This page displays my Forecasts for S-cycles # 24 and # 25.

1. The first "SG - forecast"

* The art of solar cycle prediction is unfortunately ambiguous. Our method of forecast is based mainly on SG-model (see pages "Model" and "M**iscellanea**"). SG-model is a tool for the study of solar activity, which is sensitive to factors such as: (**1) a retrospective analysis of existing data; (2) the emergence of new data, both observational and theoretical; (3) at last, the process of matching the forecast with the incoming current data (M-data and others). This last factor implies prediction ( #SGF:GL), which is made before the first M-data of a new S-cycle become available.*

Elaboration of the *SG-model *was* *beginning in 2000, when the "Origin" software had appeared in my hands.** **The first and the only published (see Pesnell, 2008) prediction (*24SGF.1*) has been done in August 2006, which corresponds to the point 3092 in M-data time series (*H m,24 *= *70** *(± 17.5) at t m,24 = 2012.96).

*24SGF.1:** {(3121+34.1)-28.7-
(32*×*1.2533*×28.7*) = (3121+57.9)-38.6-(49*×*1.2533*×*38.6) = (3121+87.8)-32.6-(31*×*1.2533*×*32.6)}. *

It fell on the decay period of the cycle #23, when, in principle, many parameters (which being requested for *SG-model*) of that cycle were known, excepting may be of the parameters of D-component (x D, 23 =3081, w D, 23 =39.1 months, H D, 23 = 20.4) and of cycle's length T 0,23 (151 months). Soon after the forecast *24SG*F.1* *has been done, the website on which the *SG-model.2006* was described, has become unavailable and I made an unsuccessful attempt to publish *SG-model.2006* in the "Solar Physics" (2008) journal. Then the new (this one) site has been developed (it has been launched in November 2008 - practically on the eve of S-cycle #24 start), on which in the real time three things are being made: *the monitoring of S-cycle #24 current data*; *an analysis of those data on the base of SG-model.2006*; and also *the SG-model revision, which has been over (mostly, but not completely) as SG-model.C (current) or simply SG-model.** ** *The results obtained above in the description of the S-cycle's shape by the SG - model prove that those results are sufficient for constructing a GL-prediction of the next S-cycle. At that point it follows to note that SG - forecast (SGF) is being made in four steps: (1) long term regular (GLL), (2) short term regular (GLS), (3) rectification of the cycle's maximum using the PMF precursor (WOC) and (4) M-data approximation (U).

* 2. Steps of predictions for cycles # 24 and 25. *

2.1. Currently (July 2018), the start time of cycle 24 is of course known (December 2008, or point 3120), but for cycle 25 is not yet. Starting to build a GLS-forecast, we first set the expected parameters of the cycle model, namely, the mean X p = 38.9 months, W = 33 months, X b = 63.5 months, X d = 92.8 months [see "DATA"- 7.2 (6)].

*2.2.* Then we consider the corresponding sections of the GL- cycle. For cycle 24, they are *D L** -type* cycles # 5 and 14, and for cycle 25 - *E L** -type *cycles # - 4 and 6, as well as an anomalous *U L**- type *cycle # 15. Cycles # 5 and 14 are similar: both long with tail; length, respectively, 151 months and 138 months; both follow the long *D H** -type *cycles # 4 and 13. The cycle # 24 also follows the long # 23 and, according to the "logic of analogues", should have an average length of ** 143 months** . Then

*t 0, 25 =*

**3262**.*2.3.* The situation with S-cycle # 25 is less certain. For lack of other estimates, we start from 3262 and estimate X P = 3262 + 38.9 = 3300.9, and from Fig. 2 (3300.9 + 588) we determine H P = 38 ± 20 and the cycle type as a transitional S-cycle. In Figure 2, you can see only four such S-cycles (# - 4, # 6, # 15 and # 25). They are clearly different, although all are in the transition region between adjacent GL-cycles. For the cycle # -4, the ending S-cycle of the Maunder GM (1645 - 1715), M-data is still missing, but it still looks like cycle # 6 (low and long). Cycle # 25 could be the same if it were not for cycle # 15, which turned out to be abnormally high and short. The transition zone between adjacent GL-cycles seems to be unstable, since at this time the speed of the meridional circulation changes from small to large (and, accordingly, switching from long DS -cycles to short US - cycles). Therefore, for the cycle # 25, the following options are possible (see Fig. 4): (1) the low and long S-cycle similar to # 6, (2) the low and short US - cycle similar to # 7 and # 16, or (3) the high and short but abnormal S-cycle similar to # 15.

* 2.4. **24SG F:GLS* (see fig.3) uses an estimate

*H* P,24*=

*34.1**(± 14). Choosing for # 24 the average parameters of those similar S-cycles, we have got (with*

*H*m,24*=

**at**

*62.9***3180 (2013/12)) :**

* 24SG F:GLS =* {((3120

**+ 38.9) - 33 -**

**) = ((3120**

*34.1***+ 63.5) - 33 - 44.6) = ((3120 + 92.8) - 33 - 30.5)} (7.1)**

(see fig.3, navy). *This is **somewhat different **forecast for K-data in comparison with* *24SG F.1*.

2.5. For # 25 we'll have three (instead of one) options plus K-shapes of cycles # 6 and # 15 as they would look in *the new M -data* (see page*"M**iscellanea**")*.

**25****SGF****: ****GLS****1 ***{(3262+42.8)-33- (38*×*1.2533*×*33) = (3262+70)-33-(60.64*×*1.2533*×*33) = (3262+102.1)-33-(38*×*1.2533*×*33)} (7.2) ** *

** ****25****SGF****: ****GLS****2 ***{(3262+38.9)-33- (38*×*1.2533*×*33) = (3262+63.5)-33-(60.64*×*1.2533*×*33) = (3262+83)-33-(38*×*1.2533*×*33)} (7.3)*

**25****SGF****: ****GLS ****3 ***{(3262+38.9)-33- (85.6*×*1.2533*×*33) = (3262+63.5)-33-(74*×*1.2533*×*33) = (3262+83)-33-(38*×*1.2533*×*33)}, (7.4)*

where the cycle's length will be 120 or 143 months. Up to now, the possibility of Grand minimum's start still is not excluded.

* 2.6. *Knowledge of the PMF maximum, as it was established after pioneering works of H.W. Babcock (1961), A.I. Ohl (1966) and J.M. Wilcox, allows to predict the height of the next S-cycle prior to observations of its first sustainable high-latitude sunspots (i.e. before the cycle's start t 0, i+1). This type of prediction is considered now as the most reliable and theoretically confirmed (see Pesnell (2008), Petrovay (2010), Cameron and Schüssler (2015)). SG-model uses such an option for WOC-forecast: <PMF M > is determined by averaging of current Avgf-data (Wilcox Solar Observatory data for the polar magnetic field strength, PMF) over the interval [x D, i - t 0, i+1 ]; then H m, i+1 = k * <PMF M >** **(see fig. 6 on page *"M**iscellanea**"*).** **It was possible during the decay phase of S-cycle #23 to evaluate <PMF M > = 55.5 (± 3) microtesla (cyan in fig.3). *Then **H** m,24 = 1.29 x 55.5 = 71.6; this is 24SGF:WOC, which is close to *

*24SG*Correction should take into account the observed t 0, i+1 and combine H m, i+1 in

**F.1**.*#*

*SGF: GLS*with H m, i+1 in

*#SGF:WOC*. For

*the easiest way to do this is to lift slightly H*m, 24 (multiplying*

**24SGF***24SG*(= 71.6/62.9), see fig.3, K-data, dark cyan). It's

**F**:GLS by 1.138*24SG*:

**F***WOC*(the maximum at December 2013).

*#SG*is an important correction of

**F**:GL/WOC*#SG*using the latest observed data (<PMF M > and t 0, i+1).

**F**: GLS

__It allows you to not only specify a maximum of S-cycle (__

__H m, i+1)__*PMF-data from Wilcox Solar Observatory (Avgf-data time series) were starting at May 1976. So, for # 22 we have H m,22 = 163.2 = 1.35 x <PMF M > (=121 (± 12)); correspondingly, for # 23 we have H m,23 = 121 = 1.2 x <PMF M > (= 101 (± 3)). Comparison of all currently available (2015) data allows to select the current compromise value of*

__, but also its shape as a whole.__**k = 1.29**(see fig.6).

2.7. Knowing 24SGF:WOC, it is possible to build U-data forecast *SGF:U* and compare it with M-data. It is supposed in *# SGF:U *that K- data represent the envelope for maximums of U-peaks, while the other parameters are taken from table 1 (page "DATA"). As data analysis shows there is a noticeable upward trend of H m, U i for U-peaks coinciding (or close) with the moments x P, i , x B, i , x D, i , i.e. moments of maximums of K-peaks shown in fig.3 by the dotted vertical lines. To emphasize this, the correspondent U-peaks will be increased by 30%; it follows that S-cycles actually should be three-humped (although for high S-cycles this tendency is less pronounced). The real U-data in fig.3 (wine line) show two of three expected peaks (compare with Karak et al., 2018).

For S-cycle #25 that steps have not yet come.

* 3. **analysis of deviations between forecasts and the current observations in the process of S-cycle evolution. ** *

3.1. It concerns both K-data and U-data. As new M-data (grey points in fig.3) are coming they can be first partially and then completely compared with forecasts, and after that "the quality of forecasts" is analyzed. Technically it includes fitting of reliable data in some time interval by SG- model. Unlike the observed M-data, smoothed (filtered) data are unreliable both at the beginning of S-cycle (because during mGM period the Sun is still not active) and at the end of the available interval (because the future M-data are unknown). By definition the Sun is becoming active (only from that moment S-cycle is described by SG-model), when K-data begin to exceed 20 (for S-cycle # 24 that moment corresponds to the point 3138). To minimize the existing uncertainty, it is possible to consider as the future M-data 24SGF:WOC or *24SGF:U*. In fig.4 an expected future M-data are shown as *25SGF:GLS (see (7.2-7.4)).* As a result, it was obtained the following current description of the whole cycle #24 (blue K-data, compare with (7.1)):

*24 SG D: K.3227 = * {((3120

**+38.9) - 33.7 - 42.2) = ((3120 + 63.9) - 32.8 - 51) = ((3120 + 84.9) - 30.8 - 16.6)}; Χ**

^{2}= 0.029 (7.5)

3.2. For June 2016 we have: *H m,24 K = 73 **at September 2013; and **H m,24 U 1 = 70.3 at Nov.2011 & *H m,24 U 2 *= 88 at Feb. 2014 (at the same time M-data also reach their maximum, 102.8). *As for description of U-data, we get:

*24 SG D: U.3204* = U 1, 24 (3120.45 - 3.3 - 4) + U 2, 24 (3131.6 - 4.93 - 8) + U 3, 24 (3141.3 - 9.3 - 19) +

U 4, 24 (3152.7 - 8.7 - *56.4*) + U 5, 24 (3160.9 - 9.9 - 43) + U 6, 24 (3169.6 - 9.4 - 42) +

U 7, 24 (3181.3 - 10.35 - *78.3*) + U 8, 24 (3191.9 - 10.35 - 55) + U 9, 24 (3202.5 - 7.9 - 36) + ... Χ^{2}= 0.037

It can be seen from Fig. 3 that the phase of the decline of cycle # 24 deviates significantly from the 24SGF: WOC, namely, the D - component in 24 SGD: K.3227 (7.5) reaches the maximum of 7.9 months earlier and is almost half as low (but that's all even within the limits of 2Ϭ).

3.3. Observations obtained in 2018 show that the activity of cycle # 24 continues to decrease gradually. In this case, the K-data goes slightly above the D-component's curve obtained in (7.5). This may mean one of two things: either the description of the K-data must be updated, or we are observing the beginning of the tail of the cycle (see the description of Fig.1). Based on the structure of the SG - model and the phenomenon of cycle # 4, we can assume that the 11-year solar cycle consists of four activity waves - required, well known P -, B -, and D - components and t (D *). Into high cycles on the ascending branches of the GL - cycles (see Fig. 2), the P-wave sets the tone; B- and D-waves go in descending order, and the t-wave is practically absent. Therefore, these cycles are short (an exception - the cycle # 11). On the descending branches of GL- cycles, all four waves take place, but the peculiarity of such S-cycles is that they either have a relatively large delayed D-wave, which "absorbs" t-wave, or a t-wave degenerates into a shapeless tail (an only exception - of course, the cycle # 4). S-cycle # 24 looks not only low (as it should be), but apparently short (which is not supposed to).

Therefore, only the long tail can “save its reputation” (its analogs S-cycles # 5 and # 14 had tails 27 and 19 months long, respectively). To date (March 2019) the tail of cycle # 24 is only 7 months. Technically, the tail is observed as the noticeable excess of the U-data and K-data over (24)SGD:K (see (7.5) and the corresponding dark yellow curve in Fig.3) at the end of the cycle.

**Nevertheless,** as it said in "a Guide book" on page *"M**iscellanea**"**, the final comparison can be made only after the current S-cycle's end.*

#### 3.4. The SG-model current expectations for the cycle 25 are shown in Fig. 4: GLS. Four 25SGF:GLS variants are shown, plus S-cycles # 6 (green) and # 15. All 25SGF:GLS-new will be compared with M new – data (see note A.5 on page "Miscellanea"). So, up to the current time, we have accomplished for S-cycle # 25 only Step (2). Nevertheless, already now (in October 2017 and later) you can make the current preliminary 25SGF: WOC (for example, <PMF max> (for the period 3205 – 3234) = 58.9; so *H** m,25 *should be 58.9×1.84 = 108 (see fig. 4:GLS)). The current elaboration of the description for S-cycles 24 and 25 (both 24SGF and 25SGF), shown in Fig. 3 and 4, is given on page "Conclusion". It should also be noted that the extrapolated K - and U - data so far correspond to the GM - forecast.

3.5. Figure 4: GLS shows that today (March 2019) the situation with the forecast of the 25th cycle is still quite contradictory. This is due to the assumed position of the cycle 25 within the Gleissberg cycle (see Fig.2). The fact is that it is in the region of the extremum (specifically, at the minimum) of the GL - cycle. Observations show that in the region of the extrema of the GL - cycle, the probability of development of anomalous S - cycles is markedly increased (see S - cycles # - 4, 4, 6, 12, 15, 19, (24), 25). Therefore, in Fig. 4: GLS, there were all four possible types of forecasts: (1) the absence of a S - cycle due to the onset of a Grand minimum (see Ref.2018/1 and current extrapolated behavior of K-data and U-data); (2) emergence of a critically low S - cycle (see 25 SGF: GLS1 and #6 K - data); (3) emergence of an intermediate S - cycle (see 25 SGF: GLS2) and (4) emergence of a high S - cycle (see 25 SGF: GLS 3 and #15 K-data). My regular forecast refers to type 3, and the inability to clarify it in one direction or another indicates a too crude understanding of the physics of the S - cycle. By the way, the current WO - forecast still remains somewhere in the middle between options (3) and (4). To compare the predictions for cycle # 25, I bring here the magnitudes and moments of the K-curves maxima shown in Fig. 4: GLS. For the low forecasts: 6K-data (* 63*/December_2026), 25SGF: GLS1 (

*/June_2026), 25SGF.1 (*

**77***/May_2024); for the intermediate forecast 25SGF: GLS2 (*

**78***/February_2026), and for the high forecasts: 25SGF: GLS3 (*

**92***/March_2025) and 15K-data (*

**129***/March_2025).*

**131**3.6. However, not everyone, unlike me, is staying before the problem of S - cycle # 25 forecast in deep thought, like Knight at the Crossroads in artwork by Victor Vasnetsov (1882). For example, Prantika Bhowmik & Dibyendu Nandy (2018) indicate that "based on our simulations, the corresponding prediction for the yearly mean sunspot number at the maximum of solar cycle 25 is ** 118** with a predicted range of

*109–139*. The maximum of solar cycle 25 will occur around 2024(±1). The range of ±1 year also includes the uncertainty in the exact timing of cycle 24 minimum which may vary by 6 months". This means that their forecast corresponds to our high forecast and short S - cycle, which is similar to cycle #15. Then they conclude that "our ensemble prediction indicates the possibility of a somewhat stronger cycle than hitherto expected, which is likely to buck the significant multi-cycle weakening trend in solar activity. Our results certainly rule out a substantially weaker cycle 25 compared to cycle 24 and therefore, do not support mounting expectations of an imminent slide to a Maunder-like grand minimum in solar activity".