On this and the following pages, the background of SG-model, data used and the procedures of their analysis, the results, forecasts for S-cycles # 24 and 25, and their relevance to the modern scenario of the solar cycle are explained.
Here is the list of data which are the sources for SG - model:
1. M - data (January 1749 - May 2015) include S - cycles # 1 - # 23 (WDC - SILSO, Royal Observatory of Belgium, Brussels).
2. U - data are (slightly) filtered M - data (include the periods > 10 months).
3. K - data are (heavy) filtered M - data (include the periods > 50 months). By the way, K-data are more smoothed than conventional 13-month boxcar averages (M 13-data).
4. Yearly averaged Y - data ( 1700 - 2015) include S - cycles # (- 4) - 23, i.e. 28 S - cycles.
5. Averaged LCN - data (9455 B.C. - 1995 A.D.) include about 30 SAEs (see Solanki et al., 2005).
6. E - data are the filtered Y - data (include the periods > 18 years).
7. Daily D - data ( 1818-1-1 - 2015) are used for Fourier analysis of the high frequency features of SSN time series.
The results of analysis of each of these time series have contributed to the solution of our problem, namely, to the quantitative description of a structure of the time series of solar activity, which is represented by the SSN time series. Our SG-model of the SSN time series is developed on the base of two assumptions that seem to me (NNK) reasonable and promising: (I) smoothed (more specific - filtered) SSN-data can be described by some superposition of Gaussian functions G(t) (2) and, in particular, (II) S-cycle (represented by K-data) can be described by the superposition of three G(t) with true to type width 2 w (for each), which is equal to the period of the first harmonic (5.5 years) of the main peak (11 years) in the Fourier spectrum of SSN time series. It follows that w in (4) should be equal in average 33 months. And everything else - as it will.
So, it is known that the main structural element of the SSN time series is the 11-year solar cycle (S - cycle). For the analysis of its form we will use K - data. Usually when it comes to a particular S - cycle, we refer to its number and settings such as the start date (t 0, i), date of maximum (t m ) and the value of this maximum (H m), and the date of the end of the cycle (this date is, at the same time, the start date (t 0, i+1) for the next cycle). SG-model allows to describe the shape of the cycle significantly more detail. Immediately it should be said that the numerical value of any parameter of S- cycle is known with some (sometimes long) delay (for example, the start of cycle #24, which took place in December 2008, became known in May 2009). Therefore, any substantive discussion on the current cycle implies some kind of forecast. In order to lay the basis of the forecast of the solar cycle, SG-model has been introduced (see page "Model") and on its basis all of the above series, including each of the 23 cycles presented in M -, U -, and K- data, were analyzed. Fig.1 displays a description of the longest S-cycle # 4.
S - cycle consists of five distinctive parts and includes 10 important reference moments. It starts at t 0, i ("the moment" 1) inside the mGM - period of low activity (the first part), which separates the neighboring S - cycles. It is possible the simple and convenient description of the mGM - period's length as the time interval, when K - data < 20. In that case the average length of mGM - period (and consequently K 0, i - data) = 35.4 ± 17.9 (months), i.e. in average it is about three years. But the observed range of its changes is [20 - 89 (months)]. After the end of mGM - period ("the moment" 2) an active part of S - cycle begins. It has duration about 8.15 years, which corresponds to a "sub-peak" of the main 11- year peak in Fourier spectrum of SSN time series. An active part of S - cycle consists, in turn, of three periods - an activity growth (the second part) , when P - component is emerging (it may include the moment x P (3), whereas the moment (4) means the start of the cycle maximum (the third part), when B - component is evolving (it includes the moments t PR (5), t m (6), x B (7) and (8), which means the start of the cycle's decay (the fourth part) - the time of D - component (it includes x D (9) and (10), which means the start of new mGM). During transition to the next mGM - period sometimes small humps at the end of cycle (the single knolls) can observe (the fifth part). The cycles with a knoll have usually a long duration. Thus, the expression (4) may differ markedly from K 0, i - data during the mGM-periods, while they are practically the same (K* i - data) in the active phase of the S-cycle (5). Therefore, to distinguish between K-data and their description using (4), let us denote (4) as #SGD: K. So, any S-cycle in SG-model can be presented by M-data (observed), U-data (filtered), K-data (filtered), #SGD: K and #SGD: U (both fitted); wherein U-data and #SGD: U (3) are practically identical. All the above applies to any S-cycle, including S-cycle # 4.
That cycle started in September 1784 ( t 0, 4 = 12 in fig.1) inside the current mGM - period. All data are shown after that moment up to May 1798 ( t 0, 5 = 176). "The moment 2" takes place at point 21. After that the active part of S - cycle # 4 begins. It is seen that an activity growth is determined mainly by P - component (x P = 50.7 what is close to t m = 56 ; at that time the moment of t PR was unknown). The moments of x B = 81 and x D = 113 correspond to the cycle's decay period. It is interesting to note that the moments x P , x B and x D approximately correspond to "the splashes" of activity, seen in M- and U- data. It explains the double peak shape observed in some S-cycles (#1, 9, 10, 11, 15, 19, 22, 23). It is S-cycle # 4 in which the most massive hump at the end of cycle (the single knoll) can be seen. It is seen very good in graph of residue = K-data - 4SGD: K, see (4, 5) and fig.1.
Due to many free parameters (9, remember J. von Neumann?), the fitting procedure is ambiguous: any S-cycle can be described equally well ( i.e. practically with the same chi-square (Χ^{2})) by several different sets of parameters. To remove that ambiguity, certain constraints were imposed on the fitting procedure itself. Here they are.
1. All w i should be close to 33 months (in average w = 33.3 (±2.5), w P = 33.2 (±2.7) , w B = 32.6 (±2.1) , w D = 34 (±2.5)). 2. Model's S-cycle length TM = x D + w D should be close to the observed length of S-cycle T 0 (Δ_{2 }= T 0 - TM = - 0.11 (±3.4) m, i.e. really close, but excluding cycles # 4, 5, 11, 12, and 14. For them Δ_{2 }= 21 (±4.8) m). 3. For pretty high, positively asymmetric S-cycles, P-component is the main component, i.e. the highest one (S-cycles # 2, 3, 4, 8, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23). 4. For pretty low, symmetric or negatively asymmetric S-cycles, B-component is the main component, i.e. the highest one (S- cycles # 1, 5, 6 (symmetric), 7, 12 (symmetric), 14, 15 and 16 (both symmetric)). 5. P- and B-components generate the cycle amplitude, whereas D-component is responsible for the cycle's length. 6. Fitting procedure concerns only the active part of S - cycle; thus the start and end portions of K-data (substantially belonging to mGM periods) are excluded from fitting (by masking, see a- and z- "components" in fig.1). The first step deals with fitting of P- and B- components to the growing and maximum phases of the S-cycle (under condition: P-component should be as high as possible for the high, positively asymmetric S-cycles, and vice versa: B-component should be as high as possible for the low, symmetric and negatively asymmetric S-cycles); the second step deals with fitting of B- and D-components to the decaying phase of S-cycle; and the third one matches both previous so as to obtain the good Χ^{2 .} 7. Fit is considered good (indistinguishable to the eye), if Χ^{2 }< 0.1, and average Χ^{2 } = 0.061 (± 0.026).
All 23 S-cycles, being fitted, have allowed to get table 1. Some results of this work are presented in tables 1 and 2, in comments to those tables and shown in fig.1 and fig.2.
Table 1: Estimations of some average SG - model parameters
PARAMETERS | X _{C} | σ | ΔX | σ | W | σ | H | σ | |
U-data (eq. 2,3) G _{U }- peaks | G _{U A}- peak | - | - | 10.8 m | 14 % | 8.48 m | 13% | > 20 | - |
G _{U 1}- peak | - | - | 12.0 m | 10 % | 6.57 m | 18% | < 20 | - | |
K-data (eq. 4,5) G_{ K }- peaks | G _{P }- peak | 40.4 m | 5.9 m | - | - | 33.2 m | 2.7 m | 82.4 | 46 |
G _{B }- peak | 64.2 m | 7.4 m | - | - | 32.6 m | 2.1 m | 55.9 | 10.3 | |
G _{D }- peak | 92.4 m | 9 m | - | - | 34.1 m | 2.5 m | 33.1 | 9.8 | |
- | - | - | - | - | - | - | - | ||
E-data (eq. 6,7) | G _{GL }- peak | - | - | 40 y | - | 28.3 y | 18% | 122 | 11% |
63.5 y | 13 % | ||||||||
LCN- data (eq. 7,8,9) | G _{GL }- peak | - | - | 46 y | 24 % | 31.65 y | 13% | - | - |
Comments: 1. The average values of main parameters of SG - model are shown in table 1. They were received by the entire array of data for all 23 S-cycles. "m" means months; "y" means years; "-" shows that the corresponding parameter can not be characterized only by the average value, or just is not relevant to the case.
2. Taking parameters from table 1, one can write symbolically the average S - cycle (<S>) as follows:
<S>= {((t 0, i +40.4) - 33.2 - 82.4) = ((t 0, i+64.2) - 32.6 - 55.9) = ((t 0, i +92.4) - 34.1 - 33.1)} (10)
3. It is also seen from table 1 that parameters w P, w B, w D, x B and x D are the most stable; other three parameters x P , H B, and H D are "intermediate" stable, and only one parameter H P looks like unstable. If to show the variation of H P,i (t) on the graph (see fig.2), one can see that H P,i (t) changes cyclically and that rhythm (cyclicity) looks like the famous Gleissberg cycle. Because the leading role in SG-model is given to P-component (see constraint # 3), we assume that it is the dynamics of the P-component is a major factor (but not the cause) of the Gleissberg cycle occurrence. Let us call this representation of Gleissberg cycle as GL-cycle.
4. Fig.2 shows 7 GL-peaks describing the variation of the H P, i (t). Parameters of SG 2 -model description of GL-cycle by the two humped curve (6,7) are shown in the right side of fig.2. The first S-cycle has the number -4, the last - number 25. "The abnormal cycles" (#12 and 19) and "cycles under forecast" (# 24 and 25) are shown as cyan circles. The latter GL3-cycle is not over, but it is not excluded that it may happen during the S-cycle #25; GL-peak #7 (GL4.1) is our forecast on the basis of the average GL-cycle; (they are shown by the gray curve). Errors for all S-cycles are taken equal to ± 14; they include both physical (including variability of unknown parameters of the poorly understood mechanism of the solar cycle) and instrumental (including ambiguity of the fitting procedure) errors. Three horizontal lines are seen in Fig. 2. They divide the whole diapason of H P, i , H B, i , and H D, i changes in four zones: > 80 (zone #4), ~ 40 - 80 (zone #3), ~20 - 40 (zone #2); and zone #1 (< 20). While H P, i may be found in any zone, H B, i belong only to zone #3, and H D, i - to zone #2 (except S-cycle # 6). Zone #1 is a critical one, where the switching to the Grand minimum (GM) is most probable. The corresponding LS-cycles occupy "the lower saddles" of GL-cycles (see # - 4, 6, 15 and 25). A regular GL-cycle (shown in grey in fig.2) has well-defined lower and upper saddles (see HS-cycles # 1, 10 and 20), ascending (see US-cycles # -3, -2, -1, 2 and 3; 7, 8 and 11; 16, 17, 18, 21 and 22) and descending sections (see DS-cycles # 0, 4 and 5; 9, 13 and 14; 23 and 24). As for the real GL-cycles (shown in orange in fig.2), they usually have a "deformed" shape, when maximums are unequal and some S-cycles (see # 12 and 19, shown in cyan) fall out of the "normal" course of GL-cycle.
5. An analysis of SSN time series and S-cycles in particular, which is describing here, has the ultimate goal of developing the solar cycle prediction techniques. So, we should have an idea about the behavior of the solar cycle's parameters. There are two kinds of such parameters: "amplitude" (H P, i , H m, i , H B, i , and H D, i) and "temporal" (t 0, i, x P,i , t m, i, x B,i , x D,i , TM,i and T 0,i ). From these 11 parameters the last one is the most mysterious. The estimate of the cycle's length (T 0, i) and conditioned by it the next cycle's start (t 0, i+1) becomes difficult, if not hopeless case. The fact is that evolution of S-cycle takes place in three relatively independent stages. Each of them can be associated with a magnetic harness at the bottom of the convective zone, "emitting sunspots". The first in time harness begins to produce sunspots at a relatively high solar latitudes (~ 30 degrees). A SSN - course in this period (an activity growth ) is described in SG-model by P-component. The second harness starts especially intensively to produce sunspots in the period of the cycle's maximum at latitudes of about 15 degrees. A SSN-course in this period ( the cycle's maximum) is described in the SG- model by superposition of P - and B - components. Then begins the downturn of the cycle (the cycle's decay) and its behavior is determined by at least two factors: production of sunspots by the third harness at the latitudes ~ 8 degrees (this factor is described in SG-model by D-component) and the effectiveness of the process of sunspot magnetic field reconnection in different hemispheres (North and South) in the equatorial zone of the Sun (this process is observed in the form of more or less rapid decline in SSN during transition from the active phase of the cycle to mGM - period). Thus it appears that the behavior of the decay phase weakly associated with the phases of growth and high activity in the S-cycle. For example, very high cycles (# 3, 4, 8, 18, 19, 21 and 22) are usually short; however, cycle # 4 was not just a long, but the longest. Distribution of S-cycle's duration has a sophisticated look, which I tried to be reduced to two types of S-cycles: the short ones, <TS> = 119.5 ± 6 months (in the range of 108 - 126) and the long ones, <TL> = 142.5 ± 6.5 months (in the range 135 - 151). So, the average duration <T 0 > = 132.4 ± 14.7 months falls within the range from 127 to 134, in which S-cycles are not observed at all. Thus, you can make more or less successful attempt to estimate the length of the current S-cycle with a precision of ± (6 -7) months, taking into account all available reasons, however, it should be understood that in the framework of the SG - model we will be able to know it for sure only after about six months after t 0, i+1 will come. The more advanced analysis shows that the length of S- cycle takes one of three values: T 0,S ~ 120 months (10 years); T 0,L ~ 138 months (11.5 years); and T 0,L+ ~ 150 months (12.5 years). To advance somehow in the question of S-cycle's length, let us assume, that GL-cycle reflects not only the amplitude variations of S-cycles, but also their temporal characteristics, i.e. the S-cycle's shape as a whole. Let us formulate this assumption as follows: "S-cycles arranged on GL-cycle similarly, have approximately the similar shapes" .
That assumption allows to distinguish three base types of S-cycles: ascending (US-cycles), descending (DS-cycles) and saddle (HS- and LS-cycles), see above, comment 4. As data show, those types correspond to the "compact" short S-cycles (U2S - # 2, 3, 8, 17, 18, 19, 21, 22; and U1S - # 7, 16; LS # 15) with T 0,S ~ 120 months; and to the "delayed" (shifted) S-cycles "without tail" ( HS-cycles - # 1, 10, 20 with T 0,L ~ 138 months; LS+ # 6; DS+ - # 9, 23 with T 0,L+ ~ 150 months) and S-cycles "with tail" (US+ - # (11), DS+ #(12), 13, 14 with T 0,L ~ 138 months; and LS+ - # 4 and DS+ - # 5 with T 0,L+ ~ 150 months). As the reference cycle, U2S-cycles can be taken with x P,i = 38 months, x B,i = 62 months, x D,i = 88 months, TM,i = 121 months, i.e. ~ T 0,S ). It's enough to enter to the forecast of the next S-cycle.
the last update: 12/31/2015