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

* The art of solar cycle prediction is unfortunately ambiguous. More information about this topic may be found in page "Appendix 3. Discussion". Our method of forecast is based mainly on SG-model (see pages "Model and Appendix1"). 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.*

** Introduction. **Development 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 (**

*24 SG*) has been done in August 2006, which corresponds to the point 3092 in M-data time series (

**F**: GL 2006*H m,24*=

*70**(± 17.5) at t m,24 = 2012.96). 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). Consequently, WO-forecast (

*24 SG*) was not possible yet and also t 0, 24 was unknown. Soon after the forecast

**F**: WO*24 SG*

**F**

*: GL 2006*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.2015.*The following describes the four-steps forecasting process and gives the current S-cycle's description (for cycles # 24 and 25) in a form that follows from the

*SG-model.2015.*

**Step I: ***This step consists in the GL-forecast of S-cycles in terms of "the logic of the development of GL-cycle". *In accordance with *SG-model.2015*, S-cycle forecast is some beforehand estimate of all its parameters. There are 11 of such parameters (9 parameters for 3 K-components, t 0, i and T 0, i ; if we would know T 0, i for the current S-cycle, we would also know t 0, i+1 ). In *SG - model.2015* the logic of the development of GL-cycle is that *it describes the **H P, i (t) - variation and that the similarly situated cycles have the similar parameters.* On this basis, 24 SGF: GL, 25 SGF: GL, 26 SGF: GL and so on (see fig.2) could be already made in 2006. For example, *24SG F:GL* (K*-data in fig.3) uses the rough estimate

*H* P,24*=

*34.8**(± 14) and the fact that*

*# 24 is S-cycle of D*

*S+- type*

*like # 5, (12) and 14. All those cycles have T M ~120 and tales Δ*

_{2 }about 20 months. Choosing for # 24 the average parameters of those similar S-cycles, we have got (with

*H*m,24*=

**at**

*59.7***3179 (2013/11)) :**

* 24SG F:GL =* {((t

**0, i**+ 35.1) - 32.8 - 34.8) = ((t

**0, i**+ 62.5) - 35.3 - 42.4) = ((t

**0, i**+ 87.3) - 34.4 - 23.6)} (11)

(see fig.3, navy). *This is the first forecast for K-data*. It is markedly different from *24 SG F: GL 2006*. And for # 25 we'll have:

*25SG F:GL* = {((t

**0, i**+ 41.4) - 31 - 20) = ((t

**0, i**+ 65.5) - 36 - 48.8) = ((t

**0, i**+100) - 36 - 20)} (12)

(see fig.4 and (*H* m,25 *= *58.4** *(± 14) at t m,25 = 2025.92). *25SG F:GL* (K-data) is shown in fig.4 as a navy curve. It is assumed that t

**0, 25**= 3261 (September 2020) and T M,25 ~136 months.

*NOTICE*. In March 2013 there was the PMF reversal (t PR, 25 ) that means the S-cycle # 25' start... To represent the current (May 2014) situation more clearly, let's again refer to fig.2. From this figure it is seen that, in general, low cycles are formed in groups, located at the junctions, or saddles, of the GL-cycles. The first such a group consists of the cycles # -4 and -3, observed immediately after the end of the Maunder minimum; the second group consists of the cycles # 5, 6 and 7; the third - of S-cycles # 14,15 and 16; and finally the fourth group consists of the cycles # 24, 25, and 26, to be forecast. When imposing the corresponding cycles of different groups to each on other, you can see that the cycle # - 4 is similar to the cycle # 6, and the cycle # -3 is (probably) similar to the cycle # 7. It is seen that S-cycles # 5, (12), 14 and 24 are close to each other, what was used for *24SG F:GL* . As for LS - type's S-cycles # - 4, 6, 15 and 25, the situation with them is more complicated and ambiguous, so, one can expect any of three scenarios:

*1- transition to the grand minimum (the start of GM period and a new SAE (9)); 2 - a critically low S-cycle, similar to # 6, and at last 3 - appearance of the relatively short S-cycle with a high*

*H B ,*

*for which cycle #15 can be regarded as the limiting case.*What will happen depends on unknown processes in the tachocline during the lower saddle of GL-cycle. True, it probably will be somewhat inconsistent with the forecast, using "GL-method", shown in figure 2. If you believe in the forecast based on the "GL-method", then the cycle # 25 is still critically low. This means that it will be close to the cycle # 6 and be on the verge of collapse into a state of Grand minimum. This feature is widely discussing now by experts (for example, see opinions of M. Giampapa and D. Hathaway), but in the framework of the data used in the SG- model, we have so far the only indication of such a scenario. This indication is connected with the "WO- method" of forecast. If to compare the courses of PMF growth after the polar reversal (t PR, i), it is evident that the course of PMF 24

*still goes much lower than all the rest. So, we have three contradictory predictions for S-cycle # 25; that situation is shown in fig.4. It is not a forecast; it is just my opinion that the curve of*

*25SG*-data) will pass somewhere between 6 SGD: K (magenta, # 6 profile) and 15 SGD: K (dark yellow, # 15 profile) lines. Step III will help to make choice between three above mentioned possibilities. Nevertheless, some recent papers give a support to the opinion that S-cycle # 25 will be a "moderate" cycle like # 24. This opinion is based on the progress in development of "WO- method" of forecast. Analyzing predictions for S-cycle # 24, W.D. Pesnell (2012 and 2016) noted prospectiveness of "solar precursor" forecast technique (see also Cameron et al. (2016) and Hathaway & Upton (2016)).

**F**(KAs for *26SG F:GL*, let us for the time refrain from it.

**Step II: ***This step consists in matching #SGF:GL with #SGF:WO. *Knowledge of the PMF maximum, as it was established after pioneering work 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.2015 uses such an option for WO-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 (=1.26) x <PMF M >** . **It was possible during the decay phase of S-cycle #23 to evaluate <PMF M > = 56.1 (± 3) microtesla (cyan in fig.3). *Then **H** m,24 = 1.26 x 56.1 = 70.7; this is 24SGF:WO, which is close to *

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

**F**:GL2006.*#*

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

*#SGF:WO*. For

*the easiest way to do this is to lift slightly H B, 24 (from 42.4 to 53.6, see fig.3, K**-data, dark cyan). Then we get (the maximum at December 2013):*

**24SGF** *24SG F*:

*GL/WO*= {((3120

**+35.1) - 32.8 - 34.8) = ((3120+62.5) - 35.3 - 53.6) = ((3120+87.3) - 34.4 - 23.6)}. (13)**

*NOTICE*: *#SG F:GL/WO* is an important correction of

*#SG*using the latest observed data (<PMF M > and t 0, i+1). It allows you to not only specify a maximum of S-cycle (H m, i+1), but also its shape as a whole. PMF-data from Wilcox Solar Observatory (Avgf-data time series) were starting at May 1976. So, for # 22 we have H m,22 = 163.3 = 1.29 x <PMF M > (=126.6 (± 3)); correspondingly, for # 23 we have H m,23 = 121.1 = 1.205 x <PMF M > (= 100.5 (± 3)). Comparison of all the currently available (2015) data allows to select the current compromise value of

**F**:GL**k = 1.26**± 0.02.

Knowing 24SGF:GL/WO, it is possible to build U-data forecast (** SGF:U**, light magenta) and compare it with M-data (see (1) and fig.3). 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. 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). It is shown in fig. 3, where the correspondent U-peaks are increased by 30%; it follows from that figure that S-cycles are actually three-humped (although for high S-cycles this tendency is less pronounced). For S-cycle #24 case it was used

*24SG*based on

**F**:U*24SGF:GL/WO*(see fig.3, dark cyan).

For S-cycle #25 this step has not yet come, but instead there is possibility to make "the back-forecast", i.e. to evaluate <PMF M,24 > on the assumption of the value *H* m,25 *= *58.4** . * It follows that <PMF M,24 > should be ~ 46.3 microtesla (*H* m,25 /1.26)*. By the way, the current PMF for December 2015 is 49 microtesla and keeps growing (see figs. 3 and 5).

**Step III ***consists in analysis of deviations between forecasts and the current observations in the process of S-cycle evolution. *It concerns both K-data and U-data.

As new M-data (grey points without error bars in fig.3) are coming they can be first partially and then completely (Step IV) compared with forecasts, and after that "the quality of forecasts" is analyzed. This procedure in its pure form has become available, starting only with S-cycle # 24. Technically it includes fitting of reliable data in some time interval by superpositions of G-peaks (2), resulting in parameters of SG- model, which describes the given S-cycle (*#SG D*), are specified. That procedure is painstaking, but it allows you to penetrate deeper into the dynamics of S-cycle. M-data are filtered to get K- and U-data. Unlike the observed M-data, a smoothed (filtered) data is 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). As the accumulation of points is proceeding, the trials of their description (it should be good) with the help of consecutive adjustments of parameters are repeating. Herewith the points with K-data < 20 (i.e. K 0 -data ) are masking (at the beginning of S-cycle); the "inconvenient points at the end" are also masking. Adjust only x P, 24 first; then add H P, 24 and so on... By the end of 2015 an interval of points [3120 - 3204] was at our disposal. To minimize the existing uncertainty, I consider as the future M-data

*24SGF:U*(light magenta points with error bars in fig.3). In fig.4 an expected future M-data are shown as

*25SGF:GL.*As a result, it was obtained the following current description of the whole cycle #24 (blue K-data):

*24SG D:Kt.3210 = * {((3120

**+37) - 31.9 - 37.8) = ((3120 + 62) - 33.6 - 52) = ((3120 + 85) - 34.2 - 22.5)}; Χ**

^{2}= 0.029

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:

*24SG D:Ut.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

Nevertheless, as it said in "a Guide book" on the page "Appendix 1", *the final comparison can be made only after the current S-cycle's end.*

*The SG-model current (2016) expectations for the cycle 25 are shown in Fig.4.* ** 25SGF:GL**-old (gray) is shown for comparison with S-cycles # 6 and # 15. 25SGF:GL-new (navy) will be compared with

*Mn*

*– data (see note*So, up to the current time (2016) we have accomplished for S-cycle # 25 only

**A.5**in the page "Appendix 1").

**Step I.**