among these points. A conclusion has been drawn that it is impossible to reconstruct polarity reversal moments in the past based on information about turning points T2. The average velocities of the latitudinal drift of the minimal, average, and maximal sunspot group latitudes have been calculated. It has been indicated that the closeness of the relationship between the first two veloc ities and the maximal activity amplitudes in the cycles differ substantially for the first (before point T2) and second (after point T2) cycle parts. The corresponding values of the correlation coefficients increase substan tially in the second cycle (after point T2). It has been established that a relationship exists between some velocities calculated in these cycles and the activity amplitudes at maximums of the next cycles. A model for predicting future cycle maximums has been constructed based on this conclusion. The probable average annual Wolf number at a maximum of cycle 24 has been determined (W(24) = 93). DOI: 10.1134/S0016793212070158

1. INTRODUCTION It is known that different states (phases), separated by moments after which the solar regime activity usu ally changes substantially, are clearly defined in each 11 year cycle. Such critical moments were for the first time revealed as parameters during studying the behavior of cyclic curves of indices, which character ize sunspot activity, and were called "reference" (Vit insky et al., 1986), "inflection" (Kuklin, 1992), and "turning" (Badalyan and Kuklin, 2000) points. Thus, the achieved results were analyzed in detail and a list of inflection points in the cycles was presented in (Kuklin, 1992) for the Wolf number. In addition, such turning points were detected in cyclic curves of coronal (in the 5303 A FeXIV green line) (Badalyan and Kuklin, 1993) and global mag netic field (Obridko and Shelting, 1992, 2003) indices. The performed studies indicate that the positions of turning points (and the duration of the correspond ing phases between them) can slightly differ from cycle to cycle. However, the moments of the corresponding turning points are generally close in time for most indices, apparently, because the processes responsible for these indices are common, although the physical mechanisms by which this phenomenon originates are still unclear.

The aim of this work is to determine and study the turning points in cyclic variations in the sunspot activ ity latitudinal characteristics, which were studied in (Miletsky and Ivanov, 2009; Nagovitsyn, 2010; Ivanov et al., 2011; Ivanov and Miletsky, 2011). Note that the parameters of the sunspot formation drift zone depend on the accepted solar cycle model. Therefore, the study of the solar cycle regularity makes it possible to better understand the solar cyclicity character and reg ularities and, finally, the cyclicity nature. 2. DATA AND THEIR PROCESSING We used the data on the sunspot characteristics pre sented in the Greenwich catalog and its continuation NOAA/USAF for 18742006 (http://solarscience. msfc.nasa.gov/greenwch.shtml). Based on these data, series of annual values of the following indices for either solar hemisphere were compiled: the sunspot group number (G), average sunspot group latitudes weighed for the sunspot area (LA), and the maximal (LH) and minimal (LL) sunspot group latitudes on a given day. Detailed calculations of these indices are presented in (Miletsky and Ivanov, 2009; Ivanov and Miletsky, 2011). From daily data, we obtained average rotation, as well as average annual, values of these indices. It is clear that the average annual values of the

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MILETSKY, NAGOVITSYN Cycles 1823 (N + S) 20 15 10 5 0 5 1950 28 24 Deg 20 16 12 8 2 3 1950 1960 2 3 1970 2 3 1980 Years 2 3 1990 2 3 2000 2010 1 1 1 1 1 1 2 1960 1970 1980 Years 1990 2000 2010 LHf13 LAf13 LLf13 18 20 19 21 22 23 D VD G

Top panel presents time variations in a series of the average annual values (cycles 1823) of the sunspot group number index (G, dot and dash line), sunspot formation zone width (D = LH LL, dashed line), and the velocity of a change in this width (VD, solid line). The bottom panel presents time variations in a series of the average rotational values (moving averaging over 13 rotations) of the sunspot activity latitudinal characteristics (cycles 1823): maximal (LHf13, doted line), average (LAf13, dot and dash line), and minimal (LLf13, solid line) sunspot latitudes.

LH and LL indices characterize the average annual position of the sunspot formation zone at the high and low latitude boundaries, whereas the difference in these values (D = LH LL) characterizes the latitudi nal extent of this zone. In addition, from the average annual LA and D values, we obtained the velocities of the average latitude equatorward drift (VLA) and vari ations in the sunspot formation zone width (VD) by means of finite difference time differentiation over three points. 3. RESULTS The bottom panel of figure presents curves demon strating (for cycles 1823) time variations in the series (obtained through the transformation of the average rotational values by using the method of moving aver ages with harmonic weights for 13 rotations) of sun spot activity latitudinal characteristics: average sun spot group latitudes (LAf13), maximal sunspot lati tudes (LHf13), and minimal sunspot latitudes (LLf13). It is clear that moments T1, T2, and T3, which are marked with numerals 1, 2, and 3 in figure and can be characterized as special (turning) points of the latitudinal cyclic curves, are clearly defined in each 11 year cycle. In this case, points T1 represent the moments of maximums of the lower and middle lati tudes (curves LLf13 and LAf13); points T2, the

moments when the lower latitude local minimums are reached (curve LLf13); points T3, the moments when the minimum of all latitudes is reached. The top panel of figure shows curves reflecting time variations in the average values of the G, D, and VD indices for the same cycles. A comparison of the plots shown in the top and bottom panels of figure makes it possible to conclude that special points T1 are always observed during the 11 year cycle growth phases (curve G) and correspond to the moments when the sunspot zone latitudinal expansion velocity becomes maximal (curve VD). Special points T2 are observed during the cycle decline phases and are located near the moments when the sunspot zone latitudinal con traction velocity is maximal (curve VD minima) and near the moments of polar magnetic field polarity reversals (Makarov and Makarova, 1996; Tlatov, 2007) (vertical solid lines). In addition, point T2 corre sponds to the tMD reference point, which is revealed on the Wolf number curve, is observed one to two years after the maximum of this index, and coincides in time with the maximums of the sunspot group power and the number of proton flares and other phenomena characterizing the so called "secondary" cycle maxi mum introduced by M.N. Gnevyshev (Vitinskii et al., 1986; Kuklin,1992; Obridko and Shelting, 1992). Sin gular points T3 are observed near minimums of the 11 year sunspot activity cycles.
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Table 1. Correlation coefficients between activity maximums in cycles and the drift velocities of the minimal, maximal, and average sunspot latitudes N&S G(n) D(n) G(n + 1) V12LL 0.21 0.26 0.18 V12LA 0.08 0.07 0.43 V12LH 0.73 0.78 0.45 V23LL 0.59 0.63 0.10 V23LA 0.44 0.45 0.56 V23LH 0.81 0.83 0.37

Table 2. Prognostic model parameters and prediction of the cycle 24 maximum Wn+1 = 0.6 + 0.57V23LAn + 1.7V12L A n + 1.5V23L A n + 3V12LAnV23LAn
2 2

R(N + S) = 0.83

W(24) = 93

We subsequently calculated the average latitudinal drift velocities (ratios of latitudinal differences to time) of the minimal LL, average LA, and maximal LH sun spot group latitudes (V12LL, V12LA, V12LH, V23LL, V23LA, V23LH) for the time intervals separating points T1, T2, and T3. Table 1 presents the linear cor relation coefficients between these values and the average annual values of the G and D indices at maxi mums of the current (n) and next (n + 1) 11 year cycles (from 12 to 23) based on data for both hemi spheres (24 points). On the T1T2 interval (see Table 1), a correlation between the maximal activity amplitudes in the cycles and the drift velocities of the minimal and average sun spot latitudes (V12LL, V12LA) is almost absent. This agrees with the results achieved in (Vitinsky et al., 1986; Hathaway, 2011); the authors stated that the average drift velocity of the sunspot zone width for an 11 year cycle is independent of the cycle amplitude. However, a substantial and significant (reliability 99.9%) correlation between activity and the maximal latitude drift velocity (V12LH) is found in the same time interval. In the T2T3 interval (after a polarity reversal of the polar magnetic field), it turns out that the activity at cycle maximums correlates with the drift velocity of not only maximal (reliability 99.95%), but also mini mal (reliability 99%) and even average (reliability 95%), latitudes. Thus, the character of interrelation ships between the latitudinal drift velocities of the minimal and average sunspot latitudes and the current activity cycle amplitude changes substantially at point T2. The value of the correlation between the above sun spot characteristic latitudinal velocities (see Table 1, bottom raw) and the activity at maximums of the next 11 year cycle G(n + 1) made it possible to use these velocities as initial variables when we constructed a model for predicting future Wolf number maximums (Wn+1) in these cycles. We found an optimal prognostic model using the Argument Grouped Consideration Method (MGUA or MGDH) (Farlow, 1984; Ivakh nenko and Myuller, 1984; Madala and Ivakhnenko, 1994; Miletsky, 2004; Miletsky and Ivanov, 2006),
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which makes it possible to select an optimal model from all possible models based on the so called "exter nal" criteria. Such external criteria can reach a mini mum at a gradual sophistication of the model, which makes it possible to select an optimal model. This method makes it possible to avoid any additional sophistication of the model and eliminate variables not bearing additional information. We sought an optimal model among all possible polynomials, the degrees of which were no higher than 3 for all points from the N and S hemispheres (24 points). As a result, using the MGUA model, we obtained an optimal model, the parameters and accu racy of which (the correlation coefficient between the model and real values R(N + S)) are presented in Table 2. We should note that we selected the drift velocities of the sunspot average latitudes before and after special point T2 (V12LA, V23LA) as optimal model variables. The last column in Table 2 presents the predicted model value of the average annual Wolf number at the cycle 24 maximum: W(24) = 93. This value corre sponds to several predictions, according to which the next cycle will be slightly lower than cycle 23 (Petrovay, 2010). 4. CONCLUSIONS The performed study indicates that special moments (turning points), when the characteristics of the latitu dinal (equatorward) sunspot zone drift change sud denly, are observed during 11 year solar cycles. The moment when a low latitude boundary minimum of the sunspot formation zone (T2) is reached, which agrees with the end of the polar magnetic field polarity reversal, is of special significance. This indicates that an interrelationship exists between these phenomena and makes it possible to reconstruct special moments in the past (specifically, polarity reversals) in the evo lution of the large scale magnetic field based on infor mation on turning points of latitudinal sunspot char acteristics (Nagovitsyn, 2010).
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MILETSKY, NAGOVITSYN Ivakhnenko, A.G. and Myuller, I.A., Samoorganizatsiya Prognoziruyushchikh Modelei (Self Organization of Predicting Models) Kiev: Tekhnika, 1984. Ivanov, V.G. and Miletsky, E.V., Width of Sunspot Generat ing Zone and Reconstruction of Butterfly Diagram, Sol. Phys., 2011, vol. 268, pp. 231242. Ivanov, V.G., Miletsky, E.V., and Nagovitsyn, Yu.A., Form of the Latitude Distribution of Sunspot Activity, Astron. Rep., 2011, vol. 55, pp. 911917. Kuklin, G.V., Inflection Points of Cyclic Curves, Soln. Dannye, 1992, no. 10, pp. 6979. Madala, H.R. and Ivakhnenko, A.G., Inductive Learning Algorithms for Complex Systems Modelling, Proc. CRC, Boca Raton, 1994. Makarov, V.I. and Makarova, V.V., Polar Faculae and Sun spot Cycles, Sol. Phys., 1996, vol. 163, pp. 267289. Miletsky, E.V. and Ivanov, V.G., Dynamic Models for Long Range and Ultralong Range Prediction of Solar Activ ity, Izv. Ross. Akad. Nauk, Ser. Fiz., 2006, vol. 70, no. 10, pp. 14431445. Miletsky, E.V. and Ivanov, V.G., Latitude Characteristics of the Sunspot Formation Zone and the 11 Year Solar Activity Cycle, Astron. Rep., 2009, vol. 53, pp. 857 862. Miletsky, E.V., Ivanov, V.G., Nagovitsyn, Yu.A., and Jung ner, H., Solar Activity in the Past: from Different Prox ies to Combined Reconstruction, Sol. Phys., 2004, vol. 224, pp. 7784. Nagovitsyn, Yu.A., Ivanov, V.G., Miletsky, E.V., and Nago vitsyna, E.Yu., The Maunder Minimum: NorthSouth Asymmetry in Sunspot Formation, Mean Sunspot Lat itudes, and the Butterfly Diagram, Astron. Rep., 2010, vol. 54, pp. 476480. Obridko, V.N. and Shelting, B.D., Cyclic Variation of the Global Magnetic Field Indices, Sol. Phys., 1992, vol. 137, pp. 167177. Obridko, V.N. and Shelting, B.D., Global Solar Magnetol ogy and Solar Cycle Reference Points, Proc. Pulkovo Conference, 2003, pp. 339344. Petrovay, K., Solar Cycle Prediction, Living Rev. Sol. Phys., 2010, vol. 7. Self Organizing Method in Modeling: GMDH Type Algo rithms. Statistics: Textbooks and Monographs, Farlow, S.J., Ed., 1984, vol. 54. Tlatov, A.G., Search for Relationship between Duration of the Extended Solar Cycles and Amplitude of Sunspot Cycle, Astron. Nachr., 2007, vol. 328, pp. 10271029. Vitinsky, Yu.I., Kopetsky, M., and Kuklin, G.V., Statistika pyatnoobrazovatel'noi deyatel'nosti Solntsa (Statistics of Sunspot Formation Activity), Moscow: Nauka, 1986. Vitinsky, Yu.I., Kuklin, G.V., and Obridko, V.N., On the Main Solar Cycle Phases, Soln. Dannye, 1986, no. 3, pp. 5356.

We also determined that the correlation between the latitudinal drift velocities and the activity level at the current cycle maximum sharply increases in the T2T3 interval, i.a., during the 11 year cycle decline. This is apparently related to the previously established close relationship between the latitudinal extent of the sunspot zone and the sunspot activity level (Miletsky and Ivanov, 2009; Ivanov et al., 2011; Ivanov and Miletsky, 2011). We found out that some parameters of the sunspot latitudinal drift velocities in individual intervals between special points (especially in the T2T3 inter val) substantially correlate with the activity amplitude at the next cycle maximum. This agrees with a previously achieved result (Miletsky and Ivanov, 2009), according to which the latitudinal characteristics of sunspot activ ity during the decline phase can bear information regarding the properties of the next 11 year cycle. We should also note that the velocity value in different intervals between different special points can be differ ently related to the properties of the next 11 year cycle. This information allowed us to construct a model for predicting maximums in the next cycles. We can conclude that the results achieved in this work make it possible to better understand the deter mining properties of 11 year cycles and lead to an increased quality in predicting the future behavior of solar activity. ACKNOWLEDGMENTS This work was partially supported by the Russian Foundation for Basic Research (projects nos. 10 02 00391 and NSh 1625.2012.2) and the Presidium of the Russian Academy of Sciences (basic research pro grams nos. 21 and 22). REFERENCES
Badalyan O.G. and Kuklin G.V. Brightness of the Green Corona and Inflection Points in Cycle 21, Astron. Rep., 1993, vol. 37, pp. 432437. Badalyan, O.G. and Kuklin, G.V., Evolutionary Regimes of the Green Line Corona Brightness during Cycles 20 21, Astron. Astrophys. Trans., 2000, vol. 18, no. 6, pp. 839859. Hathaway, D.H., A Standard Law for the Equatorward Drift of the Sunspot Zones, Sol. Phys., 2011, vol. 273, pp. 221230.

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