Wavelet entropy and the multi-peaked structure of solar cycle maximum

New Astronomy 8 (2003) 105–117 www.elsevier.com / locate / newast

Wavelet entropy and the multi-peaked structure of solar cycle maximum S. Sello Mathematical and Physical Models, Enel Research, Via Andrea Pisano 120, 56122 Pisa, Italy Received 24 June 2002; received in revised form 15 August 2002; accepted 30 August 2002 Communicated by W. Soon

Abstract Wavelet analysis of different solar activity indices—sunspot numbers, sunspot areas and flare index—allows us to investigate the time evolution of some frequency dependent functionals, like wavelet entropy, which gives useful information about the complexity level of the related signals. The main aim of this work is the analysis of the time behavior of wavelet entropy near the maximum phases of solar cycles 21–22–23 in order to further contribute to the characterization of the multi-peaked structure of solar cycle maxima and to support the current interpretation of the so-called Gnevyshev gap.  2002 Elsevier Science B.V. All rights reserved. PACS: 95.75.Wx; 96.60.-j; 96.60.Qc; 96.60.Rd Keywords: Sun: activity; Sun: sunspots; Sun: flares; Methods: data analysis

1. Introduction The recognition and characterization of a structured multi-peaked feature for the Schwabe solar cycle ( ¯ 11 year) maximum phase, date back to earlier works by Gnevyshev (1963, 1967), mainly utilizing the time behavior of different heliographic latitudinal distributions of the outside eclipses in˚ In particular, the tensity coronal line at 5303 A. author found the evidence for a bimodal structure of the coronal activity maximum of the 19th cycle, where the first peak is located at the end of the increasing solar cycle and the second one, involving lower solar latitudes only, located near the early phase of the declining activity. These first results, supported by successive works (Kopecky and KukE-mail address: [email protected] (S. Sello).

lin, 1969; Gnevyshev, 1977) including events in the photosphere and chromosphere, led the authors to conclude that there are two different processes or waves, partly superimposed in time, that may be responsible for the observed dual-peaked feature during the maximum phase of the solar activity cycles. The existence of a structured maximum in solar activity was confirmed in successive works in which we analyzed the processes involved in time behavior of large and complex active regions using data for sunspot areas, coronal magnetic energy maxima, global heliomagnetic fields and several solar-geophysical indices. More recent detailed studies on long-term galactic cosmic-ray modulation by Storini and co-workers at the Cosmic Ray Section (IFSI-CNR) of the Rome University (Storini et al., 1997), confirm the Gnevyshev’s bimodal feature during solar activity maxima, in particular for high-

1384-1076 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S1384-1076( 02 )00192-6

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energy and long-duration events, i.e., phenomena affecting heliospheric and interplanetary environment. The authors, analyzing the time distribution of cosmic-ray modulation related to intense dynamical phenomena, conclude that a ‘cluster of solar phenomena, strongly influencing the heliosphere, occurs several months before and after the observed sunspot maximum’ and suggest, in connection with Gnevyshev work, to call Gnevyshev gap (in short GG) the time interval of a reduced or depressed solar activity phenomena, probably connected to the polar heliomagnetic field reversal. A later work by Feminella and Storini (1997), contains accurate investigations of the maximum activity shape using an extended set of various indices related to different solar layers: the relative sunspot number, as photospheric activity index; the 10.7 cm radio flux, mainly as chromospheric activity index; the full solar disk ˚ solar X-ray background, as a density index 1–8 A for the quiet corona structure; and the monthly average of grouped chromospheric flares. Analyzing the time behavior of the above indices at different time-scales and for several cycles, the authors confirm the significant existence of the multi-peaked structure of the maximum phases, especially if we perform the analysis at intermediate time scales: 157 # t # 204 days. In this way we discard both the short-term fluctuations and the long-term modulations, which tend to mask the multi-peaked structure. Moreover, the multi-structured maximum phases appear as a common characteristic of all the solar atmospheric layers considered. Another important aspect of the multi-peaked structure is its relation with the intensity and long-duration or importance of the event analyzed, supporting that only particular energetic phenomena associated to the interaction between global and local magnetic fields are mainly involved in the onset of the solar maximum shape. In this context, the gap associated to the bimodal feature (GG), should occur during the space-time variability (reversal) of the general heliomagnetic field (Feminella and Storini, 1997). Another important recent contribution that clarifies the origin and the role of the multi-peaked structure and the related GG feature to time evolution of solar and geomagnetic parameters, is the work by Bazilevskaya and co-workers (Bazilevskaya et al., 2000) where there is a detailed analysis of maxima struc-

ture for solar cycles 21 and 22. The solar parameters used, sunspot group numbers, sunspot areas, grouped Ha solar flares, X-ray bursts and large-scale solar magnetic fields, was selected to represent solar processes at different scales: slowly changing and large-scale regions and fast varying small-scale structures. In particular, the authors used monthly mean data smoothed with a 7-points running average technique in order to better point out the multipeak and GG features at the proper time-scales. The results for the whole solar disk confirm the doublepeak structure of the maxima phases for both the solar cycles considered, even if in a different way for the various solar indices. Moreover, the analyses for separate solar hemispheres support the fact that the GG feature is not due to a superposition effect of single-peaked and time-shifted activity in the distinct solar hemispheres, i.e., a north–south anisotropic effect (Feminella and Storini, 1997). This separate analysis allowed also the study of time-synchronism properties of the multi-peaked structure for different solar activity indices. In particular, in the solar cycle 21 ‘the very prominent multipeaked structures exists in each solar hemisphere, alike in the time-scale but strongly shifted in phase . . . In cycle 22 the time profiles of various indices are rather synchronous in the two hemispheres, showing in general a doublepeak structure’ (Bazilevskaya et al., 2000). The geometrical features of peaks are in general different considering different indices and hemispheres. Further, a close inspection of periodicities related to different peaks near the maxima, allowed the detection of a quasi-periodic internal structure with quasi-biennal time-scales (14–18 month for cycle 21 and 23–30 month for cycle 22) showing different phase relations but amplitudes synchronous with the Schwabe solar cycle. Thus ‘the GG structure appears to be a consequence of a multi-peaked (quasi-biennal) wave superposed on the well-known ¯11 year wave’ (Bazilevskaya et al., 2000). The most recent comprehensive review on the subject by Storini and co-workers (2002) points out the great efforts spent by many researchers in different areas of solar-terrestrial phenomena in order to better analyze, characterize and to interpret the multi-peaked structure and the related GG feature. One open point is the lack of a theoretical interpretation of this solar phenomenon. A current plausible

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hypothesis is that ‘during the inversion of the polar heliomagnetic field a decrease (or gap) in the number of high energy events occurs’ (Storini et al., 2002). To support this interpretation we need more analyses on the time-space evolution of the multipolar structure of the solar magnetic field, and a more precise identification and characterization of solar activity waves, mainly detected through periodic or quasiperiodic processes acting near solar maxima. The main aim of the present work is to further contribute to the characterization of the structured maxima of some solar activity indices using the wavelet analysis technique which is particularly suitable for resolving the time-frequency evolution of complex multi-scale and transient time-series. In particular, we focus our attention on the synchronism features of the main quasi-periodic components detected through the wavelet maps, during the maximum phases of solar cycles 21, 22 and the current cycle 23. Further, using a wavelet-derived functional, called wavelet entropy, we try to detect some relations between time-evolution of the solar activity disorder content, i.e., a dynamical complexity degree expressed in terms of the number of different frequencies present in the spectrum, and the multi-peaked structure with related GG feature.

2. Wavelet analysis: wavelet entropy Since the introduction of the wavelet transform by Grossmann and Morlet, (1984), in order to overcome the window limitations of the Gabor transform to non-stationary signals, this technique has been extensively applied in time-series analysis, including the study of solar and stellar activity cycles (Ochadlick, 1993; Lawrence, 1995; Frick et al., 1997; Oliver et al., 1998). With a local decomposition of a multiscale signal, wavelet analysis is able to properly detect time evolutions of the frequency distribution. This is particularly important when we consider intermittent and, more generally, non-stationary processes. More precisely, the continuous wavelet transform represents on optimal localized decomposition of a real, finite energy, time series: x(t) [ L 2 (R ) as a function of both time, t, and frequency (scale), a, from a convolution integral:

107 1`

1 (Wc x)(a,t ) 5 ] Œ] a

t 2t E dt x(t)c *S]] D a

(1)

2`

where c is called analysing wavelet if it verifies an admissibility condition: 1`

cc 5

E dv v

21

u cˆ (v )u 2 , `

(2)

0

with: 1`

cˆ (v ) 5

E dt c(t) e

2i v t

(3)

2`

This last condition imposes: cˆ (0) 5 0, i.e. the wavelet has a zero mean. In Eq. (1) a,t [ R,(a ± 0) are the scale and translation parameters, respectively (Daubechies, 1992; Mallat, 1998). In fact, it follows from Eq. (1) that the effectiveness of the wavelet analysis depends on a suitable choice of the analyzing wavelet for the signal of interest. For our timeseries application, where we are mainly interested to track the temporal evolution of both the amplitude and phase of solar activity signals, we chose to use the family of complex analyzing wavelets consisting of a plane wave modulated by a Gaussian, called Morlet wavelet (Torrence and Compo, 1998):

c (h ) 5 p 21 / 4 e i v 0h e 2h

2 / 2s 2

(4)

where: h 5 (t 2 t ) /a, and v0 is a non-dimensional frequency. s is an adjustable parameter which can be determined in order to obtain the optimal wavelet resolution level both in time and frequency, for the characteristic time-scale of the original series (Soon et al., 1999). The limited frequency resolution imposes an half-power bandwidth of our wavelet given by: Df/f ¯ 0.12. From the local wavelet power spectrum obtained from Eq. (1), it is possible to derive various time dependent functionals. An interesting example is related to the measure of the ‘disorder’ level contained in the signal well quantified by the wavelet entropy introduced by Quian Quiroga and co-workers (1999). The concept of thermodynamic entropy is well known in physics as a measure of the system disorder. A previous measure of entropy has been introduced from the Fourier power spectrum applied to Hamiltonian systems,

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108

called spectral entropy, by Powell and Percival (1979). Using the extended technique of the wavelet formalism, it is possible to define a wavelet entropy as a function of time (Quian Quiroga et al., 1999):

Op

WSt 5 2

existing between wavelet entropy and the onset of the Gnevyshev gap.

3. Data selection

t,k

log 2 ( pt,k )

(5)

k

where: P(k,t) pt,k 5 ]]] dk P(k,t)

(6)

E

is the energy probability distribution for each scale level k at time t and:

U S DU ,

k0 1 P(k,t) 5 ]] W ], t 2cc k 0 k

2

k$0

(7)

is the local wavelet spectrum at frequency k and time t generally visualized by proper contour maps. In Eq. (7) k 0 is the peak frequency of the analyzing wavelet c (Torrence and Compo, 1998). From Eq. (5), it follows that the wavelet entropy is minimum when the signal represents an ordered activity characterized by a narrow frequency distribution, whereas the entropy is high when the signal contains a broad spectrum of frequency distribution. This last feature is a common sign related to some kind of dynamical complex behavior. Here we are mainly interested in distinguishing between ordered or regular processes, and more complex dynamical behaviors involved near the GG gap. In fact, a genuine stochastic process and a deterministic chaotic dynamics are both characterized by a broad-band spectrum, with many interacting frequencies. The wavelet entropy alone is not able to distinguish the above different cases of dynamics. Previous analyses on monthly mean sunspot numbers, clearly illustrated the usefulness of the wavelet entropy to follows the time evolution of the complexity level of the solar cycle activity (Sello, 2000). In this work, we show some results of the wavelet analysis and related wavelet entropy, using different solar activity indices in order to characterize the time evolution of some characteristic frequencies linked to the multi-peaked feature of the maximum phases of solar cycles 21–22 and the current cycle 23. One interesting aspect of this work is the determination of possible relations

The solar activity indices used for our analysis correspond to a common selection based on different atmospheric layers and time-scales: • the monthly mean international sunspot numbers extracted from the Solar Influences Data Center archive of Brussels (Cugnon, SIDC, 2002); • the monthly averages of the daily sunspot areas from the Royal Greenwich Observatory / USAF / NOAA (Hathaway, MSFC, 2002); • the monthly mean of daily flare index from Bogazici University Kandilli Observatory of Istanbul (Atac and Ozguc, 2002). The daily flare index for solar cycles 21, 22 and 23 was determined by using the final grouped solar flares which are compiled by NGDC (National Geophysical Data Center). It is calculated, for each flare, by Atac and Ozguc using the formula (Kleczek, 1952): FI 5 i t

(8)

where i represents the intensity scale of importance and t the duration (in minutes) of the flare. Kleczek first introduced the quantity FI to quantify the daily flare activity over 24 hours per day, assuming that this relationship gives roughly the total energy emitted by the flares. The data for sunspot areas and flare index were recorded both for the full solar disk and for the separate hemispheres. The solar activity processes represented by the above data are primarily connected to slow varying, large-scale structures (active regions) and to fast changing, small-scale structures (solar flares).

4. Results and discussion The principal information derived from the application of the wavelet formalism to real data is the local spectrum, Eq. (7), which allows us to resolve the time-evolution of the related frequency distribution. In Fig. 1 we show the results of the wavelet analysis applied to monthly mean sunspot numbers

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109

Fig. 1. Wavelet analysis of monthly mean sunspot numbers (SIDC) s 5 1.08.

covering the time interval: 1976–March 2002, i.e. solar cycles 21, 22 and 23. The upper panel shows the original time series in its natural units (red line) and the 12-points smoothed values (green line) with the running averaged technique:

O

1 n15 1 S˜n 5 ] Sk 1 ](Sn16 1 Sn26 ) 12 k5n25 2

(9)

where Sk is the mean value of sunspot number for the month k. The blue line is the computed wavelet entropy, properly scaled in order to clearly demonstrate its time behavior when compared with the original time series. Time is expressed in years. The central panel shows the amplitudes of the wavelet local power spectrum in terms of a colour contour map. Red corresponds to the strongest energetic contributions to the power spectrum, while blue represents the weakest components. Horizontal time axis corresponds to the axis of the upper time series and the vertical scale (frequency) axis is, for convenience, expressed in log values of cycle per year 21 . Thus the period range analyzed is from 134 days (value y 5 1) to 4.5 years (value y 5 2 1.5). The related frequency is given by: f(nHz) 5

31.7 exp( y). The corresponding period expressed in days is also shown in the vertical scale. The right panel shows the mean wavelet spectrum (an averaged and weighted Fourier spectrum) obtained by time integration of the local map. The significance of the local power map was tested using an adjustable red-noise autoregressive lag-1 background spectrum (Torrence and Compo, 1998). The white tracks correspond to local wavelet power maxima used to evaluate the beat frequency and the optimal value of s in Eq. (4) (Soon et al., 1999). An evident feature of the wavelet local power map is the discontinuity of its time distribution during different solar cycles, i.e., the periodicity content and its shape is different from one cycle to the next, as already noted in other works (Rybak et al., 2001). Here we focus the attention mainly on the characteristic features near the maximum phases of solar cycles. The changing dominant period contributions are located near 160 days for cycle 21; 150, 180 and 280 days for cycle 22; 137 days and 1.6 years for the current cycle 23. A maximum double peak structure and related GG feature is well evident for all cycles, in particular if we follow the time evolution of the smoothed sunspot values. Using the same notation as above,

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Fig. 2. Wavelet analysis of monthly mean sunspot areas (MFSC) s 5 1.39.

Fig. 2 shows the wavelet results when applied to the full disk sunspot areas time series, for the same time interval. The main period contributions are located near 160 and 240 days for cycle 21; 180 days for cycle 22; 167, 280 days and 1.6 years for the current cycle 23. In this case, the GG feature between maximum peaks is more prominent for all cycles. A detailed wavelet analysis performed by Oliver and co-workers, (Oliver et al., 1998), using a time series consisting of daily sunspot areas between 1874 and 1993, pointed out the temporal variation of the periodicity content near 160 days. An important result, confirmed in the present study, is that the periodicity only present around the maximum phases of some solar cycles (16 to 21) with a strong changing power. In particular the authors show the peak structure during the maximum of solar cycle 19 and the recent high power content during cycle 21. This analysis allowed the localization of the temporal synchronization between the appearance of the ¯ 160 day periodicity and high energy solar flare events. For a detailed comparison between our wavelet analysis and the results displayed by Oliver et al., we show an expanded-scale view of the wavelet map computed for the same frequency-time

interval in Fig. 3. We confirm the fact that the sampling considered here is adequate for studying the evolution of time-scales around 160 days. Our comparison confirms the following common features: (1) there exist a significant power around 161 days, slowly changing, which is in temporal coincidence with high-energy solar flares recorded between 1980 and 1984 (solar cycle 21); (2) this periodicity decreases after 1984 and is absent during the maximum phase of solar cycle 22, instead a longer and weaker periodicity near 180 days appears. Fig. 4 shows the results of the wavelet analysis for the full disk flare index time series, for the time interval: 1976–August 2001. The main period contributions are located near 154, 200 days and 3.38 years for cycle 21; 148, 220 days and 1.8 years for cycle 22; 134 and 284 days, for the current cycle 23. The GG feature is evident for the dual-peaked structures of solar cycle maxima 22 and 23, but less clear in the more complex multi-peaked structure of solar cycle maximum 21. Time dependence of the spectrum information, obtained by wavelet analysis, allows us to determine some synchronization features of the main period contributions related to the above different solar

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Fig. 3. Wavelet analysis of monthly mean sunspot areas (MFSC) s 5 1.39 as in Fig. 2 but using an expanded-scale view from 138 to 192 days and for the time interval: 1978–1993.

Fig. 4. Wavelet analysis of monthly mean flare index (Kandilli Observatory) s 5 1.21.

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activity indices. The strong, near 160-day periodicity, during cycle 21, appears well time-synchronized for all activity indices considered. This timefrequency coincidence, related to a Rieger-type periodicity, for sunspot areas and energetic solar flares, has been previously investigated (Oliver et al., 1998). Oliver and co-workers related the timing coincidence to a periodic emergence of new magnetic flux in compact magnetic field structures which appear near the maxima of some solar cycles (Oliver et al., 1998). This periodicity may be modulated by equatorially trapped Rossby waves and mixed Rossby–Poincare´ waves in the magnetized solar surface layer (Lou, 2000). This synchronous event is well localized near the GG feature of solar cycle 21 maximum. Further periodicities, quite synchronized during the ascending and descending parts of cycle 21, are located near 230 days. During solar cycle 22, the main synchronized periodicity near its maximum phase is located near 180 days occurring just after the onset of the GG feature. The related energetic contribution is strong for sunspot numbers but quite weak for solar flares index. For the current solar cycle 23, only the periodicity near 140-day appears well time-synchronized for all solar indices. The ‘quasi-biennal’ periodicity near 1.6 year is synchronous for sunspot numbers and sunspot areas, but this period is not detectable in the solar flares record. The information about these time synchronized frequencies near the development of solar maxima phases can help to shed more light on the possible physical mechanisms involved. The time-scale structure variability of periodicities involved during the onset of the GG feature for different solar cycles, suggests a more complex scenario than a simple modulation of a ‘quasi-biennal’ wave on the longer Schwabe cycle as previously suggested by Bazilevskaya et al. (2000). Another interesting information that can be extracted from the time-frequency wavelet analysis is the wavelet entropy. In the upper panels of Figs. 1–4, we compare the time evolution of solar activity indices and the related wavelet entropy (blue line). A prominent feature of the wavelet entropy near the solar maximum phase is the presence of a quite synchronous local maximum with the onset of the GG feature. In this respect the current solar cycle 23 appears quite anomalous, where the entropy maxi-

mum is absent or clearly preceding the GG feature, depending on the index considered. In order to study this property statistically we analyzed the near-maximum behavior of wavelet entropy for solar cycles 12 to 23 using sunspot numbers and sunspot areas (see Figs. 5 and 6). Time-synchronization of wavelet entropy is generally confirmed for different indices, with some notable exceptions, where the onset of local maximum entropy appears earlier in sunspot areas than sunspot numbers. On the basis of the current interpretation of GG feature (Storini et al., 2002), we support the fact that a reduction of the most intense activity events characterizes the onset of GG feature. The decrease of active regions energy connected to the GG feature is probably related to a general magnetic field inversion that may be discerned with a peak in the wavelet entropy. This observation suggests that, during a magnetic reversal, there is an increase of dynamical complexity probably related to an enhanced number of low intensity and comparable energy processes involved, which in turn produces a spread in the frequency distribution. In other words, the local maximum observed in the wavelet entropy near the solar cycle maximum, is consistent with the above GG-feature interpretation that invokes the inversion of the solar magnetic field. The generation and propagation of quasi bi-annual waves, related to velocity and magnetic field disturbances, near the moment of the solar magnetic field polarity reversal have been investigated by Gigolashvili et al. (1995) and Pataraya and Zaqarashvil (1995). These waves, linked mainly to a shear instability, increase the amplitudes of the magnetic field and velocity fluctuations significantly, and thus leading to an enhanced dynamical complexity. In contrast, the local maxima detected in the wavelet entropy near the solar cycle minima are certainly due to different physical processes, not involving magnetic field inversion, but with the same dynamic characteristics, i.e., numerous weak and short interacting events which are able to produce a significant spread in the wavelet frequency distribution. We further note that, during solar activity cycle minima, higher equatorial velocities occur together with torsional oscillations for the solar plasma, which are in turn connected with steeper spatial gradients and fluctuations of the differential rotation (Howard and Labonte, 1980; Balthasar et al., 1986). These ob-

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Fig. 5. Wavelet entropy vs. sunspot numbers.

servations essentially confirm the importance of the intensity of the energy carried out by the physical processes that originate the multipeaked feature near the maximum of solar cycles, as previously noted and discussed (Feminella and Storini, 1997). In particular, only dominant high-energy and long-duration events appear correlated to the successive presence of the peaks. Wavelet transform allows us to better show the invoked correlation existing between the origin and evolution of dual peak behavior in solar cycle maxima, and the time-variability of the heliomagnetic field. As already pointed out, the GG feature is not due to a superposed and time-shifted single peaked solar activity in its opposite hemispheres, or an asymmetry effect (Bazilevskaya et al., 2000). To confirm the bimodal maximum structure for both solar hemispheres, Figs. 7–10 show some sample results of the wavelet analysis applied to both sunspot areas and solar flare index. The GG feature appears as a clear feature for both solar hemispheres but its amplitude is different. More importantly, this structure is not

synchronous when we consider separate hemispheres and different solar cycles. Moreover, during the onset of the GG feature, the periodicities detected are not the same. For example, in the case of sunspot areas (Figs. 7 and 8) the main GG feature appears earlier in the northern hemisphere during cycle 21; it is quite synchronous during cycle 22; but for the current cycle 23, the GG feature is evident only in the southern hemisphere. The related main periodicities are: 164 and 134 days for northern and southern hemispheres, respectively, for solar cycle 21. Similarly, 210 and 190 days for solar cycle 22; and 164 day and 1.6 year for solar cycle 23. Furthermore, the wavelet entropy behavior is not symmetric for opposite hemispheres: generally, only one hemisphere (different for different cycles) shows a wavelet entropy local maximum. This characteristic suggests a possible phase shift involved in the onset and development of GG feature for separate solar hemispheres. For the case of flare index (Figs. 9 and 10), there is a clear multipeak structure in the northern hemi-

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Fig. 6. Wavelet entropy vs. sunspot areas.

Fig. 7. Wavelet analysis for northern hemisphere sunspot areas.

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Fig. 8. Wavelet analysis for southern hemisphere sunspot areas.

Fig. 9. Wavelet analysis for northern hemisphere flare index.

115

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Fig. 10. Wavelet analysis for southern hemisphere flare index.

sphere during cycle 21, where the main GG feature comes earlier than any similar structure in the southern hemisphere. The synchronism of GG structure is confirmed during cycle 22, whereas for the current cycle 23 there are evidences of a depressed activity only in the northern hemisphere. The related main periodicities are about: 148, 200 days and 156, 232 days for the northern and southern hemispheres, respectively, during solar cycle 21. Similar numbers are 210 and 280 days for the two hemispheres for solar cycle 22. The periods are 134 and 270 days, for only the northern hemisphere, in the solar cycle 23. We confirm again a different wavelet entropy behavior for the opposite hemispheres, although the wavelet entropy generally tend to reach high values near the GG feature. However, the limited information now available for flare activity during solar cycle 23, prevent any definite detection of the involved periodicities and the evaluation of the entropy behavior near the maximum phase. An important result of the wavelet analysis is the ability to detect common features such as time evolution, synchronization properties and shape of the spectrum content for different solar layers, as given by different solar activity indices. This information may be

also very useful to verify the reliability and accuracy of any advanced solar activity prediction method, which should aim to obtain a more complete information on time-frequency characteristics of different solar cycles.

5. Conclusions The multi-peaked structure of solar cycle maximum is a real feature of solar activity. The phenomenon is well displayed examining different solar activity indices for both full disk and separate hemispheres. Detailed analyses of the time behavior of large and complex active regions, relative to different solar atmosphere layers and including longterm galactic cosmic-ray modulation, clearly demonstrated the existence of a reduced solar activity, especially for high-energy and long-duration events, near the maximum phase of different solar cycles. This time interval, called Gnevyshev gap (GG), is probably connected to mechanisms involved in the polar heliomagnetic field inversion. In order to support the above picture and the characterization of the onset and development of the multipeaked struc-

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ture of solar maximum, we performed a detailed wavelet analysis on different solar activity indices and we evaluated the behavior of the wavelet entropy which quantifies the disorder level of related signals. Wavelet analysis allows us to follow the complex time evolution of the frequency content and its synchronization properties in the context of both different solar cycles and solar activity indices. For solar cycles 21 to 23, we found different periodicities involved in the time interval characterizing the GG feature that range from 160 days to 1.6 year. Wavelet entropy generally shows a local maximum peak quite synchronous to GG, although there are notable exceptions. This peculiar behavior appears to be consistent with the suggested GG physical interpretation, whereby a reduction of the most intense activity events, caused by the onset of a heliomagnetic inversion state, forces a general increase of dynamical complexity, probably due to an enhanced number of low intensity and comparable energy processes involved. However, many aspects of the wavelet properties of the GG feature remain unclear: (1) the different content and evolution of energy spectrum for different cycles near the multi-peaked structure; (2) the lack of a maximum peak in the wavelet entropy for some cycles, with a clear presence of GG features; (3) the asymmetric behavior of wavelet entropy for opposite solar hemispheres; etc. An extension of the present analysis including different solar activity indices and especially data from next solar cycles is needed in order to draw more statistically reliable conclusions and a more comprehensive picture on various characteristics involved in the processes driving the multi-peaked features of solar cycle maxima.

Acknowledgements Special thanks to Prof. J.K. Lawrence and to Prof. W. Soon for a careful reading of the manuscript and for many fruitful comments and suggestions.

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