* 1. INTRODUCTION*

This text is organized as a sequence of relatively independent theses and several figures.

* Solar activity (SA) is a consequence of the dynamics of the solar magnetic field (MF) in the solar convection zone (CZ) , which is characterized by:*

*= slow* (of the order of a year or more) generation of the strong toroidal field (TMF) at the bottom of CZ near the tachocline;

*= rapid *(of the order of a month) the floating up of magnetic flux tubes and decay of sunspots (SS) in the photosphere;

*= the slow *formation of the poloidal field (PMF) at the poles of the Sun;

* = *reconnection of magnetic tubes in the corona (solar flares) - about a day for coronal ejections and about an hour or less for particle acceleration (now it remains to check whether "micro flares" really heat the corona);

*= *the fact that all this is synchronized by a slow meridional circulation so that the 11-year (S-cycle) and the century Gleissberg cycle (GL-cycle) of SA are observed. When that synchronization is seriously disrupted, a Grand Minimum (GM-period) comes; when recovered - again the above mentioned SA cyclicity is observed.

2. *A**MODEL*

Described here phenomenological *SG - model of SSN time series*** **was first proposed in 2006 (see Pesnell, 2008; Kontor (2006, 2008 a, 2008 b)) as a consequence of careful consideration of the Fourier spectrum of the available series of the sunspot numbers and since then it is developing in real time.

*2.1. Overview .* Nowadays it is well known that the SA is both diverse and variable in time. We have an idea of it for the entire geological Holocene epoch (more than 10 thousand years), see

*Solanki et al. (2005)*. During all this time, the SA exists in one of two modes: passive or active. The passive mode (called GM-periods) is characterized by the fact that the 11-year S-cycle, discovered by H. Schwabe in 1844, is practically not visible in it, although it still exists, while in the active mode it is clearly visible. Considering the long-term variations of the SA, it is convenient to represent its time series as an SAE (Solar Activity Episode) sequence consisting of GM-period and the group of active S-cycles following it. In turn, each such group can be decomposed into a sequence of secular GL-cycles (it was opened in 1939), thus containing 8-10 S-cycles.

** **SAE = GM-period + GL-variation

We will not further analyze either the SAEs or the GM-periods, noting only that the current SAE began in 1645 from the Maunder minimum, which lasted until 1715 and was followed by a group of 28 active S-cycles (starting with # -3 and ending with the current # 24). But the active S-cycles, described by M-data (and these are S-cycles beginning with # 1), will be given the closest attention (M-data are the averaged monthly sunspot numbers (SSN) that were introduced by R.Wolf in 1848). This means that such characteristics of S-cycles as *height, duration* and *shape* will be considered.

2.2. Three "kinds" of SSN.** **When we consider M-data, it is found that they fluctuate noticeably, so that you can see several maxima, and the cycle time is determined by an expert decision. On the shape of the cycle, one can speak only in the most general kind. Therefore, using the Fourier spectrum of the time series SSN, two types of smoothed (filtered) data were additionally introduced: slightly smoothed U-data (f <0.1 (1 / month)) and strongly smoothed K-data (f <0.02 (1 / month) ). It can be seen that U-data is an acceptable approximation to M-data (see (1)), whereas K-data are convenient both for describing the shape of S-cycle and for its prediction.

M-data (t) ~ U-data (t) ±σ (= 11.83) (1) where Miscellanea" (paragraph A 1) and fig.9).
It was for the analysis of K-data that the SG - model of the S-cycle was developed. This is the model of superposition of Gaussians, i.e. normal Gaussian functions (2). It is used to fit K-data and thus extract information about the K- shape of each S-cycle. It turns out that all cycles (except # 4) can be well described (with chi-square < 0.1) by the superposition of three Gaussians, whose width is 2×w = 5.5 years (the period of the first harmonic of the 11-year peak in the Fourier spectrum of the SSN series). Because of the large number of free parameters (nine), unambiguous adjustment requires additional constraints and relationships between the parameters. |

*3. Appendix.*

*3.1. SG *n - *model. *G U, i - peaks have the close parameters **W*** *and distances between peaks * Δ***x** _{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

*G*

*are represented by two types of peaks, to wit by*

**U, i**- peaks*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 (

**); 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**

*S-cycles**one of the five types*of quasi - regular temporary structures (

*QRTS*), which can be distinguished in SSN time series. A typical

*G*U

*- peak (QRT*S 0

*) has the length of order of 1 year (*more specific TU = 2×w, see table 1 on the page "

*Data*").

3.*2.* *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 "base smooth K - shape" of any 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 (*

*QRT*S 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 3-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***. Then K-data can be represented as the sum of**

*-data**K 0*

*-data*and

*K * -data*. These last data > 20 and are pretty well described by SG 3- model; let us call this description #

*SGD:K*, see (3) and fig.1:

*(K _{i )}^{*}-
data ~ # SGD:K (t _{0, i}*

*; t ) = P*

_{i}(t) + B_{i}(t) + D_{i}(t)*(3)*

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 (3) that any S-cycle is a superposition of three components of kind (2) -** **with the only exception in the face of # 4, which is described by four such components. In (3)** i** denotes the same as #, namely, a serial number of the considered S- cycle.

*3.3.* All *G- peaks* ( SG- model has three types of them (*G*** U **-,

*G***-, and**

*K*

*G***- peaks)) are different from each other only by the values of their three parameters:**

*GL***(the position on the time axis),**

*Xc***(width at half maximum, usually measured in months or years), and**

*w*

**A**

*= H***·**

*w***· (**√

**(π/2))**, where

**is the height of**

*H***.**

*G - peak*Consequently, the S-cycle's shape for each S-cycle is determined by nine parameters: *X P, X B, X D* (parameters of the time position, which are counted off **t ****0, i** ); *W P, W** B, W **D* (the shape parameters, which are approximately equal and so designated by** W**), and at last

*H*

*P, H B, H D***( amplitudes of P-, B-, and D - components of S-cycle). In accordance with SG 3 -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***and to get after that the complete description (**

*P, i**) of S-cycle K - shape (3) and the forecasts (*

**#SGD:**K*) for the next S-cycles (here they are #24, #25 and so on up to # 33) in the first approximation. So,*

**#SGF:**GL

**H****P, i**- the height of Si- cycle's P component - plays a special role in SG 3- model. 3.*4. SG 2 - model of GL-cycle.* The long-term (~ 100 years) variation of Hp, i (t) is associated with the known Gleissberg cycle, which is considered in SG 2-model as a description of that variation. Then the Gleissberg cycle (GL- cycle) turns into superposition of two *G *** GL**- peaks (4):

*#GL-cycle* = *G **GL * (*t, **X C** , **w ** _{1}, A _{1 })* +

*G*

*GL*(

*t,*

*X C+*

*Δ*

*x*

*,*

*w*

*(4)*

_{2 }, A_{2 })*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 **(*QRT*S 2 *)**. *

The whole *Hp, i (t) - variation* (** GL-variation**) itself can be described by the expression (5), see also fig.2:

*GL-variation = **∑ **G **GL * (*t, (**X C, i+1 =** **X C, i +**Δ**x *_{i })*,** w *_{i }, A _{i }) (5)

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 *Δ***x *** _{i }* ~ 41 years, whereas

*Δ*

**x***~ 63.3 years (even-numbered, giving the distance between the GL-cycles). We find that in norm the temporal distance between the adjacent*

_{i+1 }

*G***- peaks within GL-cycle is less than the "length" of**

*GL*

*G***- peak (~ 55.2 years), whereas the distance between the adjacent**

*GL**approximately equal to the "length" of*

**GL-cycles**

*G***- peak.**

*GL** 3.5.* 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. Consideration of LCN-data from the point of SG-model's view shows, that

*three types (modes) of activity are realizing on the Sun:*

**depressed**(

*Grand minimum (*

*GM i*

**)**and**mGM**periods)**,***when only the rare single sunspot groups (G U 1- peaks ) are observed for the some periods (it takes approximately 10 % of time);*

**non - typical:**the low S-cycles, in which the relatively stable

**B-component**(3) dominates ( H P < H B > H D ) and the role of GL-modulation of the P-component amplitude (

**H**

**P, i**) is inessential (it takes approximately 30 % of time inside Gleissberg cycles); and, at last,

**typical**: the high (or regular) S-cycles, for which an amplitude is determined by GL-modulation (see for details [Kontor, 2006, 2008 a, 2008 b]). It takes approximately 60 % of time inside Gleissberg cycles).

3.* 6. *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 (

*QRT*S 3

*)*could be considered (the shortest from the mentioned above

*five types*of quasi - regular temporary structures is the structure, connected with the

**n (**

*solar rotatio**QRT*S - 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

last published: 3/01/2019