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## Abstract

Jan-Erik Solheim, Kjell Stordahl and Ole Humlum (hereafter SSH) published two articles in 2011 and 2012 about the relationship between the mean temperature in a solar cycle and the length of the previous solar cycle [1, 2]. For the northern hemisphere, they found a negative correlation between those two variables. A long solar cycle is followed by one with a low temperature, and a short solar cycle is followed by one with a high temperature. SSH named this the Previous Solar Cycle Length Model. For simplicity, in this note I refer to it as the Solar Cycle Model or just the model. For the same reason, I usually omit the word *mean* when referring to the mean temperature in a solar cycle.

SSH claim that their model describes a cause-effect relationship, i.e. that it has predictive power. Solar cycle 24 had just started when they wrote their articles. SSH predicted a significant temperature decrease in solar cycle 24. That solar cycle has just ended, and now it is possible to check if their prediction came true. It did not.

The temperatures fitted well with the Solar Cycle Model until the mid-1970s, but not later. The mean temperatures during the last solar cycles have been much higher than predicted by the model.

## 1 Introduction

When a solar cycle has ended, its length is known, and the Solar Cycle Model claims that it can predict the mean temperature in the next solar cycle. Solar cycle 23 ended in November 2008 after having lasted for an unusually long time, well over 12 years. SSH therefore wrote in [1]: '*We predict an annual mean temperature decrease for Svalbard of 3.5 ± 2°C from solar cycle 23 to solar cycle 24.*' Their next article [2] concentrated on the North Atlantic region including Norway and Iceland. They wrote that the Model '*provides a tool to predict an average temperature decrease of at least 1.0°C from solar cycle 23 to solar cycle 24 for the stations and areas analyzed.*'

Back in 2012 I doubted that the length of a solar cycle controls the temperatures in the next solar cycle. I therefore programmed the Solar Cycle Model myself and I downloaded the same temperature and solar cycle data that SSH had used in their analysis. I found the same negative correlation between the mean temperature in a solar cycle and the length of the previous solar cycle as SSH did. This correlation was strong till the mid-1970s, but not thereafter.

In their two articles, SSH show results when their model predicted the temperature in solar cycle 24. They do this in 16 figures, each consisting of four graphical plots. In a short chapter (3.5 in [2]), they tell that they, as a test of the model, used the model to predict the temperature in solar cycle 23 based on the temperatures measured in the solar cycles up to and including solar cycle 22. But they did not show the differences between the measurements and the predictions, neither graphically nor numerically. They did not do that, neither for solar cycle 23 nor for the solar cycles before that. In the blog post you are reading now, I do this for all the eight temperature series which I analyze. In this way I can see how well or badly the predictions for the earlier solar cycles match with the temperatures that were measured in those cycles. I show the differences between predictions and measurements graphically in the figures that follow. For solar cycle 23 the measured temperatures are higher than predicted by the model; this applies for all the eight temperature series. Appendix C shows that for six of them the temperature was higher than the upper limit of the 95 percent confidence intervals around the predictions. In this way I found that the model predicted the temperature well for the solar cycles up to and including the one that ended in the mid-1970s, but not for the cycles after that.

In 2012 I published my analysis on Skeptical Science and on the blog post Solar Cycle Model fails after mid-1970s. I showed that the temperatures in the three solar cycles after the mid-1970s were much higher than predicted by the Model, and that the same was true to an even greater extent for the temperatures measured so far in the ongoing solar cycle 24.

In 2014 I updated my analysis with more temperature data for the ongoing solar cycle 24. It showed the same as my analysis two years earlier. The temperatures measured so far in the ongoing solar cycle 24 were much higher than predicted by the model. I published the results on a Norwegian discussion forum, and I got feedback from Jan-Erik Solheim, the lead author of the two articles. His only argument impossible to counter was that we have to wait until solar cycle 24 has ended before we can conclude with respect to the prediction for that cycle.

## 2 Solar Cycle 24 has ended

Some months ago a panel co-chaired by NOAA and NASA decided that Solar cycle 24 ended in November 2019 and that solar cycle 25 started in December 2019, see Hello Solar Cycle 25. They expect Solar Cycle 25 to have the same strength as cycle 24, and they have '*high confidence that Solar Cycle 25 will break the trend of weakening solar activity seen over the past four cycles. “We predict the decline in solar cycle amplitude, seen from cycles 21 through 24, has come to an end,” said Lisa Upton, Ph.D., panel co-chair and solar physicist with Space Systems Research Corp. “There is no indication we are approaching a Maunder-type minimum in solar activity.”'*

The Norwegian organization Klimarealistene still argues that the ongoing climate change is mainly caused by changes in the solar activity, i.e. not by human activities. The three authors of [1, 2] are all members of Klimarealistene's Scientific Advisory Board. I have not seen that they have admitted that their Solar Cycle Model totally failed in its predictions for solar cycle 24. Some months ago the lead author Jan-Erik Solheim wrote on Klimarealistene's web site (Klimanytt 288) that solar cycle 25 has started without mentioning his failed predictions for the solar cycle that just ended. On the contrary, he wrote about the connection between solar activity and the climate, about the little ice age caused by low solar activity, and that it will be exciting to see if low solar activity in this century will cause a colder climate.

Because there is no sign that SSH will tell about how their Solar Cycle Model failed in its predictions for solar cycle 24, I will do so. I have repeated my analysis with updated temperatures and updated information about the solar cycles. The results are shown in the rest of this blog post.

## 3 The Solar Cycle Model run with different temperature series

SSH ran their Solar Cycle Model with temperature series downloaded in 2011. Then Solar Cycle 24 had just started. This chapter describes the results when I ran the model with temperature series downloaded in November 2020. I first ran the model with temperature series for the areas where SSH claimed that the model has predictive power. Then at last I ran it with the average of four temperature series with global coverage. As you will see, all model runs show that the predictions for solar cycle 24 were totally wrong.

The previous blog post, Local and regional temperature series, describes the temperature series that I now use and from where they are downloaded.

The figures in this chapter show the measured temperature in the solar cycles as blue circles and the predictions for these temperatures as red stars. This is for all solar cycles that have ended, i.e. up to and including solar cycle 24. The temperatures are shown as anomalies relative to the reference period from January 1881 till December 1910. The horizontal x value of the blue circles and the red stars is in the middle of the solar cycle they represent.

SSH applied the Durbin-Watson statistical test to check if there was 'too much' autocorrelation in the data. I apply the same test, and I will comment if the test was OK, almost OK or not OK. SSH also checked if the calculation of the regression line was statistical significant or not, i.e. had a p-value lower than 0.05. I do the same. In the text I will comment if these tests were OK or not. The numerical test values are given in Table 3 in Appendix 3.

In addition to what is shown in the figures, I let the model use the temperatures shown with the blue circles to predict the temperature for the ongoing solar cycle 25.

The following subchapters show the results when the model is run with different temperature series.

### 3.1 Norway and Svalbard

Figure 1: The blue circles show the mean temperature in solar cycles 10 to 24. The red stars show the model's predictions for solar cycles 15 to 24. |

### 3.2 Iceland

Figure 2: The blue circles show the average temperature in solar cycles 10 to 24. The red stars show the model's predictions for solar cycles 15 to 24. |

### 3.3 Near Longyearbyen

Figure 3: The blue circles show the average temperature in solar cycles 10 to 24. The red stars show the model's predictions for solar cycles 15 to 24. |

### 3.4 Svalbard Lufthavn

Figure 4: The blue circles show the average temperature in solar cycles 14 to 24. The red stars show the model's predictions for solar cycles 19 to 24. |

### 3.5 Vardø

Figure 5: The blue circles show the average temperature in solar cycles 10 to 24. The red stars show the model's predictions for solar cycles 15 to 24. |

### 3.6 Dombås

Figure 6: The blue circles show the average temperature in solar cycles 11 to 24. The red stars show the model's predictions for solar cycles 16 to 24. |

### 3.7 Northern Hemisphere Land only

*BEST NH land only*temperature series for the northern hemisphere as a replacement for HadCRUT3 NH.

Figure 7: The blue circles show the average temperature in solar cycles 10 to 24. The red stars show the model's predictions for solar cycles 15 to 24. |

### 3.8 Global coverage

Figure 8: The blue circles show the average temperature in solar cycles 10 to 24. The red stars show the model's predictions for solar cycles 15 to 24. |

## 4 An illustration of how the Solar Cycle Model works

Figure 9: The Solar Cycle Model predicts the temperature in solar cycle 24 based on the temperatures in and the length of the solar cycles up to and including solar cycle 23. |

Figure 10: The Solar Cycle Model predicts the temperature in solar cycle 25 based on the temperatures in and the length of the solar cycles up to and including solar cycle 24. |

## 5 Conclusion

## Appendix A. Overview of the solar cycles

*Around the world, observers conduct daily sunspot censuses. They draw the Sun at the same time each day, using the same tools for consistency. Together, their observations make up the international sunspot number, a complex task run by SILSO [World Data Center for the Sunspot Index and Long-term Solar Observations]. Some 80 stations around the world contribute their data*'.

*December 2019 confirmed as starting point of the new solar activity cycle [25]*'. This is the basis for Table 1, which I used when I programmed the Solar Cycle Model.

Table 1. Solar Cycle (SC) start, end and duration. |

## Appendix B. Mathematics used in the analysis

### B.1 Uncertainty in predictions

**both**the uncertainty of the next measurement

**and**the uncertainty of the estimate itself. I have explained this in more detail in an earlier blog post, see equation (3.3) in Confidence intervals around temperature trend lines, with reference to chapter 8.3.11 in Statistical Analysis in Climate Research written by Hans von Storch and Francis W Zwiers in 2001.

### B.2 The Durbin-Watson test for autocorrelation

*d*to be substantially larger than 2.

_{c}. The critical value depends on the significance level and on the number of observations. But d

_{c}also depends on the data. Therefore, statistical tables indicate a lower value d

_{L}and an upper value d

_{U}for d

_{c}. When d is smaller than d

_{L}, there is statistically significant positive autocorrelation. When d is greater than d

_{U}and less than (4-d

_{U}), there is no statistically significant autocorrelation, positive nor negative. When d is between d

_{L}and d

_{U}, we are not certain if the positive autocorrelation is statistically significant or not. The same holds for values of d greater than 2. When d is greater than (4-d

_{L}), there is statistically significant negative autocorrelation.

Table 2 Critical values for the Durbin-Watson test value d. The values are for α equal to 0.05 and when d is calculated using the residuals of a linear regression analysis. |

_{eff}to less than the actual number of measurements N. This approach is often used when calculating uncertainties associated with trends based on monthly temperatures. For the Solar Cycle Model, on average, 11 years pass between each observation (mean temperature), and the number of observations is therefore only between 10 and 15. It feels wrong to reduce the number of independent measurements, and I have not done so. SSH touch on this in [1, 2], but they also opt not to reduce N

_{eff}. If we had reduced N

_{eff}, the 95% confidence interval would have broadened, making the prediction look both better and worse: better, because measurements would be more likely to fall within the 95% confidence interval around the prediction; worse, because the greater uncertainty of the predictions would make them less likely to be useful.

## Appendix C. Statistics for the temperature series.

**pth [%]**is the probability of measuring a temperature as high as, or higher than, the one measured, provided that the model is correct. The calculation is based on the temperature measured in the solar cycle and the model's prediction of that temperature. The prediction is based on data up to, but not including, the solar cycle.

The three next values are statistical results when the model predicts the temperature in the next solar cycle. The prediction is based on data up to and including the solar cycle.**dw**The Durbin-Watson test value. See the explanation in the first paragraph of the appendix.**slope [°C/year solar cycle length]**. It tells how sensible the prediction of the temperature in the next solar cycle with respect to the length of the previous solar cycle is.**p-value**. The probability that random noise in the temperature measurements could result ina slope as large as, or larger than, the slope calculated. A value less than 0.05 is interpreted as statistical significant and written in black. A value larger than 0.05 is written in bold red.

Table 3. Solar Cycle Model Statistics. The column headings 20 to 24 are solar cycle numbers. See explanation in the text. |

## References

**2.**

**3.**

Dett er jo glimrende. Jeg er mektig imponert over arbeidet og innsatsen du har lagt ned i dette. Jeg synes du burde ta sikte på å publisere det i et av tidsskriftene SSH publisert i for snart ti år siden. Uansett får du aldri SSH til å innse/innrømme at deres prediksjoner var feil

SvarSlettTusen takk for oppmuntrende kommentar.

SlettJeg har tenkt på muligheten av å prøve å få inn en såkalt Commentary i tidsskriftet som SSH publiserte artikkelen i [2]. Men jeg både tror at det er for sent å gjøre det, og at jeg som en ukjent pensjonist ville hatt små muligheter til å få det publisert. Hvis jeg skulle gjort det, burde jeg ha gjort det i 2012 da jeg første gang jobbet med solsyklusmodellen til SSH. Allerede da var det klart at den basert på data frem t.o.m. solsyklus 23 bommet grovt på dens prediksjoner for solsyklus 24. Og at temperaturene målt frem til 2012 var mye høyere enn modellen predikterte basert på data t.o.m. solsyklus 24.

Jeg diskuterer gjerne videre oppfølging med deg på hapetja krøllalfa online punktum no.