Multiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts
MMultiple regression analysis of anthropogenic and heliogenic climatedrivers, and some cautious forecasts
Frank Stefani a a Helmholtz-Zentrum Dresden-Rossendorf, Institute of Fluid Dynamics, Bautzner Landstr. 400, 01328 Dresden, Germany
A R T I C L E I N F O
Keywords :Climate changeSolar cycleForecast
A B S T R A C T
The two main drivers of climate change on sub-Milankovic time scales are re-assessed by means ofa multiple regression analysis. Evaluating linear combinations of the logarithm of carbon dioxideconcentration and the geomagnetic aa-index as a proxy for solar activity, we reproduce the sea surfacetemperature (HadSST) since the middle of the 19th century with an adjusted 𝑅 value of around 87 percent for a climate sensitivity (of TCR type) in the range of 0.6 K until 1.6 K per doubling of CO . Thesolution of the regression is quite sensitive: when including data from the last decade, the simultaneousoccurrence of a strong El Niño on one side and low aa-values on the other side lead to a preponderanceof solutions with relatively high climate sensitivities around 1.6 K. If those later data are excluded, theregression leads to a significantly higher weight of the aa-index and a correspondingly lower climatesensitivity going down to 0.6 K. The plausibility of such low values is discussed in view of recentexperimental and satellite-borne measurements. We argue that a further decade of data collectionwill be needed to allow for a reliable distinction between low and high sensitivity values. Based onrecent ideas about a quasi-deterministic planetary synchronization of the solar dynamo, we make afirst attempt to predict the aa-index and the resulting temperature anomaly for various typical CO scenarios. Even for the highest climate sensitivities, and an unabated linear CO increase, we predictonly a mild additional temperature rise of around 1 K until the end of the century, while for the lowervalues an imminent temperature drop in the near future, followed by a rather flat temperature curve,is prognosticated.
1. Introduction
As heir of its great pioneers Arrhenius (1906) and Cal-lendar (1938), modern climate science (Knutti et al. , 2017)has been surprisingly unsuccessful in narrowing down itsmost prominent parameter - equilibrium climate sensitivity(ECS) - from the ample range 1.5 K-4.5 K (per CO ) asalready given in the report by Charney (1979). This sober-ing scientific yield is often discussed in terms of various in-terfering socio-scientific and political factors (Hart, 2015;Lindzen, 2020; Vahrenholt and Lüning, 2020). Yet, in ad-dition to those more “subjective” reasons for climate sci-ence to be that unsettled , there are at least two “objective”ones: the lack of precise and reliable experimental measure-ments of the climate sensitivity until very recently, and theunsatisfying state of understanding the complementary so-lar influence on the climate. Certainly, a couple of mecha-nisms have been proposed (Hoyt and Schatten, 1993; Gray et al. , 2010; Lean, 2010) that could significantly surmountthe meager 0.1 per cent variation of the total solar irradi-ance (TSI) which is routinely used as an argument againstany discernible solar impact on the climate. Among thosemechanisms, the following ones figure most prominently:the comparable large variation of the UV component with itsinfluence on the ozone layer and the resulting stratospheric-tropospheric coupling (Labitzke and van Loon, 1988; Haigh,1994; Soon, Posmentier and Baliunas, 2000; Georgieva et al. ,2012; Silverman et al. , 2018; Veretenenko and Ogurtsov,2020); the effects of solar magnetic field modulated cosmic [email protected] (F. Stefani)
ORCID (s): (F. Stefani) rays on aerosols and clouds (Svensmark and Friis-Christensen,1997; Soon et al. , 2000; Shaviv and Veizer, 2003; Svens-mark et al. , 2017); downward winds following geomagneticstorms in the polar caps of the thermosphere, penetratingstratosphere and troposphere (Bucha and Bucha, 1998); so-lar wind’s impact on the global electric current (Tinsley, 2000,2008); and the (UV) radiation effects on the growth of oceanicphytoplankton (Vos et al. , 2004) which, in turn, producesdimethylsulphide, a major source of cloud-condensation nu-clei (Charlson et al. , 1987). But even the very TSI was claimed(Hoyt and Schatten, 1993; Scafetta and Willson, 2014; Egorova et al. , 2018; Connolly et al. , 2021) to have risen much moresteeply since the Little Ice Age than assumed in the con-servative estimations by Wang, Lean and Sheeley (2005);Steinhilber, Beer and Fröhlich (2009); Krivova, Vieira andSolanki (2010). While neither of those mechanism can presentlybe considered as conclusively proven (Solanki et al. , 2002;Courtillot et al. , 2007; Gray et al. , 2010), they all together en-tail significantly more potential for solar influence on the ter-restrial climate than what was discussed on the correspond-ing one and a half pages of Bindoff et al. (2013).In view of illusionary claims (Cook et al. , 2016) of anoverwhelming scientific consensus on this complex and vividlydebated research topic, and the severe political consequencesdrawn from it, we reiterate here Eugene Parker’s propheticwarning (Parker, 1999) that “...it is essential to check to whatextend the facts support these conclusions before embarkingon drastic, perilous and perhaps misguided plans for globalaction”. We also agree with his “...inescapable conclusion(...) that we will have to know a lot more about the sun andthe terrestrial atmosphere before we can understand the na-
Frank Stefani:
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Page 1 of 14 a r X i v : . [ phy s i c s . a o - ph ] J a n ultiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts ture of contemporary changes in climate”.Thus motivated, and also provoked by recent experimen-tal (Laubereau and Iglev, 2013) and satellite-borne measure-ments (Feldman et al. , 2015; Rentsch, 2019) which pointedconsistently to a rather low climate sensitivity, we make hereanother attempt to quantify the respective shares of anthro-pogenic and heliogenic climate drivers. Specifically, we re-sume the long tradition of correlating terrestrial temperaturedata with certain proxies of solar activity, as pioneered byReid (1987) for the sunspot numbers, by Friis-Christensenand Lassen (1991); Solheim, Stordahl and Humlum (2015)for the solar cycle length, and by Cliver, Boriakoff and Feyn-man (1998); Mufti and Shah (2011) for the geomagnetic aa-index (Mayaud, 1972). Our work builds strongly on the twolatter papers, which - based on data ending in 1990 and 2007,respectively - had found empirical correlation coefficientsbetween the aa-index and temperature variations of up to0.95. Notwithstanding some doubts regarding their statis-tical validity (Love et al. , 2011), such remarkably high cor-relations might rise the provocative question of whether anysort of greenhouse effect is still needed at all to explain the(undisputed) global warming over the last one and a halfcentury. More recently, however, any prospects for such a reversed simplification were dimmed by the fact that the lat-est decline of the aa-index was not accompanied by a cor-responding drop of temperature. By contrast, the latter re-mained rather constant during the first one and a half decadesof the 21st century (the “hiatus”), and even increased withthe recent strong El Niño events.This paper aims at supplementing the previous work ofCliver, Boriakoff and Feynman (1998); Pulkkinen et al. (2001);Mufti and Shah (2011); Zherebtsov et al. (2019) by takingseriously into account both observations: the nearly perfectcorrelation of solar activity with temperature over about 150years, and the notable divergence between those quantitiesduring the last two decades. Using a multiple regressionanalysis, quite similar to that of Soon, Posmentier and Bali-unas (1996), but with the time series of the aa-index as thesecond independent variable (in addition to the logarithmof CO concentration), we will show that the temperaturevariation since the middle of the 19th century can be repro-duced with an (adjusted) 𝑅 value around 87 per cent. Sucha goodness-of-fit is achieved for specific combinations of theweights of the aa-index and of CO that form a nearly lin-ear function in their two-dimensional parameter space. Bestresults are obtained for a climate sensitivity in the range be-tween 0.6 K-1.6 K (per 2 × CO ), with a delicate dependenceon whether the latest data are included or not. Derived fromempirical variations on the (multi-)decadal time scale, thisclimate sensitivity should be interpreted as a transient cli-mate response (TCR), rather than an ECS. Our range cor-responds well with that of Lewis and Curry (2018), 0.8 K-1.3 K, but is appreciably lower than the “official” 1.0 K-2.5 Krange (Knutti et al. , 2017). The lower edge of our estima-tion will be plausibilized by recent experimental (Laubereauand Iglev, 2013) and satellite-borne measurements (Feld-man et al. , 2015; Rentsch, 2019). It is also quite close, al- though still higher, then the particularly low estimate of lessthan 0.44 K, as advocated by Soon, Conolly and Conolly(2015) after comparing exclusively rural temperature datain the Northern hemisphere with the TSI. The upper edge,in turn, is not far from the spectroscopy-based estimation byWijngaarden and Happer (2020).With the complementary share of the Sun for global warm-ing thus reaching values between 30 and 70 per cent, anyclimate forecast will require a descent prediction of solar ac-tivity. This leads us into yet another controversial playingfield, viz, the predictability of the solar dynamo. While theexistence of the short-term Schwabe/Hale cycles is a truismin the solar physics community, the existence and/or stabil-ity of the mid-term Gleissberg and Suess-de Vries cycles arealready controversially discussed, and there is even more un-certainty about long-term variations such as the Eddy andHallstatt “cycles”, which are closely related to the sequenceof Bond events (Bond et al. , 2001).In a series of recent papers (Stefani et al. , 2016, 2017,2018; Stefani, Giesecke and Weier, 2019; Stefani et al. , 2020a,b;Stefani, Stepanov and Weier, 2020), we have tried to developa self-consistent explanation of those short-, medium- andlong-term solar cycles in terms of synchronization by plan-etary motions. According to our present understanding, thesurprisingly phase-stable 22.14-year Hale cycle (Vos et al. ,2004; Stefani et al. , 2020b) results from parametric reso-nance of a conventional 𝛼 − Ω dynamo with an oscillatorypart of the helical turbulence parameter 𝛼 that is thoughtto be synchronized by the 11.07-year spring-tide periodic-ity of the three tidally dominant planets Venus, Earth, andJupiter (Stefani et al. , 2016, 2017, 2018; Stefani, Gieseckeand Weier, 2019). The medium-term Suess-de Vries cycle(specified to 193 years in our model) emerges then as a beatperiod between the basic 22.14-year Hale cycle and some(yet not well understood) spin-orbit coupling connected withthe motion of the Sun around the barycenter of the solar sys-tem that is governed by the 19.86-year synodes of Jupiter andSaturn (Stefani et al. , 2020a; Stefani, Stepanov and Weier,2020). Closely related to this, some Gleissberg-type cyclesappear as nonlinear beat effects and/or from perturbationsof Sun’s orbital motion from other synodes of the Jovianplanets. Finally, the long-term variations on the millennialtime-scale (Bond events) arise as chaotic transitions betweenregular and irregular episodes of the solar dynamo (Stefani,Stepanov and Weier, 2020), in close analogy with the super-modulation concept introduced by Weiss and Tobias (2016).Being well aware of the conjectural nature of this syn-chronized solar dynamo model, we nevertheless dare to makea cautious prediction of the aa-index for the next 130 years,based on some simple 3-frequency fits to the aa-index dataover the last 170 years. While our choice for three frequen-cies is motivated by the dominance of one Suess-de Vriesand two Gleissberg-type cycles, we employ different ver-sions of fixing or relaxing their frequencies, which leads to acertain variety of forecasts. A common feature of all of themis, however, a noticeable decline of solar activity until 2100,and a recovery in the 22nd century. Those predictions for Frank Stefani:
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Page 2 of 14ultiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts the aa-index are then combined with three different scenar-ios of CO increase, including an unfettered annual increaseby 2.5 ppm and two further scenarios based on hypotheticaldecarbonization schemes. The models thus obtainedare then blended with the different combinations of weightsfor the aa-index and CO as derived before in the regres-sion analysis. For the “hottest” scenario we predict an addi-tional temperature increase until 2100 of less than K, whileall other combinations lead to less warming, partly even tosome imminent cooling, followed by a rather flat behaviourin which decreasing solar activity and a mildly increasingtrend from CO compensate each other to a large extend.
2. Multiple regression analysis
In this section, we perform a multiple (or better: dou-ble) regression analysis of the temperature data (dependentdata) on the geomagnetic aa-index and the logarithm of theCO (independent data). We do this in an intuitive and easilyreproducible way by showing the fraction of variance unex-plained ( FVU ) , i.e. the ratio of the residual sum of squaresto the total sum of squares, whose minimum is then identi-fied. From those FVU ’s we will derive the corresponding 𝑅 value, both in its usual and in its adjusted variant. Let usstart, however, with a description of our data base. While reliable CO data are available for quite a longtime, we decided to restrict our data base to the time fromthe middle of the 19th century, for which both temperaturedata and the aa-index are readily available. et al. ,2019). Actually, these data are not gravely different from thecombined sea/land surface temperature (HadCRUT), apartfrom some slight but systematic divergence during the lasttwo decades. At this point, our preference for HadSST is alsosupported by their better agreement with the UAH satellitedata (starting only in 1978) with their significantly broaderspatial coverage.The HadSST data are shown as open circles in Fig. 1a,together with two exemplary centered moving averages withwindows 11 years (full line) and 23 years (dashed line), whichwe will frequently refer to in this paper. These curves showthe typical temporal structure comprising a slow decay be-tween 1850 and 1905, a rather steep rise between 1905 until1940, again a mild decay until 1970, followed by a steep in-crease until 1998. As for the last two decades, we first see the“hiatus” between 1999 and 2014, being then overwhelmed Figure 1:
Data of the HadSST sea surface temperatureanomaly Δ 𝑇 (a), the aa-index (b), and log of the ratio ofthe CO concentration to the reference value of 280 ppm (c).The annual data between 1850-2018 are complemented by cen-tered moving averages with windows 11 years and 23 years, asalso utilized by Mufti and Shah (2011). The sources of thedata are described in the text. by the recent strong El Niño events. We will come back tothose latest years further below. The aa-index measures the amplitude of global geomag-netic activity during 3-hour intervals at two antipodal mag-netic observatories, normalized to geomagnetic latitude ±50 ◦ (Mayaud, 1972). Inspired by the work of Cliver, Boriakoffand Feynman (1998) and Mufti and Shah (2011) who pointedout the remarkable correlation of up to 0.95 between (timeaveraged) temperature anomalies and the geomagnetic aa-index, we will use the latter data as a proxy for solar ac-tivity. A viable alternative would have been to use sunspotdata, or various versions of the TSI, as exemplified by Soon,Conolly and Conolly (2015). Given, on one side, their gen-erally high correlation with sunspots numbers (Cliver, Bori-akoff and Bounar, 1998), and, on the other side, their high re-liability based on precise measurement down to 1844 (whichavoids some ambiguities concerning the correct variabilityof the TSI (Soon, Conolly and Conolly, 2015; Connolly et al. ,2021)), we focus here exclusively on the aa-index, leaving Frank Stefani:
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Page 3 of 14ultiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts aa data are shown in Fig 1b. Already by visual inspection,between 1850 and 1990 we observe a remarkable similarityof their shape with that of temperature, while after 1995 theaa-index steeply declines, whereas the temperature contin-ues to increase. We will have more to say on that divergencefurther below. data The CO After having presented the data that actually will be usedin the remainder of this paper, we make a short break toconsider whether these are indeed the most relevant data.Our reliance on the aa-index might indeed be questioned, asother studies (Vahrenholt and Lüning, 2020) have claimed astrong temperature dependence on ocean-atmosphere varia-tions, such as the Pacific Decadal Oscillation (PDO) (Man-tua and Hare, 2002) and the Atlantic Multidecadal Oscilla-tion (AMO) (Wyatt, Kravtsov and Tsonis, 2012), with theirsimilar time structures governed by a sort of 60-70-year “cyclic-ity”. On the other hand, there is also evidence for direct cor-relations of the aa-index with regional features, such as theNorthern Annular Mode (NAM) (Roy et al. , 2016). In orderto make contact with those possible links, in Fig. 2 we showexemplarily the 23-year averages of the AMO and the PDOdata, together with the previously shown aa-index (appropri-ately shifted and scaled) and Δ 𝑇 . It is clearly seen that theaa-index and Δ 𝑇 have a particularly parallel behaviour until1990, say. There is also some similarity with AMO, whilePDO has a different time dependence.In Table 1 we quantify those relationships in terms of theempirical correlation coefficients 𝑟 for different data combi-nations. We do so for different end points of the time inter-val, namely 2008, 2003 and 1998 (note that these are the cen-tered points of the last moving average interval, into whichthe annual data from up to 11 years later are included). Ourfirst observation is that both the correlations of CO and ofthe aa-index with Δ 𝑇 have similar 𝑟 -values in the order of0.9, which is a first indication for their comparable influ-ences. However, there are some subtleties to discern: thecorrelation for CO acquires its highest value ( 𝑟 = 0 . ) forthe full time interval, and decreases slightly to 0.869 whenthe time interval is shortened by 10 years. By contrast, thecorrelation of aa-index with Δ 𝑇 has only a value 𝑟 = 0 . Figure 2:
Comparison between the centered moving averagesover 23 years of the four data sets Δ 𝑇 , aa-index, AMO-indexand PDO-index. Note the remarkable parallelity of Δ 𝑇 andaa-index until 1990, and the divergence thereafter. The AMO-index has also some similarity with Δ 𝑇 , while the PDO indexis significantly different. for the full interval, but grows to 0.95 for the restricted inter-val. This latter result confirms that of Mufti and Shah (2011)obtained for a similar period, and by Cliver, Boriakoff andFeynman (1998) for a still shorter interval. Given the visualsimilarity of AMO and Δ 𝑇 in Fig. 2, their correlation issurprisingly small, but would increase if the overall upwardtrend of Δ 𝑇 were subtracted. The PDO seems to show norelevant correlation with Δ 𝑇 . Table 1
Empirical correlation coefficients 𝑟 between different data sets,each of which represents a centered moving average over 23years. Note the large values for the correlation both betweenCO and the aa-index with Δ 𝑇 , compared to weak or barelyexisting correlations of the AMO- and PDO-index with Δ 𝑇 .Correlated data 1867-2008 1867-2003 1867-1998CO with Δ 𝑇 Δ 𝑇 Δ 𝑇 Δ 𝑇 -0.131 -0.001 0.126aa with PDO 0.050 0.049 0.074 In Fig. 3 we show some further correlation dependen-cies, this time on the time shift 𝛿𝑡 between the two respec-tive data, which possibly could give a clue about intrinsicdelay effects. One curve shows the correlation between theaa-index and the AMO index, whereby the aa-index at ear-lier times is correlated with AMO at later times. While thecorrelation is not large, we see at least a clear maximum at 𝛿𝑡 = 11 years, as if the AMO-index lagged behind the aa-index by this delay time. The second curve shows the corre-sponding relationship between Δ 𝑇 and AMO, with the cor-relation reaching a maximum of 0.3 at 𝛿𝑡 = 6 years. Whilethis looks like a sort of inverted causality (AMO lags behind Δ 𝑇 ) it could simply mean that Δ 𝑇 is indeed governed bythe aa-index, which also determines the AMO-index at latertimes.We also show the corresponding curves for aa-index and Frank Stefani:
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Page 4 of 14ultiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts
Figure 3:
Correlations between various data, each representinga 23-years centered moving average, in dependence on the timeshift 𝛿𝑡 between them. In each case, the first item indicatesthe earlier data set, the second the later one. The aa- Δ 𝑇 correlation starts at the value 𝑟 = 0 . for 𝛿𝑡 = 0 and reachesa maximum of 𝑟 = 0 . for 𝛿𝑡 = 10 years. While this mightinsinuate a 10-years causal time shift between aa-index and Δ 𝑇 , it is more likely connected with the canceling effect of thelatest decade with its poor correlation. In the second curve,for which we have completely omitted the last 10 years of bothdata, we get 𝑟 = 0 . at 𝛿𝑡 = 0 , increasing to a maximum of 𝑟 = 0 . at 𝛿𝑡 = 3 years. Compared to those large 𝑟 -values,the correlation of Δ 𝑇 and AMO is much smaller, viz 0.3 at 𝛿𝑡 = 6 years (AMO lagging behind temperature). Still smalleris the correlation of aa-index and AMO, with a maximum of 𝑟 = 0 . for 𝛿𝑡 = 11 years. Δ 𝑇 , in the two versions for the full time interval and the10-year shortened one. For the full interval, 𝑟 starts at thevalue 0.8 for 𝛿𝑡 = 0 , but reaches a maximum of 0.915 for 𝛿𝑡 = 10 years. While on face value this seems to indicate a10-years lag between the aa-index and Δ 𝑇 , it is more likelyconnected with the implied cancellation of the last 10 yearsduring which the aa-index was decreasing. In order to testthis, in the second curve we have omitted the last 10 years ofboth data completely. Here we find an 𝑟 = 0 . at 𝛿𝑡 = 0 ,which still increases to a maximum of 𝑟 = 0 . at 𝛿𝑡 = 3 yr.This sounds indeed like a reasonable time delay betweencause (aa-index) and effect ( Δ 𝑇 ). After those preliminaries, we start now with the multipleregression analysis. For that purpose, we model the temper-ature data using the ansatz Δ 𝑇 model = 𝑤 aa ⋅ aa + 𝑤 CO ⋅ log (CO ∕280 ppm) with the respective weights 𝑤 aa and 𝑤 CO for solar and CO forcing, and compare them with the measured data Δ 𝑇 meas .This procedure is very similar to that of Soon, Posmentierand Baliunas (1996) who had used though, instead of aa , thelength of the sunspot cycle, the averaged sunspot number,and a more complicated composite as proxies of solar irradi-ance. While Soon, Posmentier and Baliunas (1996) translateall these proxies into some percentage of TSI variation (fix-ing their “stretching factor” by the best fit of modeled and ob-served temperature history), we stick here to the somewhatweird unit K/nT for 𝑤 aa , without specifying in detail the physical mechanism(s) underlying the solar-climate connec-tion. In case of a dominant Svensmark effect, say, this unitwould indeed have an intuitive physical meaning, whereas incase of a dominant UV radiation impact on the ozone layer(plus subsequent coupling of stratosphere and troposphere),it would just represent a very indirect, co-responding proxy.One of the non-trivial questions to address beforehand iswhich time average should be used. Evidently, the structuresof temporal fluctuation of the three data are quite different.As for the independent data, the CO curve is the smoothestone, so the results of any regression will be widely inde-pendent on the widths of the averaging window. Much morefluctuating is the aa-index, with its dominant 11-year period-icity which had been taken into account, though in differentways, by Cliver, Boriakoff and Feynman (1998) and Muftiand Shah (2011). Cliver, Boriakoff and Feynman (1998)have been working both with a decadal average and the so-called aa-baseline ( 𝑎𝑎 min ), i.e. the minimum value whichgenerally occurs within one year following the sunspot min-ima. The empirical correlation coefficient of 𝑟 = 0 . be-tween 𝑎𝑎 min and the 11-year average temperature turned outto be even better than that between two decadal averages( 𝑟 = 0 . ). Mufti and Shah (2011), in turn, worked with 11-year and 23-year averages, which will also serve as a firstguidance for our study, though later we will consider the de-pendence on the widths of the averaging windows in a morequantitative manner.The dependent variable, i.e. Δ 𝑇 , is also characterizedby significant fluctuations, although not with the dominant11-year periodicity of the aa-index whose climatic impact isthought to be smoothed out by the large thermal inertia of theoceans. By contrast, Δ 𝑇 is strongly influenced by short-termvariations due to the El Niño-Southern Oscillation (ENSO)and volcanism, which had been considered in the multipleregression analysis of Lean and Rind (2008). Our deliber-ate neglect of those short-term variations, and the focus on(multi-)decadal variations, thus requires some averaging onthe decadal time scale.In addition to that distinction between different averag-ing windows, we will also consider three cases with differentend-years of the utilized data. In the first case, we take intoaccount all data until 2018. As evident from Fig. 1, it is inparticular the last decade which shows the strongest discrep-ancy between the decreasing aa-index and the partly stagnant(“hiatus”), partly increasing temperature (in particular dur-ing the last El Niño dominated years). Hence, in order toasses the specifics of this divergence between the two data,and to compare them with the high correlations found byCliver, Boriakoff and Feynman (1998) and Mufti and Shah(2011), we will also consider two shortened periods with endyears 2013 and 2008, respectively.Let us start, however, with the full data ending in 2018.For the two moving average windows (abbreviated hence-forth as “MAW”) of 11 years and 23 years, Figs. 4a and4b show the fraction of variance unexplained ( FVU ) , i.e.the ratio of the residual sum of squares to the total sum ofsquares, in dependence on the respective weights of the aa- Frank Stefani:
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Page 5 of 14ultiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts index ( 𝑤 aa , on the abscissa) and the logarithm of CO ( 𝑤 CO ,on the ordinate axis). The latter value provides us immedi-ately with a sort of instantaneous climate sensitivity of theTCR type. The minimum FVU min = 0 . is obtained forthe weights’ combination 𝑤 aa = 0 . K/nT and w CO =1 . K in case of
MAW = 11 years, and
FVU min = 0 . is obtained for 𝑤 aa = 0 . K/nT and w CO = 1 . K incase of
MAW = 23 years. From the ellipse-shaped contourplots of
FVU we see that those minima reflect a dominat-ing influence of CO over the aa-index. The two red curvesin Fig. 6c show now the corresponding temperature recon-structions based on those optimized values of 𝑤 aa and w CO ,for MAW = 11 years (full line) and for
MAW = 23 years(dashed line). For both lines we obtain a reasonable fit ofthe general upward trend of Δ 𝑇 , but a poor reconstruction ofits oscillatory features. This clearly corresponds to the com-parable high value of w CO compared to that of 𝑤 aa . Anyputative higher share of 𝑤 aa would lead to a drastic decreaseof the reconstructed Δ 𝑇 for the last two decades, resultingin forbiddingly large FVU values when compared with therelatively high observed Δ 𝑇 in this late period.This brings us to the question of what happens if we ex-clude the latest “hot” 5 years (with their strong El Niño in-fluence), thus restricting the date until 2013 only. The corre-sponding results are shown in Fig. 5. Obviously, the optimalweights’ combination now shifts away from CO to aa , withvalues 𝑤 aa = 0 . K/nT and w CO = 1 . K for
MAW =11 years and 𝑤 aa = 0 . K/nT and w CO = 1 . K for
MAW = 23 years. Evidently, the resulting (green) temper-ature reconstruction curves in Fig. 5c appear now more os-cillatory.In Fig. 6 we show the corresponding plots for the casethat we use the end year 2008, which basically correspondsto the database of Mufti and Shah (2011) (2007 in their case).Evidently, the regression for this shortened segment leads toa significantly stronger weight for the aa-index. The min-imum
FVU is than obtained at 𝑤 aa = 0 . K/nT and w CO = 1 . K for
MAW = 11 years, and at 𝑤 aa = 0 . K/nTand w CO = 0 . K for
MAW = 23 years. Due to the domi-nance of 𝑤 aa the (blue) reconstruction curves in Fig. 6c (inparticular that for MAW = 23 years) show now a significantoscillatory behaviour.While those three examples provide a first illustrationof how sensible the solution of the regression reacts on thechoice of the end year, and the widths of the
MAW , thelatter dependence will now be studied in more detail. Forthat purpose, we analyze first the coefficient of determina-tion 𝑅 , which is related to the previously used FVU accord-ing to 𝑅 = 1 − FVU . The corresponding dashed curvesin Fig. 7a have a rather universal shape, starting from val-ues of 0.78...0.84 for MAW = 3 years to around 0.94 for
MAW = 39 years. The monotonic increase of 𝑅 , which atfirst glance might suggest the use of high values of MAW ,should be treated with caution. The reason is that a signif-icant share of this increase is just due to the increasing ra-tio of explanatory terms 𝑝 (in our case two: 𝑤 aa and w CO )to the “honest” number of data points 𝑛 . The latter is not Figure 4:
Regression analysis for the full data (with end year2018). (a)
FVU in dependence on 𝑤 aa and w CO for MAW =11 years, with
FVU min = 0 . reached for 𝑤 aa = 0 . K/nTand w CO = 1 . K. (b) The same for
MAW = 23 years, with
FVU min = 0 . reached for 𝑤 aa = 0 . K/nT and w CO =1 . K. (c) 11-year and 23-year moving averages for the original Δ 𝑇 (purple) and for the reconstructed Δ 𝑇 (red) when usingthe optimized values of 𝑤 aa and w CO from (a) and (b). identical to the number of considered years, 𝑁 y , but - dueto the moving average - approximately equal to 𝑁 𝑦 ∕MAW (better estimates, using the lag-one auto-correlation 𝑟 (Love et al. , 2011), might still be worthwhile). To correct for thiseffect, we use the so-called adjusted 𝑅 , which is, in general, 𝑅 = 1 − FVU × ( 𝑛 − 1)∕( 𝑛 − 𝑝 − 1) , hence in our specialcase 𝑅 = 1 − FVU × ( 𝑁 𝑦 ∕MAW − 1)∕( 𝑁 𝑦 ∕MAW − 3) .This adjusted 𝑅 , as a much more telling coefficient of de-termination than 𝑅 , is shown with full lines in Fig. 7a. Themonotonic increase still seen for 𝑅 gives now way to a morestructured curve. For the green (data until 2013) and theblue curve (data until 2008) we observe a local (if shallow) Frank Stefani:
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Figure 5:
Regression analysis for the reduced data (end year2013). (a)
FVU in dependence on 𝑤 aa and w CO for MAW =11 years, with
FVU min = 0 . reached for 𝑤 aa = 0 . K/nTand w CO = 1 . K. (b) The same for
MAW = 23 years, with
FVU min = 0 . reached for 𝑤 aa = 0 . K/nT and w CO =1 . K. (c) 11-year and 23-year moving averages for the original Δ 𝑇 (purple) and for the reconstructed Δ 𝑇 (red) when usingthe optimized values of 𝑤 aa and w CO from (a) and (b). maximum around MAW = 25 years. The red curve (datauntil 2018) has a very flat plateau between
MAW = 11 and
MAW = 27 years, with an extremely shallow maximum at
MAW = 25 years. We consider those local maxima around
MAW = 25 years (
MAW = 27 years for the blue curve) as asort of best fits. Although 𝑅 still rises slightly for the high-est MAW values, the corresponding number of explainedvariables becomes then too small to allow for a decent re-construction of the structure of the Δ 𝑇 curve.The corresponding dependencies for 𝑤 aa and w CO areshown in Fig. 7b and 7c, respectively. Fig. 7d depictsthe same solutions in the two-dimensional parameter space.In this representation, all three curves (red, green and blue) Figure 6:
Regression analysis for the reduced data (end year2008). (a)
FVU in dependence on 𝑤 aa and w CO for MAW =11 years, with
FVU min = 0 . reached for 𝑤 aa = 0 . K/nTand w CO = 1 . K. (b) The same for
MAW = 23 years, with
FVU min = 0 . reached for 𝑤 aa = 0 . K/nT and w CO =0 . K. (c) 11-year and 23-year moving averages for the original Δ 𝑇 (purple) and for the reconstructed Δ 𝑇 (red) when usingthe optimized values of 𝑤 aa and w CO from (a) and (b). form a sort of common, weakly bent line which appears toconnect the extremal values of w CO ≈ 1 . K on the ordinateaxis and 𝑤 aa ≈ 0 . K/nT on the abscissa. This commonline is just a reflection of the ellipsoidal shape of the
FVU as shown in Figs. 3-5 which, in turn, just reflects the sig-nificant ill-posedness of the underlying inverse problem. Toput it differently: due to the high correlation coefficients of Δ 𝑇 both with CO as well as with the aa-index, the separateshares of the two ingredients are very hard to determine.However questionable our fixation on the shallow max-ima at MAW ≈ 25 years might ever be: Fig. 7 also provesthat any alternative choice of a higher
MAW would not changethe final solution drastically. In either case, both 𝑤 aa and Frank Stefani:
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Results of the regression in dependence on the widthof the moving average window (
MAW ). (a) 𝑅 and its adjustedversion 𝑅 , each for the three time intervals ending in 2018,2013, 2008. The (shallow) local maxima of 𝑅 around MAW =25 years are indicated by full symbols. (b) 𝑤 aa in dependenceon MAW . The full symbols are the values corresponding tothe local maxima in (a). (c) same as (b), but for w CO . (d)Regression result in the two-dimensional parameter space of 𝑤 aa and w CO . Note the universal shape of the solution given aslightly bent, but nearly linear function connecting the extremalvalues w CO ≈ 1 . K on the ordinate axis and 𝑤 aa ≈ 0 . K/nTon the abscissa.Frank Stefani:
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Figure 8:
Original Δ 𝑇 data, averaged over 25 years, and re-constructions based on the optimal combinations of 𝑤 aa and w CO from Fig. 7. The dashed segments of the green and bluecurves indicate those time intervals that did not enter in therespective regressions. From the red over green to blue curve,we see an ever improving reconstruction of the oscillatory be-haviour, and an ever increasing divergence with the observeddata at later years. forcing of 0.358 W/m , corresponding to 2.63 W/m (per 2 × CO ), i.e. to 0.71 K (with 3.7 K/(W/m )) or 0.82 K (with3.2 K/(W/m )), in nearly perfect agreement with Feldman et al. (2015).These three papers are widely consistent among each other,giving sensitivities in the range between 0.64 K and 0.82 K,which is close to the lower edge of our estimate. We alsonote that this end of our regression, which corresponds toapproximately 70 per cent “for the sun”, is similar to the 50-69 per cent range as once found by Scafetta and West (2007,2008).This said, we should also note that Wijngaarden and Hap-per (2020) have found the much higher value of 1.4 K value(at fixed absolute humidity) which would fit to our upperlimit of 1.6 K value which is, in turn, significantly lower thantheir value 2.2-2.3 K, as inferred for fixed relative humidity.The range of our estimates is slightly sharper as the range0.4 K to 2.5 K of Soon, Conolly and Conolly (2015) (with thehigh value deemed unrealistic by the authors), and slightlywider than the 0.8 K-1.3 K range of Lewis and Curry (2018),but in either case quite consistent with those estimations.
3. Predictions
In the preceding section we have derived a certain plau-sible range of combinations of the respective weights of theaa-index and the logarithm of CO by means of regressionanalysis of data from the past 170 years. In the following wewill leave the realm of solid, data-based science and enterthe somewhat “magic” realm of predictions. Given all theunderlying uncertainties concerning the future time depen-dence of the aa-index, of CO , and of further climate factorssuch as AMO, PDO, ENSO, volcanism etc., any forecast hasto be taken with more than one grain of salt. This said, we will at least do some parameter studies, by allowing the un-known time series of the aa-index and CO , and their respec-tive weights, to vary in some reasonable range. Let us startwith the aa-index. This subsection is definitely the most speculative one ofthis paper as it is concerned with forecasts of the aa-indexfor the next 130 years. There is no doubt that the aa-indexis strongly correlated with the sunspot number (SSN), withtypical correlation coefficients of around 𝑟 = 0 . when av-eraged over one cycle Cliver, Boriakoff and Bounar (1998).Neither is there any doubt that the SSN and the aa-index areboth governed (though in a non-trivial manner) by the solardynamo. So any prediction of the aa-index boils down to aprediction of the solar dynamo which many researchers be-lieve to be impossible, at least beyond the horizon of the verynext cycle for which reasonable (though not undisputed) “pre-cursor methods” exist (Svalgaard, Cliver and Kamide, 2005;Petrovay, 2010).In order to justify our audacious forecast for the aa-index,we have to make a little diversion on the solar dynamo and itsshort-, medium- and long-term cycles. The reader should bewarned, though, that our arguments do not reflect the main-stream of solar dynamo theory. We think nevertheless thatthe last years have brought about sufficient empirical evi-dence that justifies at least a cautious try.Let us start with some recent evidence concerning thephase stability of the Schwabe cycle, a matter that was firstdiscussed by Dicke (1978). In Stefani et al. (2020b) wehave reviewed the pertinent results derived from algae datain the early Holocene (Vos et al. , 2004) and from sunspotand aurorae borealis observations, combined with C and Be data, from the last centuries. Without going into thedetails it was shown that the Schwabe cycle is very likelyphase-stable at least over some centuries, with a period be-tween 11.04 years and 11.07 years. While certain nonlin-ear self-synchronization mechanisms of the solar dynamocannot be completely ruled out as an explanation (Hoyng,1996), the external synchronization by the 11.07-years pe-riodic spring tides of the (tidally dominant) Venus-Earth-Jupiter system provides a suspiciously compelling alterna-tive. Based on previous observations and ideas of Hung(2007); Wilson (2008); Scafetta (2012); Wilson (2013); Okhlop-kov (2016), we have corroborated a model (Weber et al. ,2015; Stefani et al. , 2016, 2017, 2018; Stefani, Giesecke andWeier, 2019) in which the weak tidal forces of Venus, Earthand Jupiter serve only as an external trigger for synchroniz-ing (via parametric resonance) the intrinsic helicity oscilla-tions of the kink-type Tayler instability in the tachocline re-gion. The arising 11.07-yr period of the helicity parameter 𝛼 ultimately leads to the 22.14 year period of the Hale cycle.Building on this phase coherence of the Hale cycle, later(Stefani et al. , 2020a; Stefani, Stepanov and Weier, 2020)we exploited ideas of Wilson (2013); Solheim (2013) to ex-plain the mid-term Suess-de Vries cycle as a beat period be-tween the 22.14-yr Hale cycle and the 19.86-yr synodic cy- Frank Stefani:
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Figure 9:
Climate predictions until 2150. (a) 23-year mov-ing average of the aa-index, and three 3-frequency fits to it,extrapolated until 2150. (b) Three scenarios for CO concen-tration. (c) Temperature forecasts for the 1st CO scenario(dashed curve in (b)), for the three pairs of 𝑤 aa and w CO re-sulting from regression with end year 2018 (red), 2013 (green),2008 (blue). Each of the coloured bundles comprise the threedifferent 3-frequency fits from (a). (d) Same as (c), but forthe second CO scenario with a mild decarbonization scheme(dotted line in (b)). (e) Same as (c), but for third CO scenariowith a radical decarbonization (dash-dotted line in (b)). cle of Jupiter and Saturn which governs the motion of theSun around the barycenter of the solar system. Note that,apart from first ideas (Javaraiah, 2003; Sharp, 2013; Sol-heim, 2013; Wilson, 2013), the spin-orbit coupling that isnecessary to translate the orbital motion of the Sun into somedynamo-relevant internal forcing, is yet far from understood.In our model, the Suess-de Vries acquires a clear (beat) pe-riod of 193 years which is in the lower range of usual esti-mates (but see Ma and Vaquero (2020)). The situation withthe Gleissberg cycle(s) was less clear: those appeared asdoubled and tripled frequencies of the Suess-de Vries cy-cle, but also as independent frequencies resulting from beatperiods of other synodes of Jovian planets with the Schwabecycle (see Fig. 10 in Stefani, Stepanov and Weier (2020)).Further below, this vagueness of the Gleissberg cycle(s) willbe factored in when fitting and extrapolating the aa-index.Lately, in Stefani, Stepanov and Weier (2020) we have triedto explain the transitions between regular and irregular inter-vals of the solar dynamo (the “supermodulation” as definedby Weiss and Tobias (2016)) in terms of a transient route tochaos.With those preliminaries, we will now fit the aa-indexover the last 170 years in order to extrapolate it into the fu-ture. We assume that we have safely left the irregular periodof the solar dynamo, as reflected in the Little Ice Age whichcan be considered as the latest link in the (chaotic) chain ofBond events (Bond et al. , 2001). Guided by our (double-)synchronization model, and encouraged by Ma and Vaquero(2020) who had indeed derived an 195-yr cycle in the quiet(regular) interval from 800-1340, we keep this Suess-de Vriesperiod fixed to 193 years in all fits. Concerning the Gleissberg-type cycle(s) we will be less strict, though. In the first ver-sion, we fix the two (half and tripled) periods of 96 and 65years, for the second one we only fix the 96 years period,and for the third one we will keep both Gleissberg cyclesundetermined.In the interval between 1850-2150, the results of thosethree different 3-frequency fits are presented in Fig. 9a, to-gether with the (fitted) 23-year averaged aa-index data be-tween 1850-2007. Evidently, all three curves approximatethe original data reasonably well, and all show a similar ex-trapolation with a long decay until 2100, and a recovery af-terwards. This behaviour is quite similar to the predictionof Bucha and Bucha (1998) (their Figure 14) as well as withthe variety of predictions by Lüdecke, Weiss and Hempel-mann (2015). The differences between the three fits mainlyconcern the high frequency part; whereas the version withthree frequencies being fixed is the smoothest, the other twofits comprise some stronger wiggles. This simply reflects thefact that the Gleissberg cycle is more vague than the Suess-de Vries cycle. In the following we will always work withall three curves obtained, hoping that they constitute a rep-resentative variety for the future of the aa-index. Thus prepared, we consider now three different CO sce-narios (Fig. 9b), for each of which we further take into ac- Frank Stefani:
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Page 10 of 14ultiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts count the three predictions for the aa-index as just discussed,as well as the three optimal combinations of weights for theaa-index and CO as derived in the previous section. Delib-erately, we restrict our forecasts until 2150, admitting thatneither the CO trend nor the aa-index are seriously pre-dictable beyond that horizon.Let us start with the simple case of an unabated linearextrapolation of the recent CO trend to rise by 2.5 ppm an-nually (upper curve in Fig. 9b), which would bring us toa value 736 ppm in 2150 (still steeper trends are not com-pletely ruled out, but perhaps not that realistic given the re-cent worldwide reduction commitments). For this scenario,Fig. 9c shows three red, green, and blue bundles of curves,each comprising the three different fits of the aa-index asdiscussed above. The red bundle (“hot”) corresponds to thered solution in Fig. 7, with a rather high CO -sensitivity of1.5 K and an aa-sensitivity of 0.017 K/nT. The green bun-dle (“medium”) corresponds to the green solution of Fig. 7based on sensitivities of 1.2 K and 0.022 K/nT. The blue bun-dle (“cool”) corresponds to 0.6 K and 0.032 K/nT.We see that in 2100 the “hot” variant leads to a temper-ature anomaly of Δ 𝑇 = 1 . K which is . K above the av-erage Δ 𝑇 ≈ 0 . K from the first decade of this century. The“medium” and “cool” curves are flatter, leading to Δ 𝑇 =0 . K and Δ 𝑇 = 0 . K, respectively. However, for thosecases to be of any relevance, an imminent drop of the temper-ature would be required to reach the corresponding curvesbefore they can continue as flat as shown.The second scenario (dotted line in Fig. 9b) assumesa slowly decreasing upward trend of the CO concentrationtowards the end of the 21st century, with a hypothetical timedependence CO = [382 + 2 . 𝑡 − 2007)∕(1 . + ( 𝑡 − 2007)∕100)] ppm reaching a final value of 529 ppm in 2150. The resulting red,green and blue curves in Fig. 9d show a rather flat behaviour.The last (and rather unrealistic) CO scenario (dash-dottedline of Fig. 9b) assumes a radical decarbonization path with CO = [382 + 2 . 𝑡 − 2007)∕(1 . + ( 𝑡 − 2007)∕50)] ppm . The resulting temperature curves (Fig. 9e) are basically con-stant throughout the end of our forecast horizon.
4. Conclusions
This work has revived the tradition of correlating so-lar magnetic field data with the terrestrial climate as pio-neered by Cliver, Boriakoff and Feynman (1998) and Muftiand Shah (2011). Just as these authors, we have found anempirical correlation coefficient between the aa-index and Δ 𝑇 with remarkably high values ranging from 0.8 until 0.96,which points to a significant influence of solar variability onthe climate. Our modest innovation was to employ a multi-ple (double) regression analysis, with the logarithm of atmo- While we do not refer here to IPCC’s Representative ConcentrationPathways, this case has some similarity to their RCP 6.0, at least until 2100. spheric CO concentration as the second independent vari-able, whose pre-factor corresponds to an (instantaneous) cli-mate sensitivity of the TCR type. For a lengths of the cen-tered moving average window of 25 years we have identifiedoptimal parameter combinations leading to adjusted 𝑅 val-ues of around 87 per cent. Depending on whether to includeor not include the data from the last decade, the regressiongave climate sensitivity values from 0.6 K up to 1.6 K (per CO ), and values from 0.032 K/nT down to 0.017 K/nTfor the corresponding sensitivity on the aa-index.Ironically, if interpreted as . . K, the derived cli-mate sensitivity range turns out to have (nearly) the sameample 50 per cent error bar as the “official” ECS value ( . K), which we had criticized in the introduction. Yet, inview of the impressive 95 per cent correlation (obtained forthe restricted period 1850-2008) we believe the correct cli-mate sensitivity to be situated somewhere in the lower half ofthat range. This “bias” is also supported by the fact that dur-ing the last years an intervening strong El Niño, a positivePDO, and a positive AMO, have all conspired to raise thetemperature to significantly higher values than what wouldbe expected from the sole combination of aa-index and CO .With the upcoming switch to La Niña conditions, and theimminent return of the PDO and AMO into their negativephases, we expect a significant temperature drop for the com-ing years, which then might re-establish the strong correla-tion between aa-index and Δ 𝑇 . At any rate, the next decadewill be decisive for distinguishing between climate sensitiv-ity values in the lower versus those in the upper half of thederived range. In this sense we are more optimistic thanLove et al. (2011) who believed that we “...would have topatiently wait for decades before enough data could be col-lected to provide meaningful tests...”. Given the high cor-relation for most of the past 170 years, and the good corre-spondence with the results of recent satellite-borne measure-ments (Feldman et al. , 2015; Rentsch, 2019), our “bets” areclearly on the lower half of that range.Based on those estimates, we have also presented somecautious climate predictions for the next 130 years. Withthe derived share of the solar influence reaching values be-tween 30 and 70 per cent, such predictions depend criticallyon a correct forecast of the solar dynamo (in addition tothat of CO , of course). Following our recent work towardsa self-consistent planetary synchronization model of short-and medium-term cycles of the solar dynamo, we have ex-trapolated some simple 3-frequency fits of the aa-index to thedata from the last 170 years into the next 130 years. Apartfrom intrinsic variabilities of such forecasts (mainly con-nected with the vagueness of the Gleissberg-type cycle(s)),we prognosticate a general decline of the aa-index until 2100,which essentially reflects the 200-years Suess-de Vries cy-cle. Such a prediction presupposes that we have indeed leftthe irregular solar dynamo episode (corresponding to theLittle Ice Age, the latest “Bond event”) and that we will fur-ther remain in a regular phase of solar activity (Weiss andTobias, 2016; Stefani, Stepanov and Weier, 2020), similarto that between 800 and 1340 (Ma and Vaquero, 2020). Of Frank Stefani:
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Page 11 of 14ultiple regression analysis of anthropogenic and heliogenic climate drivers, and some cautious forecasts course, we have to ask ourselves whether our prediction of adeclining aa-index could be completely wrong, with the Suneventually becoming even “hotter” in the future, thus addingto the warming of CO . While such a scenario cannot becompletely ruled out, we consider it as not very likely, giventhat the solar activity at the end of the 20th century was per-haps the highest during the last 8000 years (Solanki et al. ,2004), and that it has declined ever since.As for the CO trend, we have considered three scenar-ios, comprising an unfettered 2.5 ppm annual increase un-til 2150, as well as one soft and one radical decarboniza-tion scheme. Even in the “hottest” case considered, we findonly a mild additional temperature rise of less than 1 K untilthe end of this century, while all other cases result in flat-ter curves in which the heating effect of increasing CO iswidely compensated by the cooling effect of a decreasingaa-index. Whatever the rationale of the advocated K goalmight be, it will likely by maintained even without any dras-tic decarbonization measures. Apart from that, we also ad-vise that any imminent temperature drop (due to the turn ofENSO, PDO and AMO into their respective negative phases)should not be mistaken as, and extrapolated to, a long-lastingdownward trend (Abdussamatov, 2015).In this work, we have focused exclusively on a quasi-instantaneous, i.e. TCR-like climate sensitivity on CO . Asfor ECS, we agree with Knutti et al. (2017) who opined that“(k)nowing a fully equilibrated response is of limited valuefor near-term projections and mitigation decisions” and that“(t)he TCR is more relevant for predicting climate changeover the next century”. In view of the millennial relaxationtime scale underlying the concept of ECS, we fear that - per-haps much too soon - the huge Milankovic drivers will cooldown mankind’s hubris of being able to significantly influ-ence the terrestrial climate (in whatever direction). Data availability date were obtained from ftp://data.iac.ethz.ch/CMIP6/input4MIPs/UoM/GHGConc/CMIP/yr/atmos/UoM-CMIP-1-1-0/GHGConc/gr3-GMNHSH/v20160701 . Declaration of competing interest
The author declares that he has no known competing fi-nancial interests or personal relationships that could have ap-peared to influence the work reported in this paper.
Acknowledgment
This work was supported in frame of the Helmholtz -RSF Joint Research Group "Magnetohydrodynamic instabil-ities”, contract No HRSF-0044. It has also received fundingfrom the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovation pro-gramme (grant agreement No 787544). I’m very grateful toAndré Giesecke, Sebastian Lüning, Willie Soon, Fritz Vahren-holt and Tom Weier for their valuable comments on an earlydraft of this paper.
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Frank Stefani: