1. The current Conclusion (optimistic)
1.1. Introduction. The most general notion used here is solar activity. It includes a generation of largescale toroidal (TMF) and poloidal (PMF) magnetic fields (solar dynamo), their inter conversions, the slow (magnetic diffusion) and fast (reconnection or solar flares) their dissipation and a whole set of the other observable solar phenomena (the 11year solar cycle (Scycle) of sunspot number (SSN); Grand minimum (GM) of solar activity; the Gleissberg cycle (GLcycle) and so on). The fact is that a mathematical basis for understanding the solar activity is the system of Navier  Stokes and magnetohydrodynamics equations , which consists of nearly two dozen equations. It is clear that such a complex system of equations can not be solved in the forehead, and it is required the significant simplifications of it (leading to a loss of some essential physics). Such simplifications, for example in the theory of the solar cycle, indeed have been made, see Ossendrijver, 2003; Nandy, 2009; Parker, 2009; Olemskoy, 2014. However, for the elaboration of a theory, the prompts or constraints that it receives from observations are necessary. These are observed effects that can not be ignored. Such effects are Hale's magnetic polarity law; Spörer’s law; Scycle's length; some helioseismic results; Joy's law; GM existence and a number of other. Science develops when curious observers and experimenters throw up facts that dodgy theorists explain with the help of their theories and even get consequences, which after are searched for by these very observers and experimenters. Studies of solar activity are not an exception to the rule.
1.2. Data. In this situation, I am in the crowd of observers who dream to put out of countenance those arrogant theorists. But I do not know if they will notice me (I certainly mean the results presented). I started in 1998 from that to describe the shape of any solar cycle (represented by (yearly averaged) Ydata) by a onehumped asymmetric smooth curve (it was a lognormal curve). The result was unsatisfactory. Then I made the transition from smoothed to filtered data (using Fourier series) and received along with the regularly observed Mdata, U  data and Kdata. Udata are slightly smoothed in comparison with Mdata and may be used as some approximation (1) of abruptly changing Mdata. Kdata are more smoothed than the generally accepted smoothed M13data and it allows to get the good (indistinguishable to the eye) description of the active phase for any Scycle. Another preliminary result was the realization that the first harmonic (5.5 years) of the main (11year) peak in the Fourier spectrum of SSN time series can be related to w (width parameter) of Gpeaks (2), superposition of which describes the shape of the S cycle represented by K data. From consideration of 23 observable shapes of S cycles it followed that the smallest number of Gpeaks, included in this superposition, is 3. It meant that the number of free parameters should be equal to 9. It was the price to pay to obtain the low Χ^{2}, i.e. the good (indistinguishable to the eye) description of the Scycle shape. It allowed to get the table 2 of all parameters for all 23 available cycles. SGmodel assumes that the solar cycle is a superposition of, at least, three onehumped symmetric curves (Gpeaks (2)). The page "Data" displays the average values of main parameters of SG  model (table 1). This provides a description of the regular shape of a particular Scycle, its regular forecast, and even the regular SSN time series. In using the term "regular", I mean such a state of solar activity, which is not distorted by stochastic/chaotic fluctuations of SGmodel's parameters (which, of course, take place in reality).
Table 2 of Scycle's parameters (#SGD)
A B C D E F G H I J K L cycle #
75 48.8 36 40.6 73.9 33 48.1 103.5 33 39.5 135 0.052 1
210 34.3 32 72.6 54.6 33 52.4 79.8 29.4 41.1 108 0.059 2
318 36.7 29.3 116.5 57.5 29.7 63.9 87.3 26.5 28.6 111 0.087 3
429 37.7 32 109.8 60.4 35.7 65 95.6/132.2 33.7/35.8 49/19.1 164 0.088 D* 4
593 40 31.8 25 70.3 36 37.7 92.8 31.2 18 151 0.037 * 5
744 48.5 33 14.4 73.5 33 36.4 100.6 33 14.2 149 0.075 * 6
893 47.6 33.9 39.9 75.7 31.5 45.2 95.7 27 39 126 0.015 7
1019 36.1 29.9 107.8 56.9 26.9 62.3 83 32.6 41 116 0.08 8
1135 44 37.5 82.5 69.5 38.8 59 109.7 33.4 37.6 149 0.05 9
1284 43 31.7 70.4 66.3 32.7 53.6 103.8 35 39.6 135 0.07 10
1419 39 32 103 62 30.6 59.3 92 33 26 141 0.06 * 11
1560 32.8 33.9 35.5 58.1 31.3 50.3 84.1 34.4 25.8 135 0.035 * 12
1695 35 34.8 63.4 60.4 35.7 46.7 103 35.9 19 143 0.052 13
1838 33.6 31.5 36.8 59.3 34.9 41.1 85.6 33.6 30.4 138 0.007 * 14
1976 41.3 34.4 60 63.2 33.9 52 92.9 32.5 14 120 0.033 15
2096 35.2 33 44.1 57.6 33 51.5 85.2 33 25 122 0.017 16
2217 43 32.1 79.7 65 31.7 66.3 97 31.7 37.7 125 0.056 17
2342 43 32.5 125.1 65 29.6 64.6 90 30.5 46.3 122 0.038 18
2464 41 33.5 167.6 64.2 31.8 84.3 95.5 32.5 33.1 126 0.084 19
2590 40 35.4 80.3 67 37 66.7 103 38.6 38.6 140 0.04 20
2730 39 32.4 120 59 28.5 72.5 83 34 61.6 123 0.029 21
2853 34 31.2 122.4 54 28.1 68.2 75 34.4 52.9 116 0.026 22
2969 45.5 39.6 90.9 72.9 35.5 64.3 111.5 38 20 151 0.046 23
col (A)  t 0, i (months from the beginning of Mdata series (January 1749)); col (B)  x P,i , col (C)  w P, i, col (D)  H P,i ; col (E)  x B,i , col (F)  w B, i, col (G)  H B,i ; col (H)  x D,i , col (I)  w D, i, col (J)  H D,i ; col (K)  T 0,i ; col (L)  Χ^{2}. All parameters (except H P,i ; H B,i ; H D,i ; and Χ^{2 }: they are dimensionless) measured in months.
1.4. SG  model scope. Not trying to convince a reader of any special benefits of the proposed approach and not hoping to protect myself from the inevitable criticism (which is not always constructive and often is directed by the human factor), I represent here some results.
First of all, it is H P,i (t) variation (see col (D) in table 2 and fig.2), which can be described (in accordance with the adopted approach) by the sequence of GLpeaks (2), the temporal distance between which changes periodically (from long one to short one and vice versa). Consideration of fig.2 allows to see in this sequence of a "Bactrian camel" of the Gleissberg cycle (GLcycle).
It is seen from fig.2 that Scycles in their positions on GL cycle (which is actually represented in fig.2 as one of the possible images of the Gleissberg cycle) can be ascending of the type U (including maximum), or descending of the type D, or saddle ones (wherein the saddle cycles occupy either the upper (#1, 10, 20) or the lower tier (#  4, 6, 25?).
It is obvious but very important that each of those types of Scycles is characterized both its peculiar shape, and duration ((T 0,i ), see the page "Data"). From the point of view of possible deformations of Scycle's shape, they can be "compact" (it concerns mostly the ascending cycles), "shifted" and / or "tailed" (this applies to both descending and saddle cycles, but not to all).
It is this last result allows transition of Scycle's forecasts from the method of "averages" ("avmethod", when only H P,i (t) is predicted while other parameters are taken average, see table 1) to the method of "analogs" ("anmethod", when only parameters of related similar cycles are compared). Experience has shown that it significantly improves distinctness of some forecasts, while for others the "avmethod" is more preferable .
The joint analysis of Kdata and Udata would complement the Scycle forecast #SGF:GL.L/S, represented only by Kdata (see fig. 3, 4), by the Scycle forecast #SGF:U, represented by Udata (see fig. 3). The Upeaks, approximately coinciding with the moments x P,i , x B,i , and x D,i , can be sometimes increased by 1.3 times as compared with other U peaks, the height of which is determined by the value of Kdata (K data represent the envelope for maximums of Upeaks). Using (1), #SGF:U may be compared with the current Mdata (a different approach is proposed in Karak et al., 2018).
Another result is the possibility of working out a mutual option of "GL and WO forecast". SGmodel shows that PMF reaches the maximum <PMF max > during the period between x D,i and t 0,i+1 . Averaging PMFdata (Wilcox Solar Observatory Polar Field Observations, Avgf  data) for that period, we receive <PMF max,i >, which can be connected with H m,i+1 .
1.5. A diverse phenomenon of solar activity is apparently a side effect of a plasma dynamical system, which is formed in the solar convection zone (and on its borders) and probably is in a state of weak chaos. SSN time series is one of the observable manifestations of solar activity. And although, to my knowledge, the chaotic origin of SSN time series is not yet proven, its behavior, at first glance, looks very irregular. However, on closer examination, one can find in SSN time series some regular temporal structures, which are described in the SGmodel by three types of Gpeaks (2),
namely, G GLpeaks with w ~ 28 years (they describe the regular behavior of Gleissberg cycles);
GKpeaks with w ~ 33 months (they describe the behavior of regular 11year solar cycles),
and GUpeaks with w ~ 30 weeks (they describe a regular component of the SSN fluctuations with the period of 1.3 years).
Since the MHD system of the solar convection zone that generates the solar cycle (and much more) is not only complex, but also difficult to observe, the SG  model relies not so much on theory as on the analysis of available and continuously incoming data (in fact it is Mdata and < PMF>  data). So, SGmodel is not a physical, but a phenomenological model, in which the elementary methods of mathematical statistics are used. There is no visible physics in it, but basic physical concepts about the origin of solar cycle lay down in its base. The issue here is not that I, being a physicist, want to do without physics, but the fact that the physics of the main elements of SGmodel is still insufficiently known. Indeed, there is no appropriate physical model of the Sun's Great Conveyor Belt (SGCB), which apparently determines the behavior of the Gleissberg cycles; there is no discussion about threecomponent structure of the 11year Scycle; there is no physical model of the 1.3year oscillations, which are clearly presented in the U  data.
The fact is that (according to some sources) the horizon of predictability for the solar cycle is only a few years, and so any attempt to build a longterm forecast of Scycle looks unprofessional. From the other side, to have such a forecast is very interesting (and in some cases, very important). Here the President Eisenhower's aphorism, applied to such a complex system as the human society, comes to mind: "plans are useless, planning is priceless". Replacing one notion by another close one, it would be to said with respect to the solar activity: "forecasts are useless, forecasting is priceless". Indeed, "wittingly doomed" attempts to forecast Scycles have led to detection of a number of connected features of solar activity, which ultimately lead to better understanding of the Scycle and properties of the SSN time series. Working with the SGmodel for more than 10 years, I was convinced in this from my own experience. The hope is that the SSN time series is only weakly chaotic, and therefore it is possible to consider SSN time series as a regular process with a slightly fluctuating parameters of the elements of solar activity (in the SGmodel it is Gpeaks) and relatively rare anomalous deviations from regularity (indeed, of 23 Scycles, only three (cycles # 12, 15 and 19) are really anomalous). Fig.5 illustrates the view of the recent solar activity and its forecast from the standpoint of the SGmodel on the background of solar research projects and the projects of manned missions in the hither heliosphere.
1.6. Results. So, what do we learn applying SG model to SSN time series?
I must say that in the study of solar activity, there are two aspects  practical and theoretical. They, of course, closely related, but there are differences. The practical aspect aimed to use the beneficial properties of the SA and to protect from its negative effects, and its main mean is the expanding of monitoring of solar activity, space weather and so on (Kontor and Somov, 1998). At the same time the theoretical aspect aimed on the solution of solar mysteries, one of which is the nature of the solar cycle. This puzzle exists more than a century and it was not resolved even during such a creative 20th century; therefore it moved into the 21st century, along with associated general physical problem of turbulence and mathematical problem of solving of the Navier  Stokes equations. Generally speaking, it is known that science adheres (until a certain time) some paradigms and science is subject to fashion. For the solar cycle a paradigm is "MHD  dynamo" (see Ossendrijver (2003) and Charbonneau (2010)) and a current fashion is "Flux Transport Dynamo" (FTD)  models (see Choudhuri (2012), Olemskoy (2014)), "in which the toroidal field (TMF) is produced by differential rotation in the tachocline, the poloidal field (PMF) is produced by the Babcock  Leighton mechanism at the solar surface and the meridional circulation (SCB) plays a crucial role". And this role is to arrange the solar activity (particularly for us  the structure of SSN time series) in a large scale (across the whole convective zone) and in the long term (about 10 Scycles, or one Gleissberg cycle). Working with the SGmodel, we are able to indirectly monitor changes of TMF in the tachocline (variations of sunspot numbers (SSN)), changes of PMF on the surface of the convective zone (using WSO observations) and, as we shall see below, changes of the mode of meridional circulation (or the Great Sun's Conveyor Belt (SCB or simply CB)).
Let's start with small scale and will move towards ever larger scale. We miss the shortest from the mentioned above five types of quasi  regular temporary structures, connected with the solar rotation, and immediately turn to the U data, which can be described by G U, i peaks ((2), see also table 1). These peaks have a width at the base T_{U }=2·w = ~ 17 months, but partly overlap and their closeness C = [ΔX/(2·w)] = ~ 0.64. Origin of G U, i peaks could be spatial (if you imagine that TMF in the tachocline splits into bundles and meridional circulation (CB) occurs across them (see Kontor and Khotilovskaya (1988) and Kontor (1991)), temporal (connected with transverse vibrations of the tachocline (see Howe et al. (2000)), or mixed.
The next quasiregular temporary structure is 11year Scycle, which consists of three G K peaks (see tables 1 and 2). Every G K peak is "responsible" for the process of TMF floating to the solar surface as a result of the CB impact on the tachocline, firstly in its highlatitude part (Pcomponent), then in its middle part (Bcomponent) and finally in its lowlatitude part (Dcomponent). It is possible that the properties of sunspot groups and the corresponding them active regions are different for those different parts of the tachocline. Here it should be noted the observed interaction U and K peaks, which manifested in the noticeable (especially for not very high Scycles) increase of the Upeak height when K peaks reach the times of their maximums (see figs 3, 4 and 5).
The next quasiregular temporary structure is secular Gleissberg cycle. Most impressively it looks like a variation of H P, i (t), GLcycle, see fig. 2. During GLcycle, 11year Scycle shows all possible amplitudes, lengths and shapes. By the way, there are few possibilities: by and large three. Another feature of the GL cycle is a temporary distance between the maximums of the neighboring GLpeaks (T CB ), which changes regularly from big ~ 63 years (between GLcycles) to relatively small ~ 41 years (inside GLcycle). It is tempting to identify T CB with the turnover time of meridional circulation (CB) and to explain Gleissberg cycle as a quasiregular, but multistep process of Sun's Conveyor Belt interaction with the tachocline, which results in the inter conversions of TMF and PMF and is accompanied by the solar activity.
During the evolution of GL cycles there are two types of abnormalities in Scycles. The first type of anomalies is the appearance of S cycles that differ significantly from the expectations of SG model (for example, #12, 15 and #19, see fig. 2). The second type of anomalies is a temporary disappearance of the normal Scycles, and with them, and GLcycle (it is well known Grand minima). This indicates violations of the regular regime of solar activity, but the observed anomalies are relatively rare. FTD models attempt to explain Grand minima by means of critical reduction of the CB velocity (Karak (2010)), which would lead to a substantial increase of T CB .
The last quasiregular temporary structure of SSN time series is the ~1000  year SAEs, which include adjacent periods of very low (Grand minima) and relatively high (Gleissberg cycles) solar activity (see page "Model").
1.7. Predictions. To develop SGmodel, it took a fairly large database, which, nevertheless, is too small to entrust the forecast of the Scycle to neural networks and, in general, to artificial intelligence (AI).
So, the first forecast on its base became available beginning with Scycle # 24. This was the period of the recession of the GL.3.2peak (see fig.2), which covers Scycles # 22, 23 and 24. Working with SG  model, I have made three predictions: (GLS, WOC and U) for Scycle #24, (GLS) for Scycle # 25, and (GLL) for Scycles # 26  33. This was being done sequentially step by step.
Time steps of SGF:
1996, May (2969)  Scycle #23 starts (t _{0,23}).
1998
 Ydata; description of the Scycle
shape by a log  normal function.
2000
 the start of SGmodel elaboration.
2004, March (3065)  the first estimate of
Scycle # 23 shape:
23SGF.1 => {2969 + 34.1  36.3  90 = 61.2  34.2  63 = 100.8  38.8  36}
The consequence of this was:
=> an estimate of T _{0,23} (=152 months) by the "analog
method" (#23 is Scycle of D_{H}type like the cycles # 4, 9, 13);
=>"the refinement" of X _{D,23} =152 – 38.8 =113.2 months instead of 100.8 months (3082, or September 2005);
=> extrapolation of GL.3.2 peak and addition of GL.4.1  peak (see fig.2):
GL.1cycle [17333177=178127118]= GL.2cycle [183820100=187534106]= GL.3cycle [194728.5112=198828128];
and the average GL.4cycle: 2053  28  107 (Ydata)
=> an estimate of t _{0,24} (=3120, or December 2008) and the first estimate of Scycle # 24 shape: 24SGF.1=> {3120 + 34.1  28.7  32 = 57.9  38.6  49 = 87.8  32.6  31}, [with H*m,24 = 70 at 3168 (2012/12];
=> an estimate of t _{0,25} (=3247, or July 2019) and the first estimate of Scycle # 25 shape: ,
25SGF:GLL => {3247+ 34.1  32.5  16 = 61.2  38.6  49= 91.1  32.6  7}.
These estimates could be made in 2004 only if we understood the SG  model then as it is today. But in fact they were made only by 2006, and only partially. This is due to the fact that the understanding of the SG model was improving gradually during almost the entire cycle 24. At the same year a forecast of H _{m,24} (= 74.1), based on the "Ohl's method" has been made (Kozhemyakina and Chernyshova, 2006). Since this "precursor method" has proven itself well, I adapted it later into the prediction procedure under the name WOC  forecast.
2009, May (3125)  t _{0,24} became known, there were made 24SGF:GLS (7.1) [with H**m,24 = 64.1 at 3180 (2013/12] and 24SGF:WOC [with H***m,24 = 73.3 at 3180 (2013/12] (see fig.3). After that, the observed (M data) and smoothed (U  data and K  data) began to arrive. The mGM  period ended about 3138  2010, May, and the active period of S  cycle # 24 has begun. At the growth phase of activity, the K  data's time course goes slightly higher than the forecast. 2011  Approximately in the middle of the growth phase (3155 or November 2011), a significant peak (H m 1 = 97) of M  data is observed (it also can be seen in U  data). It happened three month before X _{P, 24} . 2013  Then we observed t _{PR, 24} (3171 or March 2013) and the cycle's maximum (H m,24 = 73.6 at 3177 (2013/09], which is very close to 24SGF:WOC.
2014  Shortly after the maximum of the Kdata, we can see the second maximum of the Mdata (H m 2 = 103 at 3182, or February 2014), which is close to X _{B, 24} (=3184). Then the rather rapid phase of activity decline in cycle # 24 begins. 2016  Now (2017) Scycle # 24 is coming to the end, entering to the mGM  period at 3216 or December 2016. Comparison of available K  data with 24SGF:GLS, 24SGF:WOC and 24SGF:U shows that the increasing phase and the maximum of the cycle are close to 24SGF:WOC. As for the decline phase, we see a significant discrepancy between the course of the forecast and K  data. It consists in the fact that the forecast goes noticeably higher than Kdata. It means that the forecast relatively D  component was mistakable. It will be possible to see soon, on what else it will influence. However, simple expectation is not suitable for humans. Therefore, I have adjusted the current K  data by SG  model for moment 3227
2017 =>(November 2017).
24 SGD : K.3227 => {3120+ 38.9 33.7  42.2 = 63.9  32.8  51 = 84.9  30.8  16.6} (7.5)
Now the only riddle remains: when will the cycle # 24 end?
Today (4/11/2018) there is nothing special, but it is possible to see from fig. 3 that Kdata go exactly according to 24 SGD: 3227.
If for all the early S  cycles (including # 24), the "old Mdata" have been used (see page "Miscellanea"), beginning from Scycle # 25 the "new Mdata" are in use. It is shown in fig. 4. After the transition to the new M  data (May 2015), it is necessary to refine the GL  cycle, as shown in Fig. 2. Instead of the above description of the GL  cycle, we get:
GL*1cycle [42532478=980311118]= GL.2cycle [170132399=2157337110]= GL.3cycle [2953350112=3448348121];
and the average GL.4cycle: [4201  335  155 = 4697  335  155]; (M  data, new)
GL.ln.1 [4600.3774=9900.135114]= GL.2 [17000.08103=21700.075113]= GL.3 [29800.058114=34800.051112];
and the average GL.ln.4  cycle: [4200  0.04  153 = 4670  0.04  168] (Mdata, new.)
As a result we have H P, 25 = 38 (± 20) and three variants for 25SGF:GLS (see fig.4): 25SGF: max, 25SGF:av, 25SGF:min, namely, {3306  33  58/38/18 = 3330  33  66.2/60.6/55 = 3357  33  20}.
The time interval [ x D,24  t 0,25 ] is important in several points. It concerns the actual moments x D,24 and t 0,25 , the possible cycle # 24' tail and length, the course of mGM, <PMF max > , and at last the very possibility of forecasting for 25SGF:WOC and 25SGF:U. The question arises: what with all this abundance of opportunities to do? Figure 4 shows that 25SGF: GLS1 slightly exceeds cycle # 6 and is close in height to 25SGF:GLL. This is group 1. This is also adjacent to 25SGF: GLS2, it differs only in that it is short. So, this "loose" group 1 includes four options for low cycle # 25.
Group 2 includes two close variants of a high short anomalous cycle of type # 15.
In this situation, you can still resort to the "current" WOCforecast, i.e. to use data on the incomplete interval [X D  t 0, next]. For example, according to the data for June 2018 H m +, 25 = 108. This can be interpreted in such a way that the WOCforecast passes just between groups 1 and 2, making the situation for today as uncertain as possible. Therefore, we should patiently wait for the beginning of the cycle # 25.
As for forecasts for the later cycles # 26  33, these are #SGF:GL.av following the course GL.4 cycle (fig. 2). For U  type's S  cycles (#26, 27, 30, 31) the length is taken To, i =120 months, whereas for D  type's S  cycles (#28, 29, 32, 33) the length is taken To, i =143 months. So, these are the strictly regular S  cycles.
_{ Table 3: Status of SGforecast for four nearest Scycles is shown in parentheses}
# 
t _{0} 
X _{P} 
W_{P} 
H_{P} 
X _{B} 
W _{B} 
H _{B} 
X _{D} 
W _{D} 
H _{D} 
X _{D}+W _{D} 
T _{0} 
Data 

23 
2969 
45.5 
39.6 
90.9 
72.9 
35.5 
64.3 
111.5 
38 
20 
149.5 
151 
old 

24 
3120 
38.9 
33.7 
42.2 
63.9 
32.8 
51 
84.9 
30.8 
16.6 
115.7 
(142) 
old 
3227 
25 
(3262) 
(44) 
(33) 
(38) 
(68) 
(33) 
(60.6) 
(95) 
(33) 
(20) 
(128) 
(143) 
new 

26 
(3405) 
(40) 
(33) 
(91) 
(64) 
(33) 
(75.5) 
(93.5) 
(33) 
(48) 
(126.5) 
(120) 
new 

2. The final Conclusion (pessimistic)
201922: Comprehension of the model. I will try to start from the beginning (ab ovo). From ancient times, on the sun people visually observed spots, sporadically. They were like blots, soiling the perfect surface of the our luminary.
Scycle’s discovery. Somewhere in 1610 they began to look at the sun through telescopes. The spots became more visible, and in 1844 Schwabe noticed an 11year solar cycle, which I designate as Scycle.
W number. In 1848, Wolf introduced the number of sunspots W = k × (10×g + f). I.e. all clearly visible spots (f) and groups of spots (g), for which it cannot be said with certainty how many spots there are in them, are counted. Wolf decided that, on average, the number of unobservable spots in a group (due to their indistinguishability) = 10. Then, putting k = 1 (as is being done now) for a single spot, we get W(1) = 11 (although in fact there is only one); for two spots W(2) = 12 or 22; for three, W(3) = 13 or 23 or 33 (depending on their mutual position in the groups), etc. Thus, being a function of two variables, W reflects not only the number of sunspots, but also their mutual coordinates on the solar surface. If we fix f, then we will have W in the range [(10 + f)  11 f]; and if g  [11 g  ~ 20 g and even more]. What is their physical meaning? True, Waldmayer later showed that W is proportional to the total area of sunspots and, consequently, to the flow of strong TMF. This result is partly qualitative; in addition, the degree of homogeneity of the time series of solar activity (SA) thus obtained is unclear. In general, without hesitation, I took the W  series as the baseline data for SA, although I could take a closer look at the G (group number)  series ...
SA/SSN time series. To date, several time series of solar activity have been received, of which, apart from Wseries, F10.7  series (the index F10.7 was introduced in 1963), G  series (not inferior in length to W — series) and LCN  series (averaging over Scycle (10 years) for a period of about 11 thousand years) can be noted. As for Wseries itself, it is available in three versions: W(Y)  series (average annual W, available from 1700); W(M)series (average monthly W, available from January 1749) and W(D)  series (daily W, available from January 1, 1818). Working with these series, you can get the following classification of temporary structures of solar activity (quasi  regular temporary structures of SSN time series (QRTS i)) during the entire Holocene (in descending order of their characteristic duration):
QRTS i  QRTS 3, called SAE = GMperiod + GLvariation. Its characteristic duration is about 1000 years. It consists of a “Grand minimum of activity” (GMperiod) and several Gleissberg cycles following it (GLcycles), which form the GLvariation. Since both the GMperiod and the GLcycle are QRTS (2) (that is, they have a duration of about 100 years), we find that the “typical” SAE consists of one GMperiod and nine GLcycles. And although the LCN  series shows dozens of SAEs, we present in more detail the last current SAE, which began in 1645 as the Maunder minimum (1645  1715) and includes so far only three Gleissberg cycles. The next (third) structure of the SA is the 11year Scycle (QRTS (1)).
Fourier analysis. At this point, it’s time to point out the Wseries analysis method we use. Although it is not original, I apply to the “mysterious” W  series, Fourier transform known from the XIX century and get its amplitude Fourier spectrum. I divide the full range of the frequency spectrum {0  f n} into several subranges whose boundaries are the minima of the spectrum.
GL – cycle. The lowestfrequency subband of the W(Y)series spectrum includes periods which are more then ~ 17 years. It has several peaks, the most notable of which are peaks at 103 years and 56 years. I would like to think that the main peak (103 years) indicates the average duration of the Gleissberg cycle, and its first harmonic (~ 56 years) indicates that the GLcycle is twohumped (more precisely, that the cycle with a period ~ 56 years is visible within the Gleissberg cycle).
S – cycle’s details. The next (second) subrange covers the periods {17  4.17} years in the spectrum of the W(D)  series. In this subrange, three peaks are clearly visible: the main peak of the Scycle (~ 11.3 years), some side peak with a period of 8.6 years, and the first harmonic with a period of 5.6 years. At the same time, both the main peak and its harmonic forked. Instead of ignoring the shape of the main peak and these two small peaks (as is often the case), I assign a certain value to them: the first small peak indicates the duration of the Scycle active phase (~ 8.6 years), while the second indicates that inside the active phase of the Scycle, a cycle with a period of 5.6 years is observed (more precisely, the range of periods from 63 to 71 months), which is on average ~ 66 months. Moreover, the bifurcation of the main peak indicates, in my opinion, the existence of either short (~ 120 months) or long (~ 143 months) S  cycles.
16 – month U – peaks. The next (third), even higher frequency sub range of the Fourier spectrum (it covers periods {4.17  2.56} years) does not show noticeable peaks, whereas in the {2.56  0.83} year range you can see seven distinct, though not large peaks. After that, against the background of the general decline of the highfrequency part (fourth) of the Fourier spectrum, you can see only one clear peak corresponding to a period of 27 days (i.e, the synodic period of the sun rotation.
K and U data. Fourier analysis allows filtering, i.e. smoothing the observed data, and I took this opportunity to get, along with the observed W(M)series, two of its smoother versions, which were called Kdata and Udata. Based on the results of considering the form of the Fourier spectrum briefly described above, in K  data only periods longer than 4.17 years (frequencies less than 0.02 (1 / month)) were taken into account, whereas in U  data  periods greater than 0.83 years (frequencies lower than 0.1 (1 / month)). This meant that U  data are smoother than the observed M  data, while K  data are smoother than U  data. This corresponds to the smoothing of M  data by 5 and 25 points (we find that the widely used smoothing by 13 points gives a curve that is smoother than U  data, and less smooth than K  data). All this was done "by eye" and can not claim special severity.
SG – model. The result of the Fourier analysis of the SSN time series was the selection of K  and U  data as objects to describe the shape of the Scycle using the SGmodel. This turned out to be a difficult task for me, which took many years (practically the development of the SG model began in 2000) and required the repeated use of the trial and error method. The idea of the SG model is to describe the characteristic contours of the corresponding time series in the form of the superposition of several successive “solitons” represented by a normal Gaussian curve (G peak (2), see the page “Model”). Then the selected periods of the Fourier spectrum (T = 1 / f) are considered as the “lengths” of the base of the corresponding Gpeaks having a halfwidth w = T / 2. With this approach, it turned out that to describe the K  shape of the Scycle (the contour observed in K  data) requires a 3 G (K) peaks with w = 33 months (SG (3) model); to describe the Ushape of the Scycle (the shape observed in Udata) requires ~ 10 G (U) peaks with w = 8 months (SG n model); and to describe the shape of the Gleissberg twohumped cycle (GL  cycle), naturally, it takes 2 G (GL)  peak with w = 30 years (SG (2) model), see Table 1 [page “Data”, 7]. So, the matching of the SG model with the Fourier analysis data of the W  series made it possible to describe such known structures as the Gleissberg and Schwabe cycles, as well as the “1.3 year variations”, respectively G (GL) , G (K) , and G (U)  peaks. All this (as mentioned above, except for the SG model itself) is more or less known and therefore, before delving into the details of the SG model, it's time to see how the problem of solar activity is interpreted theoretically, more precisely, the problem of the solar cycle, which is actually source of the observed SА.
Physics of Scycle. Since Hale (1908), sunspots have been known to be associated with strong toroidal magnetic fields (TMF). Somewhat later (1919), Larmor suggested that the rotating Sun could generate them. Cowling (1933) then examined an axisymmetric dynamo and pointed out the difficulties associated with it; in addition, he introduced the ω effect, which is responsible for the generation of a strong TMF from a weak dipole magnetic field (PMF) by means of the differential rotation of the Sun. In the 40s, Alfven developed magnetic hydrodynamics, for which he received the Nobel Prize. In the mid50s, the star of Parker rose, because of his having built the first closed model of the solar cycle and the theory of the solar wind, which the Parker Solar Probe, launched in August 2018, is called to test today. In the 60s, Babcock and then Leighton developed the solar cycle model, which has come to the yard today and is actively studied by modern means (it was named the transport BLmodel). According to this model, largescale PMF is formed, roughly speaking, in the photosphere, while smallscale TMF is somewhere near the bottom of the convection zone. Also in the 1960s and later, the concept of magnetic reconnection began to be developed (this concept first explained solar flares, then the coronal mass ejection (CME) mechanisms, and now it only remains to check whether micro flares heat the corona), methods of helioseismology and applications of chaos theory. Currently, the following phenomena are considered in the solar cycle mechanism. It is believed that the mutual generation of TMF and PMF is at the heart of the solar cycle mechanism. The toroidal field is amplified from the poloidal one due to the ωeffect at the lower boundary of the solar convective zone (CZ), called tachocline (introduced in 1992). Then they are assembled into small magnetic tubes, which float up due to MHD instabilities, change the helicity due to the αeffect and are observed on the photosphere as sunspots. Subsurface smallscale convection, largescale meridional advection and turbulent magnetic diffusion in accordance with the Leighton mechanism gradually destroy sunspots, stretch their toroidal magnetic fields and lead to the formation of largescale polar PMF, which leads firstly to polarity reversal (PR, which occurs during the period of maximum activity of S  cycle), and then to the formation of the maximum PMF of the opposite sign (which is observed at the time of the beginning of the new S  cycle in the period of minimum activity between adjacent S  cycles (mGM  period)). In order for the 11year cyclicity of solar activity not to be interrupted, it is required to deliver a new PMF to the bottom of CZ (to the tachocline region), where a new TMF will begin to form from it.
It is believed that global meridional circulation / advection (Global Solar Conveyor Belt (GSCB, introduced in 2006), which makes one revolution in about 50 years), turbulent magnetic diffusion, diamagnetic pumping and, perhaps, something else are involved in this delivery (see Karak et al., 2018). As for the tachocline, transverse oscillations with a period of about 1.3 years (Howe et al., 2000), different modes of Equatorial Magnetohydrodynamic Shallow Water Waves (Zaqarashvili, 2018) and probably some other types of oscillations reflecting in the Wseries dynamics can be excited in it. In any case, T. Zagarashvili (2018), apparently in a fit of enthusiasm, believes that "TMF influences on the dynamics of shallow water waves in the solar tachocline. A subadiabatic temperature gradient in the upper overshoot layer of the tachocline causes significant reduction of surface gravity speed, which leads to trapping of the waves near the equator and to an increase of the Rossby wave period up to the timescale of solar cycles». He finds “that the toroidal magnetic field splits equatorial Rossby and Rossby gravity waves into fast and slow modes. For a reasonable value of reduced gravity, global equatorial fast magneto Rossby waves (with the spatial scale of equatorial extent) have a periodicity of 11 years, matching the timescale of activity cycles. The solutions are confined around the equator between latitudes ±20°–40°, coinciding with sunspot activity belts. Equatorial slow magnetoRossby waves have a periodicity of 90–100 yr, resembling the observed longterm modulation of cycle strength, i.e., the Gleissberg cycle. Equatorial magnetoKelvin and slow magneto Rossbygravity waves have the periodicity of 1–2 years and may correspond to observed annual and quasibiennial oscillations. Equatorial fast magnetoRossbygravity and magnetoinertiagravity waves have periods of hundreds of days and might be responsible for observed Riegertype (154 days) periodicity. Consequently, the equatorial MHD shallow water waves in the upper overshoot tachocline may capture all timescales of observed variations in solar activity, but detailed analytical and numerical studies are necessary to make a firm conclusion toward the connection of the waves to the solar dynamo». If all this were true, it would mean that the theory does not contradict the Gpeaks appearing in the SGmodel, and it would remain to find out how much and why they are exactly such as they are. If any of the elements (TMF generation in the tachocline area and its “correct” floating up to the photosphere, the formation of a new largescale PMF near the solar surface and its delivery to the tachocline) of this complex MHD system will be seriously affected by any instabilities, then the mysterious GM  period begins, when CZ will take about a century to restore a more or less regular (including an 11year cycle and its modulation in the form of the Gleissberg cycle) cyclical nature of solar activity. So, in order to understand all this, and, moreover, to learn how to make forecasts of "the solar weather", there is a lot of work.
SGmodel's details.
 K  data series represents a continuous sequence of 24 Scycles, each of which (in the first approximation) is characterized by the sequence number i (# 1, 2, ..., 24); duration (T o, i) and maximum height (H m, i). The shortest is Scycle # 2, the longest is Scycle # 4, the highest is Scycle # 19, and the most (critically) low is Scycle # 6.
 As for the observed shape of the Scycle, it is significantly different at the beginning of the cycle, during its “middle” and at the end of the cycle. In essence, this means that the Scycle spends most of the time (~ 8.5 years) in the active phase (when the “level” of solar activity exceeds W = 20), whereas at the beginning (when the cycle is only “flaring up”) and at the end of the cycle (when the cycle is already "fading out") activity is low and is not described by the SG model. I call this period between cycles miniGM or just mGM.
 Now the main question arises: how to describe the active (middle) phase of the S  cycle by function (having the minimum number of free parameters), which adequately describes the observed shape of the Scycle. I know, for example, four such functions that were proposed in the 90s of the last century, which have no more than four free parameters and are shown in Fig.1 from my archive. Which one to use is obviously a matter of taste; I, for example, quite successfully used the lognormal function (LN (t)), when described the shapes of cycles represented by W(Y)  series. Further, I noticed such features of the K  forms of some cycles, which suggested that these cycles consist of several domeshaped components. However, considering the shape of the cycles represented by K  data, I had a desire to describe them not only acceptable, but good, i.e. so that the observed and modeled shapes of the cycle were almost indistinguishable by eye and using at the same time simple symmetric functions. As experience has shown, it is enough for this that at fitting χ2 would be <0.1. Given the presence in the Fourier spectrum of the peak with a period of 66 months, I came to the idea of describing the shape of the Scycle by a superposition of several normal Gauss functions with a halfwidth w = 33 months (SG  model means "superposition of the Gaussians"). But how many exactly? Some short cycles can, in principle, be described by two Gaussians; the longest cycle # 4 (Fig. NNK  68) is described completely by four Gaussians (components); all other cycles  by three (P , B , D  components). Thus, we choose three for the sake of generality and obtain the SG (3)  model. By the way, the presence of several components makes it possible to describe any asymmetry of the cycle shape, and besides some long cycles have “shapeless” tails, which I tend to regard as a degenerative fourth component.
 Fitting algorithm used (see page "Data", 4).
The results of the description of the shape of S  cycles by SG (3)  model. As it can be understood from what has already been said, S  cycle in SG (3)  model has two types of shape: the Kshape and the Ushape (the observed Mdata can be called Mshape). Here are the results of analysis of Kshape , which is characterized by such general parameters as the beginning of the cycle (t o, i), the moment of reaching the maximum (t m, i), the maximum itself (H m, i) and the duration of the cycle (T o, i). Consideration of cycles # 1, 2, ..., 23 showed that the Kshape of each cycle is determined by its structure, i.e. the ratio of the parameters of each of its three components (each component has in turn three parameters). The resulting structure for each cycle is shown in Table. 2. From this table it can be seen that the halfwidth parameter for all components is approximately the same and is really close to w = 33 months. The parameters of the position of the components (x p, x b, x d) determine the “model length” of the cycle (Tm = x d + w), which is noticeably smaller than To, i for cycles with a tail (cycles # 5, 6, 11, 12, 14). Wherein, all the cycles can be divided into two equal groups: short (~ 120 months, with a length of less than the average (132 months), specifically, less than 127 months) and long (~ 143 months, with a length greater than the average, specifically, more than 134 months). Another group of parameters is the height of the components (Нp, Нb, Нd). The “main” of them is Нp, which in cycle # 6 is equal to 14.4, and in cycle # 19  167.6; for the remaining cycles Н p, i (for now) is within this range. The heights of the P and Bcomponents correlate well (Нb = 35 + 0.28 × Нp), whence we get three consequences: (1) the height range of the Bcomponent is already {35  82}; (2) the cycle is called high if Нp > Нb, otherwise  low; (3) the condition Нb < 35, apparently, corresponds to the collapse of the Scycle and the transition of solar activity to the state of a “grand minimum of activity” (GM  period). As for the Dcomponent, in a sense, it behaves “by itself”. From SG (3)  model it follows that if you plot the graph Н p, i as a function of calendar time, then this graph can be described by a twohumped Gleissberg cycle curve (GLcycle) corresponding to the peaks of the Fourier spectrum of the W (Y)  series, as mentioned above, see fig. 2. Considering the curve of three consecutive GLcycles, shown in Fig. 2, it can be noted that the peculiarities of the shape of S  cycles depend on their “position” on this curve. Thus, short cycles (both low and high) “lie” on the ascending branches of GL peaks (# 2.3; 7, 8; 16, 17, 18; 21, 22); exceptions are cycles # 11 (lies on the ascending branch, but has a tail and therefore is long), # 15 (according to its position it must be long and low, but it is short and high), # 19 (according to its position it must be long, but really it is short and "too" high). In turn, long cycles (both low and high) “lie” on the descending branches of GL peaks (# 4, 5; 9, 13, 14; 23) and in the saddles between the GL peaks (# 1, 6, 10, 20 ); exception is the cycle # 12, which is too low. Thus, of the 23 cycles, three (# 12, 15, and 19) are anomalous, because they “do not sit on” the GLcycle, and the cycle # 11 is atypical (it should be short but has a tail, and so, it is long).
Now we need to philosophize a little. When approaching the analysis of the shape of the Scycle, we first used not observations (Mdata), but strongly smoothed Kdata. To imagine at least the Kshape of the Scycle, we had to introduce the SG(3)model. The application of this model to Kdata made it possible to quantitatively present the structure of Kshape for each of the cycles analyzed. It turned out that Kshape (in contrast to its simple structure, which is just a superposition of three "normal solitons") of the S  cycle is quite diverse. Since SG(3) itself is a regular model (that is, it does not contain random variables), the resulting diversity must be due mainly to physical reasons. One of the consequences of the use of the SG(3) model was the discovery of the “century course” of the heights of the P  components in successive S  cycles, which could be described by the SG(2)  model of the Gleissberg cycle. This result, in turn, allows you to deal with the variety of Kshapes, if we assume that it is determined by a different arrangement of Scycles on the GLcycle curve (see Fig. 2), which reflects the dynamics of physical processes on the Sun that generate solar cycles .
In light of the above, let's take another look at Fig. 2. Roughly speaking, it begins with the end of the Maunder minimum, the most famous GM  period (1645  1715). During the transition from the mode of “depressed” SA to the mode of developed SA, an Scycle # (4) was observed, a critically low cycle, apparently similar to Scycle # 6, with Hb = 36.4 (see the corollary (3) of the fact that Hp & Hb correlates). After that, the first GL1.1peak (S  cycles # (3), (2), (1), 0, 1) of the first GL1cycle (S  cycles # (3), ( 2), (1), 0, 1, 2, 3, 4, 5, 6) began to develop . It was the lowest and there is little to say about it, because at that time there was only W(Y)  data. The GL1.2 peak was already “normal” (compared to the others) and had a length of 2 × w = 54 years. On its ascending branch there are two shortest (high) S cycles (# 2 and 3; their total length is 219 months), and on the downward one  there are long cycles # 4 (high), 5 and 6 (low) with a total length 464 months. I assume that the ascending branch of the GL  peak corresponds to the period of relatively high speed GSCB, and the descending  its relatively low speed (twice as less). Moreover, with a high circulation rate, it is probably possible to achieve a higher initial strength of the generated magnetic field, which also leads to an increase in the outgoing magnetic flow ( Wnumber will be greater). So, if the speed of the meridional circulation is high, this leads to ever higher short Scycles. After a sharp decrease in the velocity of the meridional circulation, which we actually observe as a vertex of the GL peak, occurs (as a result of some instability), the length of Scycles increases and they height gradually decreases.
But that is not all. The GL cycle begins to develop from the moment the GSCB speed increases. After two or three short Scycles (after 2030 years), a certain “growth limiter” is turned on and Scycles become long (the first GL peak is formed). However, after a couple of Scycles, the speed again briefly (another pair of Scycles) increases and only after the “growth stop” is reenabled, the second GL peak is formed and the GLcycle ends. In this case, the temporal distance between GL peaks inside one GL cycle is approximately 42 years, while the distance between neighboring GL peaks of different GL cycles is approximately 65 years. It remains unclear why one inclusion of a “growth limiter” is missing, and the GLcycle consists of 10, rather than 5 Scycles. But and that's not all. It turns out that the GL  cycle is unstable inside itself. This is manifested in the appearance of anomalous Scycles. We have already said that at the end of each GLcycle a critically low Scycle can appear, which serves as a precursor of the GMperiod (see Scycle # 6). However, the next candidate for this role (see Scycle # 15) turned out to be short and too high, which according to our concept is a sign of the premature switching on of fast meridional circulation. Now we have to wrestle with how Scycle # 25 will be. Another abnormal cycle is Scycle # 12, which, when slowing down the meridional circulation (it is long), prematurely became too low. As for the Scycle # 19, instead of lengthening and decreasing, it continued to grow (becoming the highest today) and did not think of lengthening. True, after his “crackdown” everything was back to “normal”. In connection with all this, I have to admit that the complex solar MHD system (described in the theory by the viscous Navier – Stokes equations), producing solar activity (SA), has, in addition to regularity and variability, also weak chaos (with strong chaos almost everything would be irregular and unpredictable).
If I allowed myself a little higher to philosophize, now is the time of mysticism. I mean the problem of predicting the solar cycle. Once Freeman Dyson expressed in the spirit that “if something is predictable then it is not science”. This can be understood in such a way that the forecast is the result, not the subject of science. And since the solar cycle has not been studied enough, it is premature to predict it. But reasonable arguments rarely stop enthusiasts, and attempts to predict the Scycle have already been made for the last 10 cycles, starting with # 15, when there could not be any adequate theory of the cycle. Moreover, at present, a successful forecast is considered as a way to draw attention to the elusive features in the behavior of the solar cycle. It must be said that the first successful forecast of the height of the cycle was made by A.I. Ohl for the cycle # 20 just during developing of the BL  model. It has been recognized today as the most reliable "precursor method". My time came in 2006, when I made one of the successful predictions of the # 24 cycle. Despite this, during this whole (still current) cycle, I tried to bring my GL  method to the mind, showing this process on my website. The result was a combined GL / WOC method, which I use to predict the shape of the oncoming cycle # 25 and the next few cycles. The method consists of four stages: (1) the initial forecast of K  shape of the cycle (GLL step); (2) the final forecast of Kshape of the cycle (GLS step); it is followed by (3) the "specifying" forecast of the height of the cycle (WOC step); immediately after this comes (4) the forecast of the U  shape of the cycle. Step 1 can be made several cycles in advance (its lead time depends on the position of the moment of the forecast on the current GL  cycle), see the forecast for cycles # 26  33, shown in Fig.2. Step 2 is done during the decline of the current cycle at the moment when the parameters of its P component become known (see Fig.4 for # 25) and the last (before the forecast) adjustment of the current GL  peak appears (see Fig. 2). Steps 3 and 4 are done no later than the date when the start of the new cycle becomes known. Immediately after this, a new stage begins: a comparison of the forecast with observations (which is doing now relatively S  cycle # 24, see Fig.3)...
When I started browsing the site, I suddenly realized that the forecast was very approximate in principle, and in connection with this I wondered: why should I even make a SA forecast, whom and why could it be useful? Let's leave aside the scientific aspect of forecasting. It remains the fact that the SGmodel allows to fully describe the approximate shape of the upcoming Scycle. It will have growth, high and low phases; Uforecast what is a variation that approximates M  data, and includes expected cycle peaks (see Karak et al., 2018); finally, the forecast of the progress of the new PMF and the end date of the cycle. But, as Figure 3 shows, a coincidence, if it does, will be only qualitative: the phases, and even more so the peaks, will not coincide in time; PMF  forecast and the end of the cycle will also not be accurate ... It will be like a fuzzy forecast, which is blurred by various random factors (± Ϭi) and therefore cannot be used specifically. But on the basis of the solar cycle, you can build, as I showed it at one time, forecasts of various aspects of the “solar and space weather”: the development of active regions (ARs), the intensity of flare activity and Xrays, the appearance of coronal emissions (CMЕs) and their influence on the Earth magnetosphere, variations of galactic cosmic rays (GCR) intensity, etc.
But there is another unpleasant side of this case: we cannot satisfactorily predict abnormal cycles (neither # 12, nor # 15, nor # 19) and periods of GM. Why, then, to engage in such predictions and what should actually be done? There is an answer: the SG  forecast described here is an orienting result showing what we should (and approximately when) expect in the future. The same are the forecasts of solar and space weather, built on the basis of SG  forecast. They will satisfy some practical needs that do not require urgent action. In the same situations, when an immediate response is required, the only reliable means is to continuously monitor the situation. Fig. 5 shows some of the probes, leading continuous observations of the sun in real time. Alas, there are no “Helio stationary satellites” (HSS) among them, which we wrote about, 20 years ago (Kontor and Somov, 1998).