The phenomenological SG - model of SSN time series was first proposed in 2006 (see Pesnell, 2008; Kontor (2006, 2008 a, 2008 b)) as a result of careful consideration of the Fourier spectrum of the available series of the sunspot numbers and since then it is developing in real time. It can be described as follows. The most convenient SSN time series to be considered first is the monthly averaged sunspot numbers (SSN), or M-data (1749 - 2015), or M(t). It can be represented as:
M-data (t) ~ U-data (t) ±σ (= 11.83) (1)
where M(t) is 0 or positive; "~" means some approximation; U(t) is the filtered M(t), corresponding to the frequency range [0 - f U:(1/10 (months))] in the Fourier spectrum; and the last σ - term is an estimation of random scattering of M(t) around U(t), which is taken as a constant and equal 11.83 (for details, see page "appendix 1" (paragraph A 1) and fig.9).
As for U(t), it is regarded in SG-model as some superposition of Gaussian (or normal) functions, or G - peaks (2)
, where G U, i - peaks have the close parameters W and distances between peaks Δx i = (x c, i+1 – x c, i), and a considerable variability of their amplitudes, H i = A i / (w i · (√(π/2))) . U-data analysis shows, that GU, i - peaks are represented by two types of peaks, to wit by G U 1- type and G U A- type. G U 1- peaks have a small amplitude (H U 1 < 20) and so, they correspond to the single simple sunspot groups. G U A- peaks have amplitudes H U A > 20, which are changing in the broad range from ~ 20 to ~ 200, and so, they correspond to the active phase of the solar cycle with multiple sunspot groups of different complexity observed on the Sun. So, U-data is an alternating sequence of groups, consisting of G U 1- peaks or G U A- peaks. Groups of small G U 1 - peaks (they have the average length about 2.5 years) describe the periods of very low solar activity ("microscopic grand minimums", mGM) between the "active periods" of 11-year Schwabe cycles (S-cycles); in their turn, S-cycles (their active phase has the average length about 8.5 years) are represented by groups of G U A- peaks with the changing amplitudes (see fig. 5 in [Kontor (2006)] ). It should be noted that the G U - peaks are one of the five types of quasi - regular temporary structures (QRTS), which can be distinguished in SSN time series. A typical G U - peak (QRTS 0 ) has the length of order of 1 year ( more specific TU = 2 w, see table 1 on the page "Data").
Analysis of K-data (K(t)), which are the very smoothed (filtered) M-data (they correspond to the frequency range [0 - f K:(1/4.17 (years), or 1/50 (months)] in the Fourier spectrum), allows to find out both the envelope of G U A- peak's maximums (H U A,i ) in each S-cycle and the "real shape" of the every S-cycle . Note here that S- cycle itself is the second from the mentioned above five types of QRTS of SSN time series. A typical S - cycle has the length of order of 10 years (QRTS 1 ). K-data are the result of M-data (pretty heavy) filtering , i.e. a smooth continuous quasi-periodic curve, which can be regarded as a sequence of S-cycles. However, it is known that this sequence of well observed S-cycles is sometimes interrupted by GM-periods during which solar activity is low, and the question of the existence of S-cycles is problematic. As SG-model has been developed to describe and forecast the so-called well-observed S-cycles, it is used when solar activity is developed enough, i.e. when the K-data exceeds a certain threshold (for U-data this limitation is less significant). Let us call periods, when the K-data <20, as GM - or mGM - periods (the latter are "gaps" between the well-observed S- cycles) and denote the corresponding to those periods K-data as K 0 -data. Then K-data can be represented as the sum of K 0 -data and K * -data. These last data > 20 and are pretty well described by SG- model; let us call this description SGD (t), see (4).
K*i -data ~ # SGD (t 0, i ; t) = P (t) + B (t) + D (t) (4)
where t 0, i is the date of the S i -cycle (t) observed (or predicted) beginning and t 0, i < t < t 0, i +1 . It follows from (4) that any S-cycle is a superposition of the three components with each component of kind (2). In (4) i denotes the same as #, namely, a sequence number of the considered S- cycle. Then, see (5):
K-data = ∑ [K 0,i (t) + K* i (t)] (5)
All G- peaks ( SG- model has three types of them (G U -, G K -, and G GL- peaks)) are different from each other only by the values of their three parameters: Xc (the position on the time axis), w (width at half maximum, usually measured in months or years), and A = H · w · (√(π/2)), where H is the height of G- peak.
Consequently, the S-cycle's shape for each S-cycle is determined by nine parameters: X P, X B, X D (which are counted off t 0, i ), W P, W B, W D (which are approximately equal and so designated by W), and H P, i, H B, H D ( amplitudes of P-, B-, and D- components of the S-cycle). In accordance with SG-model, eight from nine free parameters can be considered as approximately constant, and only one parameter (H P, i) noticeably varies from one S-cycle to another. So, it is necessary to find out only the envelope of H P, i and to get after that the complete description (#SGD) of S-cycle shape (4) and the forecasts (#SGF: GL) for the next S-cycles (here they are #24 and #25) in the first approximation. So, H P, i - the height of Si- cycle's P component - plays a special role in SG- model. The long-term (~ 100 years) variation of Hp, i (t) is associated with the known Gleissberg cycle, which is considered in SG-model as description of that variation. Then the Gleissberg cycle (GL- cycle) turns into a superposition of two G GL- peaks (6)
#GL-cycle = G GL (t, X C , w 1, A 1 ) + G GL (t, X C+ Δx , w 2 , A 2 ) (6)
GL- cycle itself is the third from the mentioned above five types of quasi - regular temporary structures of SSN time series. A typical GL - cycle has the length of order 100 years (QRTS 2 ).
The whole Hp, i (t) - variation (GL-variation) itself can be described by the expression (7):
GL-variation = ∑ G GL (t, (X C, i+1 = X C, i +Δx i ), w i , A i ) (7)
The mathematical expression for that envelope can be received from analysis of some SSN time series segment, which covers three Gleissberg cycles. It is the yearly averaged Y-data (1700 - present), and heavy smoothed (filtered) Y-data (E-data) corresponding to the frequency range [0 - f E : (1/17.86 (years)) in the Fourier spectrum] (see for details [Kontor, 2006, 2008 a]). Observations show that if the number of GL-cycles in GL-variation is integer, the odd Δx i ~ 40 years, whereas Δx i+1 ~ 60 years (even-numbered, giving the distance between the GL-cycles). We find that in norm the temporal distance between the adjacent G GL - peaks within GL-cycle is less than the "length" of G GL - peak (~ 60 years), whereas the distance between the adjacent GL-cycles longer than the "length" of G GL - peak.
To receive an idea about variability of the amplitude of S-cycle's modulation, which is determined by GL-variation, it is necessary to consider SSN time series, which is longer than WSN time series, for example, LCN time series (9455 B.C. - 1995 A.D., see Solanki et al., 2005). LCN-data is close to E-data because they have the close Nyquist frequencies (see fig.8). Consideration of LCN-data from the point of SG-model's view shows, that three types of activity are realizing on the Sun: the Grand minimums (GM i ), when only the rare single sunspot groups (G U 1- peaks ) are observed for the periods ~ hundred years (GMA-type); the low S-cycles, in which the stable B-component (4) dominates ( H P < H B > H D ) and the role of GL-modulation (6) of the P-component amplitude (H P, i ) is inessential (BDA-type); and, at last, the high (or regular) S-cycles, for which an amplitude is determined by GL-modulation (similar to (6), see for details [Kontor, 2006, 2008 a, 2008 b]). It is GLA-type of solar activity. Therefore any SSN time series (WSN time series, LCN time series and others) can be expressed as:
where SAE i ( t ) is a segment of SSN time series, which starts from GM i (t 0, i - the date of GM i beginning); after that, some low S-cycles (BDA), and a number of high S-cycles, grouped in GL - cycles (GLA), are observed. SAE i (t) is finished when the next GM i+1 starts. So, the largest reliably known temporal structure of SSN time series is the "Solar Activity Episode",
SAE = GM-period + GL-variation (9).
For example, WSN time series begins from 1610 A.D., but the new (and still current) SAE started at 1645, when the Maunder Minimum begins (see, for example, fig.1 in Kontor (2006)). It is followed by critically low S-cycle # -4 and after that three GL-cycles are observed (the critically low S-cycles are observed in "the GL-cycle's junctions" (cycle # 6 and, probably, # 25). So, the current SAE is not finished yet. SAE are the fourth from the mentioned above five types of quasi - regular temporary structures of SSN time series. Their lengths have a very large spread (from 200 to 2000 years), but as a reference the SAE length of order 1000 years (QRTS 3 ) could be considered (the shortest from the mentioned above five types of quasi - regular temporary structures is the structure, connected with the solar rotation (QRTS - 1 ); consequently it has the typical length about 27 days, or of order 0.1 year).
SG - model is just an instrument for investigation of SSN time series' and, specially, S-cycles' details, but we will see that it helps to understand qualitatively the temporal structure of SSN time series and variations of S-cycle shape in framework of contemporary Flux Transport Dynamo models. Mainly 3 of 5 structures are analyzed - QRTS 1 or S-cycle presented by K-data (see fig.1); QRTS 2 or GL-cycle as an envelope of H P, i (t) variation (see fig.2) and, at last, QRTS 0 or U-data (they could be the result of the transverse oscillations of the solar tachocline).
N. N. Kontor, e-mail: sgnnk@Live.com
first published: 11/27/2008