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Astron. Astrophys. 328, 670-681 (1997)

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4. Results of wavelet analysis

As discussed in Sect. 3, a one-month averaging chosen for performing the wavelet analysis is not completely arbitrary as it corresponds to the average lifetime of a sunspot group. We focussed our wavelet analysis on time-scales larger than the lifetime of individual sunspots.

4.1. General structure of solar activity 1610 - 1995

The general result of the wavelet analysis is presented in Fig. 4 for the whole period 1610-1994. The signal is shown on the lower panel, and the spectral density (modulus of wavelet coefficients) dependence for a given period (vertical axis) and epoch (horizontal axis), in the upper panel. The spectral density corresponds to the intensity of black. Let us consider a given epoch, say 1850. We notice two black layers that are well-pronounced. One corresponds to the Schwabe cycle (11-y cycle), the other to the Gleissberg cycle (about 100 years).

[FIGURE] Fig. 4. Wavelet transform for solar activity 1610 - 1994; Morlet wavelet modulus.

Fig. 5a presents the global wavelet spectrum as a function of the period T. For comparison, we also give the Fourier spectrum (Fig. 5b). Because the absolute value of Fourier and Morlet transforms is not very pertinent, we present the result in arbitrary units so that they can be separated on the plot. Both curves show a maximum that is related to the 11-year period (the Schwabe cycle). However, the Fourier transform contains many additional maxima that are mainly data processing artefacts. Only one of the additional maxima survives from the Morlet analysis. This maximum corresponds to a time scale of about 100 years, i.e. it can be connected with the so-called Gleissberg cycle. The 11-year power may be connected with a quasi-periodic change of solar activity; the physical meaning of the Gleissberg cycle will be discussed further, with the help of a more sophisticated wavelet analysis.

The spectra in Fig. 5 are plotted on a log-log scale, in which any power-law function produces a linear dependence. Out of the peaks, the Fourier [FORMULA] and Morlet [FORMULA] spectra can be described by some power law functions, [FORMULA], [FORMULA], where [FORMULA] and [FORMULA] are estimated from the slope of the linear fit for the corresponding part of the spectrum. One gets [FORMULA] between maxima and [FORMULA] for [FORMULA] years; [FORMULA]. Such power law-dependencies are typical for stochastic, turbulent motions. However, one should stress that the Fourier spectrum index, [FORMULA], is smaller than the corresponding index of the Morlet spectrum, [FORMULA]. This is because the contribution of noise in the data reduces the spectral index while wavelet transform tends to cleaning the data. A pure white noise would produce a flat spectrum.

[FIGURE] Fig. 5. Solar activity spectra; a Fourier spectrum, b wavelet Morlet spectrum; c spectra of errors series generated as white noise with the dispersion of real data; d effect of inner frequency in Morlet function on the wavelet spectrum

To assess the significance of the wavelet signal, we generated several time-series of white noise with the same dispersion as that of the real time-series (three of these time-series are denoted in Fig. 5c as E1, E2, E3). Then we applied the wavelet analysis to the statistical noise time-series. It is clear from Fig. 5c that the wavelet signals present in the real time-series are significant for timescales longer than 1 year.

In contrast with the Fourier analysis, the wavelet result depends on the choice of the wavelet. We used the Mexican hat (12), (see Fig. 1a), and the wavelet coefficients are shown in Fig. 6. The corresponding global spectrum contains the same two peaks, as in the Morlet spectrum. This confirms the robustness of our global wavelet analysis. We also played with the value of [FORMULA] that characterizes the Morlet wavelet: no significant difference is visible on the spectra (Fig. 5d).

[FIGURE] Fig. 6. Wavelet transform for solar activity 1610 - 1994; Mexican hat.

4.2. Properties of the main cycles

The Schwabe cycle is visible for almost all epochs except during the deep Maunder minimum (1650 - 1680). The deep Maunder minimum should be considered separately (see below). The Schwabe periodicity is remarkably stable. While the intensity of this 11-year cycle has changed by several orders of magnitude, its duration has changed by several years only, i.e. by 10 % to 20 %. The black layer period corresponding to the Schwabe cycle has finite thickness, i.e. the duration of a given cycle can be estimated only with limited accuracy. However, there are three pronounced deviations of the cycle length present in the time series, one at the end of the Maunder minimum (1680 - 1712), one during the so-called Dalton minimum (1790 - 1820), and one at the beginning of the 20th century, near 1900; the latter event can be compared with the example given in Appendix (Fig. 12) when there is a change in the length of the cycle. The corresponding Schwabe cycles are also less intensive.

To investigate the time-behavior of the main solar activity cycle in detail, we plotted the time-scale T, which corresponds to the maximum spectral intensity at a given time t in the wavelet representation of the 8-16 year band (Fig. 7a). The layer is thick enough but the position of the maximum varies with the cycle length. This has been demonstrated in Appendix (Fig. 12), where we give the wavelet representation of a signal for a small frequency change. To check this result further, we calculated the T(t) function using the ridge extraction techniques developed by Torresani (1995), and obtained the same result (see Fig. 7a).

In the plot of Fig. 7b we show the intensity of the wavelet transform (modulus of wavelet coefficients) tracing the maximum defined before. From Fig. 7 we distinguish several episodes, one corresponding to the Maunder minimum (itself divided in two parts - a deep minimum and the end of the minimum), another corresponding to the Dalton minimum and a small event mentioned above near the year 1900. Within the deep Maunder minimum, our algorithm detects a weak (though hardly significant) signal near [FORMULA] years. The three other zones indicated on Fig. 7a correspond to an increase of T. There is a time around 1750 where T increases without equivalent decrease of [FORMULA] in Fig. 7b. However, when looking at the initial data (Fig. 3), the first maximum in sunspot group number is significantly smaller than its neighbours. This corresponds to a single 11-year cycle, and thus does not show up in Fig. 7b. We can conclude that before each solar activity minimum T increases, and is accompanied with a low level of solar activity that persists as long as T increases; the deeper the minimum, the higher the T level before the event. Let us stress that the variation of T near 1900 was also associated with a decrease in the strength of the cycle.

[FIGURE] Fig. 7. Length (a) and cycle strength (b) versus time. The thin line on the panel a traces the local maximum of wavelet coefficients on the phase plan, and the thick line is obtained by the ridge extraction techniques.

The three events mentioned above follows a 100-year periodicity, which produces the second intensive layer seen in Fig. 4 (and the peak in the spectrum on Fig. 5a): this corresponds to the Gleissberg cycle. It is tempting to connect its duration with the typical time intervals between the Maunder minimum, the Dalton minimum and the event near 1900. This idea can be illustrated by using the Mexican hat transform, where white vertical layers isolate epoch of the grand deviations from that of normal activity (Fig. 6). The next deviation should be expected soon since the last one occured about 100 years ago. But no great minimum is to be expected because the level of T corresponding to the present level is not high. Along this line of thought, sunspots of new cycle (cycle 23) have been observed in 1995 and do not exceed 20 degres latitude. This could be an indication that cycle 23 would be weaker than the previous ones.

Physical properties of the grand deviations in terms of sunspot number, asymmetry, and rotation have been studied (Nesme-Ribes et al. 1994), and a possible scenario for the chaotic long-term magnetic evolution has been proposed by Nesme-Ribes et al. (1994).

4.3. Wavelet analysis for Maunder minimum-type episodes

The intensities contrast between the Maunder minimum and nearby epochs is very large (Fig. 4). So we assume that the properties of the grand minimum is somehow reflecting the characteristics of the nearby epoch. This is why we consider several time intervals: the whole Maunder minimum (1643 - 1712), and the end of the Maunder minimum (1695 - 1737). For comparison, we also study the Dalton minimum (1796 - 1845). Results of the wavelet technique are shown for the end of the Maunder minimum (Fig. 8), and for the Dalton minimum (Fig. 9). These events show strong similarities and are characterized by a small increase of the cycle length, with a substantial decrease in the cycle strength. As we consider only monthly sunspot group data, we cannot describe the main features occuring at the end of Maunder minimum, namely the recovery of the north-south symmetry which is typical of the normal state of solar activity (Jennings, 1991; Sokoloff and Nesme-Ribes, 1994).

[FIGURE] Fig. 8. Morlet wavelet transform for the end of the Maunder minimum (1695 - 1737): modulus
[FIGURE] Fig. 9. Morlet wavelet transform for the Dalton minimum (1796 - 1845): modulus.

A quite different result was obtained for the deep Maunder minimum (Figs. 10 and 11). There is no signal corresponding to the Schwabe cycle. Such a specific feature does not appear in any other wavelet transform domain. Suppose that we are dealing with the modulus of a sinusoidal, one or two of the period having a zero value (Appendix). The wavelet transform would produce the same pattern as that seen on Figs. 10 and 11. So our interpretation is that sunspot groups did not show any clear 11-y periodicity during the deep Maunder minimum (1670 - 1690). The sparse sunspots occuring at that time do not follow any periodic process, and should be considered as white noise. The 11-y periodicity was recovered after 1690, at the end of Maunder minimum. Comparison of Figs. 10 to 11 hints that the culmination of events referred to as the Maunder minimum occured near 1665-1670.

[FIGURE] Fig. 10. Morlet wavelet transform for the deep Maunder minimum (1643 - 1712): modulus.

[FIGURE] Fig. 11. Morlet wavelet transform for the deep Maunder minimum (1643 - 1712): phase.
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© European Southern Observatory (ESO) 1997

Online publication: March 26, 1998

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