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Solar Active Regions and the Global Temperature AnomaliesAs a matter of fact, I saw the Peak Inversion pattern in NOAA’s Global Temperature Anomaly curve already two years ago, and reported about it in a German BLog post. Although in 2019, only a single peak could be observed at the end of the GTA curve, and that was a hall mark of a GTA maximum in 2016, namely up to 1.31 °C above the average of the last century. Now in 2021, it is evident, that this single peak evolved again into the Peak Inversion pattern. And again the GTA inversion fell together with a SAR Minimum, now between the end of Solar Cycle 24 and the start of Solar Cycle 25. The red curve shows the monthly NOAA time series of global surface temperature anomalies from January 1880 to July 2021 with Δ𝜗 on the left hand yaxis ranging from 0.71 °C in January 1893 to +1.31 °C in March 2016. On the same timescale the blue curve shows the daily time series of total solar’s active regions in millionths of a hemisphere (µhsp), as it can be obtained from Solar Cycle Science, and we can see 13 peaks in (141 + 7/12) years, namely 141.6/13 = 10.9 years per cycle, i.e. the 11year solar cycle, see Fourier analysis of the daily time series of Solar Active Regions. I understand, most of you still see no correlation, perhaps the following screencast of the subsequent curve analysis may give some more insight  the application of the Digital Fourier Filter on the SAR series has been discussed in a previous article. The high frequency cut of 0.001/d corresponds to a cycle duration of 1000 days = 2.74 years, and 4·2.74 = 10,96 years. That means 1000 days are almost exactly a quarter of the solar cycle period. The screencast starts with GTA and SAR files as it were obtained from NOAA and Solar Cycle Science respectively. The timescales of the files were converted to decimal years as has been described here and here.
Note, the vertical guides are being made sticky by pressing the Here is the generated diagram, click on it in order to open the PDF file in high resolution. In general, we can observe spikes in the red GTA curve at the end of each of the falling edges and immediately at each beginning of the rising edges of the peaks in the blue SAR curve. However, there is more to discover, and we need to have a more detailed view on the gaps between the solar cycles. Let’s start with the gap between Solar Cycle 24 and the start of the currently advancing Solar Cycle 25.
Here is the generated diagram, click on it in order to open the PDF file in high resolution. 24/25 → 1.67 µhsp/d  73.6 °C·d
The blue curve is the first derivative of the Fourier filtered daily time series of Solar Active Regions. This shows the slopes of the curve. Minima and maxima of the slopes correspond to Inflection points in the SAR series and slope values of 0.0 correspond to minima and maxima in the curve itself. At the inflection point, i.e. the steepest fall of the peak of Solar Cycle 24, the slope value is 1.67 µhsp/d (pronounce: The red curve is the SavitzkyGolay smoothed monthly time series of NOAA’s Global Temperature Anomalies. SG smoothing is generally used for improving the signal to noise ratio, while at the same time not affecting too much the signal. Usually the peak heights become depleted a bit, which is the expend for the peaks becoming a bit broader. By this the peak’s integral stays almost unaffected. A window of 13 data points and a polynomial order of 4 has been applied. Higher polynomial orders give less depletion, but result also in less smoothing. Generally, it is beneficial for the signal to noise ratio to choose a higher polynomial order and run the SG smoothing twice (or even more times). Here the smoothing was done twice. Again, in the GTA curve we observe the double peaks plus two smaller inner satellite peaks, and perhaps a tiny middle peak in the gap between SAR cycle 24 and 25. The interesting fact is that the GTA peak starts to evolve exactly at the steepest fall of the SAR curve. The GTA peak has been integrated with the aid of the secant tool of the CVA application as shown in the screencast above. The integral over 457 days is 73.6 °C·d, i.e the average is 73.6/457 = 0.16 K (°C). The earth is covered by 70.8 % (361·10^{12} m^{2}) with water which has a specific heat capacity of 4.18 kJ/kg/K. For a very coarse estimation of the energy which is involved to cause an average temperature increase of 0.16 K over 457 days, we could assume that said water would be warmed up to a depth of 2 m. So, we are talking about an energy surplus of:
With that the average solar power surplus reaching the earth over said 457 days would be (W = J/s):
This coarse estimation does neither account for atmosphere&land nor does it consider any dissipation. In the following the results of the same analysis of the other 12 gaps between Solar Cycle 24 down to 12 is presented. 23/24 → 1.94 µhsp/d  33.2 °C·d 22/23 → 2.79 µhsp/d  59.4 °C·d 21/22 → 2.97 µhsp/d  11.9 °C·d 20/21 → 1.08 µhsp/d  171.6 °C·d 19/20 → 3.21 µhsp/d  53.7 °C·d 18/19 → 1.99 µhsp/d  67.1 °C·d 17/18 → 1.25 µhsp/d  97.4 °C·d 16/17 → 1.88 µhsp/d  33.0 °C·d 15/16 → 1.04 µhsp/d  65.5 °C·d 14/15 → 1.26 µhsp/d  94.2 °C·d 13/14 → 1.28 µhsp/d  142.8 °C·d 12/13 → 1.22 µhsp/d  18.7 °C·d DiscussionWith a few arguable exceptions, GTA peaks start to evolve at the very times of the inflection points of the falling edges of the SAR cycles. Usually a second and more GTA peaks can be observed in the gap between two SAR cycles.
There seems to be a rough negative correlation between the SAR peak heights/slopes (falling edge) and the GTA peak integral. The higher the SAR peaks and the steeper the f.e. slopes, the smaller the GTA integrals. For sure there are other influences, though. Furthermore, this quick & dirty integration over a secant base is quite error prone. Once the underlying correlations are more clear, it might be worth the effort to redo the integrations using more reasonable baselines. It is known that the sun flips the polarity of its magnetic field at the maximum of each cycle. Generally changes of magnetic fields induce an electromotive force in electrical conductors. Said polarity reverse must induce enormous high electrical currents in the various plasma zones surrounding the sun. Perhaps, this produces excessive amounts of resistive heat which eventually hits the earth. Copyright © Dr. Rolf Jansen  20210822 10:28:35 Discussion on Twitter: 1434160518126768135 