Abstract
Severity of warming predicted by climate models depends on their Transient Climate Response (TCR). Intermodel spread of TCR has persisted at ~ 100% of its mean for decades. Existing observational constraints of TCR are based on observed historical warming response to historical forcing and their uncertainty spread is just as wide, mainly due to forcing uncertainty, and especially that of aerosols. Contrary, no aerosols are involved in solarcycle forcing, providing an independent, tighter, constraint. Here, we define a climate sensitivity metric: timedependent response regressed against timedependent forcing, allowing phenomena with dissimilar time variations, such as the solar cycle with 11year cyclic forcing, to be used to constrain TCR, which has a linear timedependent forcing. We find a theoretical linear relationship between the two. The latest coupled atmosphereocean climate models obey the same linear relationship statistically. The proposed observational constraint on TCR is about \({{1}}/{{3}}\) as narrow as existing constraints. The central estimate, 2.2 ^{o}C, is at the midpoint of the spread of the latest generation of climate models, which are more sensitive than those of the previous generations.
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Introduction
Comprehensive climate models, which incorporate more and more known relevant physics, have become more sensitive, as measured by their warming response to doubling atmospheric CO_{2} experiments in the latest generation, 6th Coupled Model Intercomparison Project (CMIP6)^{1}. Tighter observational constraint is needed to determine the plausibility of the more sensitive models^{2}. The climate science community has devoted over four decades to constrain a canonical climate sensitivity metric, the Equilibrium Climate Sensitivity (ECS), which is the global surface warming at equilibrium to doubling atmospheric CO_{2} concentration. At equilibrium there should not be any ocean heat uptake; extrapolating from our current state with large but inadequately measured ocean uptake to that equilibrium state involves uncertainties. Another climate sensitivity metric, the Transient Climate Response (TCR), is defined as the global surface temperature response to 1% per year increase in atmospheric CO_{2} at the time of doubling. The idealized forcing is to standardize the forcing in the model runs for this experiment. The model produced warming in response to it depends on each model’s climate sensitivity: the more sensitive the model is, the higher TCR it produces. TCR is more relevant for projections and mitigation decisions within a century, and more closely related to the social cost of carbon^{3}, compared to ECS. It has been estimated that halving the range of uncertainty for TCR has a present economic value of $10 trillion^{4}.
Combining multiple lines of evidence can often achieve a reduction of uncertainty for the estimates of climate sensitivity and that strategy has been successfully applied to ECS^{5} by combining three lines of evidence, with historical warming being one of them, and the other two being from paleo climates. There is however a lack of truly independent lines of evidence for TCR other than historical warming, because, unlike ECS, TCR involves a specific timedependent forcing, and there are no other known analogs. Here, by proposing a regressed climate sensitivity metric, dissimilar timevariations of forcing can be incorporated, and this allows more independent phenomena to be included as lines of evidence. We demonstrate here its use in constraining the TCR utilizing the observed response to solarcycle forcing.
The existing line of evidence uses the observed historical global warming^{6,7,8} to directly constrain the TCR of model simulated warming. The dominant uncertainty in this approach lies in the different forcings used in different models to reproduce the observed historical warming. Largely because of the uncertain aerosol forcing in models, model simulated warming is not a good discriminator of model sensitivities, because most climate models succeeded in simulating the same (known) centennial warming, despite their widely different climate sensitivities (albeit more so for ECS than for TCR)^{9}. Recent improvements in estimating the constraint involve turning the historical warming in models into an “Emergent Constraint” for TCR by seeking a linear relationship for shorter, post1975 decades that had more constant aerosol loading, with empirically fitted slope and intercept^{10,11}. On the other hand, climate models have not been tuned to simulate the solarcycle forcing. As a result, solar responses are found to differ widely from model to model, with larger response generally for higher TCR. Consequently, this phenomenon may provide a more distinct discriminator for model TCRs, and—without the “shared bias” of attempting to simulate the same wellknown observation—may be more suitable for use in “Emergent Constraints”. Furthermore, solar cycle has a separate external forcing than CO_{2}. The response to this external forcing is amplified by the climate feedback processes of the Earth system. It is hoped that by examining the solarcycle response at the Earth’s surface, we can estimate the effect of the climate feedback involved, and hence the climate sensitivity of the Earth system. This is a different approach than some of the current ones, for example, of using the effect of Arctic snowalbedo^{12}, or tropical low clouds^{13,14} to provide a constraint on the climate sensitivity (ECS in the cited cases). Using a component of the feedback process to constrain ECS or TCR requires an additional assumption that the other feedback components are “unbiased”, e.g. not compensating^{9}. This structural uncertainty has so far not been assessed.
Nevertheless, arguments against using the solarcycle forcing are: (a) its cyclicintime forcing is very different than the linearly increasing forcing of the TCR experiment; (b) there is a difference in the lag of response to forcing introduced by the deeper ocean because of the difference in time scales in the two phenomena; (c) whether the two phenomena experience the same climate feedbacks, and (d) methods used to extract the expected small solarcycle signal need to be particularly effective in reducing contamination by internal variability and other forcings. A modified climate sensitivity metric, which is the timedependent response regressed against timedependent forcing, is defined here to overcome problem (a). We will use both 2layer emulators of climate models and the climate models themselves in discussing why we think difficulties (b) and (c) can be overcome. To address (d), we will describe and test a more sophisticated spacetime method to extract the solarcycle response.
Results
Solarcycle response extracted from instrumental record
The 11year solar cycle, also called the 11year sunspot cycle, is a quasiperiodic phenomenon associated with dark spots on the surface of the Sun. The brightening effect in the surrounding faculae overcompensates the dimming effect of the dark spots, producing more radiation during the solar maxima (solar max) than solar minima (solar min). This variation between solar min and solar max in the Total Solar Irradiance (TSI) has been measured by orbiting satellites since late 1978 to be variable between cycles but close to 1 W m^{–2}, in a disk facing the sun at the top of the atmosphere. (TSI is defined as the spectrally integrated power from the sun received at the top of the Earth’s atmosphere. \({{{{{\rm{TSI}}}}}}/4\) is the power per unit spherical area of the Earth). The relative accuracy (the variation between solar max and solar min) is high, but because of changing satellites, the longterm variability in TSI is more uncertain and has been a subject of debate^{15}. Nevertheless, the secular trend is small^{16}, and we are concerned only with the relative variation. TSI before the satellite era has been reconstructed based on proxies of the solar dark and bright magnetic regions. The uncertainties, mostly involving the trend, are discussed in Methods, and found to be small.
Using the most recent observed surface temperature dataset HadCRUT5^{17}, which provided 200 ensemble members to assess various aspects of observational uncertainty, we calculate the global mean solarcycle response \({\kappa }_{{SOLAR}}\) as the regression coefficient against TSI, denoted by \(\left\langle {T}_{{SOLAR}}\left(t\right) {TSI}\left(t\right)\right\rangle\). (See Methods for how regression is calculated analytically and numerically).
The mean value is obtained as the ensemble mean (over the 200 surface temperature sets) for the modern period 1956–2014 (see Fig. 1), involving 5 complete solar cycles, with brackets denoting the “very likely” range (5–95%):
The two groups used in the Linear Discriminant Analysis (LDA) analysis are the solar max group and the solar min group. The years with TSI above (below) the local mean are classified into the solar max (min) group. The local mean is determined using Empirical Mode Decomposition^{18,19} (see Methods). The latest climate models, CMIP6, ended their historical runs after 2014. Since we want to use the same period for both observation and model outputs, 2014 is the last year we can use. To avoid known systematic bias of our method from having uneven number of years in the solar max group and solar min group when we extract the solar signal using our LDA method, complete solar cycles need to be used (with at most oneyear difference between the two groups). It is the nature of the solarcycle phenomenon that there usually are more years in solar min group than in the solar max group, as solar max is usually reached rather quickly. Starting the period earlier than 1956 will increase the number of excess years in the solar min group, until 1946 (with 34 maxes and 35 mins). However, global observational data prior to 1950 are sparser. Therefore, we select 1956–2014 for the present study and the maximum number of solar cycles that we can use is five.
In addition, Fig. 1 shows the null distribution obtained using the method of bootstrap with replacement, by randomly scrambling the years of the observed data relative to the TSI^{20}. Autocorrelations up to 11 years is taken into account by randomly sampling, with replacement, blocks of the time series each of length L = 11 years using the Lblock method^{21}. The extracted solarcycle signal in Eq. (1) is statistically significant at over 99% confidence level. Furthermore, compared to the “null distribution” obtained from the solarcycle “signals” in the 48 CMIP6 PreIndustrial Control runs known to have no solarcycle forcing, the observed signal is statistically significant at 100% confidence level. See Fig. 2. That is, no solar response is found by our method when it should not exist. Sensitivity to major volcano eruptions is discussed in “Methods” and Fig. 3 and found to be small.
Solar cycle and TCR
As the only known phenomenon with a wellmeasured radiative forcing in the decadal range, the observed global response to the 11year solar cycle can serve to constrain TCR for the following reasons:^{22}

(1)
The spatial patterns of the tropospheric and surface temperature response to 2% solar forcing and to 2×CO_{2} are nearly the same in general circulation models^{23}, as originally proposed by Manabe and Wetheral^{24} for persistent forcing. The observed latitudinal pattern of the 11year solar cycle is also nearly the same, within observational uncertainty, as that predicted for CO_{2} induced global warming and TCR^{25}. As for the latest climate models, we compare in Fig. 4 the spatial structure of the TCR, calculated based on 41 CMIP6 models that have archived TCR runs in the open domain, with the spatial structure of the solarcycle response calculated based on 51 CMIP6 models that have the solar cycle forcing in their historical runs. They are very similar, though with different amplitudes, consistent with Manabe and Wetheral^{24}. The common features include: general warming over the globe, Arctic amplification of warming, more warming of the continents than over the oceans. And more warming over the Arctic than over the Antarctic, the latter being more affected by the cold ocean upwelling than radiation from above.

(2)
Effective Radiative Forcing (ERF) is defined as the effective radiative forcing at the top of the atmosphere after the stratosphere and troposphere has adjusted to the forcing while holding the surface temperature unchanged. A forcing agent with the same ERF as CO_{2} should yield the same surface warming. ERF for the solarcycle forcing has been calculated by the 6th Assessment Report (AR6)^{9}, putting it on the same footing as CO_{2} when measuring their effects on surface warming, despite their very different long and shortwave behaviors in the stratosphere. This will be discussed in more detail later.

(3)
The fast climate radiative feedback mechanisms relevant for both phenomena appear to be the same, as deduced from their tropospheric patterns; the responses would both have been smaller without these climate feedbacks. We previously inferred from observations^{26}, and diagnosed explicitly using an aquaplanet model^{23}, that the radiative and dynamic feedback mechanisms for the solarcycle response are: evaporative, water vapor plus lapse rate, cloud, and albedo feedbacks, same as those at play in the response to CO_{2} forcing. The response to the solar radiative forcing contains these climate gains, which is measured by TCR^{27}. This is likely the reason why when we normalize the smaller solar response by its smaller forcing, we find a linear correspondence with the TCR normalized by its larger forcing.

(4)
Observational data analysis of the tropospheric solarcycle response illustrates the pathway of the response from where the forcing is the largest to where the response is the largest^{26}. Net radiative forcing at visible and infrared frequencies, which is the bulk of the solarcycle forcing, is largest in the tropical region and yet the response at the surface is smallest there. The warming is largest over the polar region where the forcing is the smallest. The tropical ocean surface is warmed little because of evaporative feedback. As water over the oceans is evaporated, heat is mostly carried aloft as latent heat instead of being absorbed by the ocean; the large ocean inertia does not come into play significantly to delay the response. We do not find more than a few months of lag. The latent heat of evaporation is deposited near 200–300 hPa in the tropical troposphere. There, downgradient poleward heat transport aloft followed by thermal downwelling to the cold polar surface under the stable static stability of the polar lower troposphere combine to produce the characteristic spatial pattern of polar amplification of surface warming, the same mechanism as that for CO_{2}induced warming. The mechanisms were analyzed^{23} using an aquaplanet model and persistent forcing; it was found that the two phenomena share similar amplification mechanisms for the fast climate feedbacks in the troposphere although their stratospheric response is very different. We will show here that the relationship continues to hold for cyclic solar forcing.
A different climate sensitivity metric
For timevarying forcing, we define a measure of sensitivity to external forcing as the regression of response to forcing, denoted as \(\mu=\left\langle T\left(t\right)F\left(t\right)\right\rangle\). The response is taken to be the globalmean surface temperature change, T, excluding unforced variability. The forcing, F, is the globalmean ERF^{9}. For solarcycle forcing and response, we have \({\mu }_{{SOLAR}}=\left\langle {T}_{{SOLAR}}\left(t\right){F}_{{SOLAR}}\left(t\right)\right\rangle .\)
\({\mu }_{{SOLAR}}\) is related to \({\kappa }_{{SOLAR}}\) through a scaling factor:
where from AR6^{9}, \(\frac{1}{b}\equiv \left\langle {F}_{{SOLAR}}\left(t\right){TSI}\left(t\right)\right\rangle=\frac{0.72\left(1\alpha \right)}{4}\), α being the planetary albedo. See later and Methods for assignment of uncertainties. Therefore, instead of finding \({\mu }_{{SOLAR}}\) using regression, we simply multiply b to \({\kappa }_{{SOLAR}}\), which we have already obtained above, to get \({\mu }_{{SOLAR}}\).
Normalized TCR
For the TCR experiment, a climate sensitivity metric can be defined as the global warming response regressed against its ERF:
We additionally define the normalized TCR as:
This quantity is not the same as \({\mu }_{{TCR}}\), but it is only slightly less (see later). When the regressed quantity cannot be calculated from the archived models, the normalized TCR is used.
Since the ERF from doubling CO_{2}, \({F}_{2\times C{O}_{2}}\), is not an observed quantity, observations cannot constrain TCR itself, only the normalized TCR. AR6^{9} assessed that the ERF for doubling CO_{2} has a \(\pm\! 12\%\) uncertainty. Using the normalized TCR avoids including this rather large uncertainty in our metric for TCR. We recommend models in the future report their TCR response regressed against the forcing, because the regression of the response to the linear forcing is better than simple ratios in reducing the contamination by other climate variability, in addition to absorbing the 12% uncertainty in the hypothetical forcing. We also recommend comparing the spreads in TCR by using the spreadtomean ratio, where the numerator is the spread (5 to 95% range unless otherwise specified) and the denominator is the mean:
Equation (4) can be applied directly to our normalized TCR, without having to first converting it to TCR.
Previously, direct observational constraints on TCR were provided by the “historical difference method”, considering the ratio of historical change in globalmean temperature^{6,7,8} since the preindustrial period, \(\Delta T\), and in radiative forcing, \(\Delta F\):
The uncertainty from the historical change method is large because (i) the temperature data in the beginning of the industrial era is sparse; (ii) the difference may include internal variability; and (iii) more importantly: “the single largest contribution to formal error in calculated TCR is, however, due to uncertainty in \(\Delta F\)”^{8}. And the largest component of this uncertainty is due to aerosol forcing^{7}. There is no aerosol involved in solarcycle forcing, which we will use as a separate constraint.
Expected linear relationship
We propose to use \({\mu }_{{SOLAR}}\) to constrain \({\mu }_{{TCR}}\). We expect a linear relationship between the two because both are amplified by the same climate gain factor on decadal time scales. This can be shown analytically in a 2layer “emulator”. Such emulators^{9,28} were extensively used in AR6. It consists of a surface layer of temperature T, with heat capacity C_{,} which is being forced by \(F=Q\left(t\right)\), radiating to space above and losing heat to the deeper ocean layer below^{28,29,30}, whose temperature is denoted here as \({T}_{d}\) with heat capacity \({C}_{d}\):
Numerical solutions are shown in Fig. 5. Exact analytic solutions are available for a linear forcing and for a periodic forcing, given by Geoffroy et al.^{28}, who also gave parameter values obtained by calibrating the 2layer emulator with CMIP5 models. The solution consists of an instantaneous response to forcing plus slow and fast adjustments. Because of the disparity of time scales, the upper layer quickly adjusts, while the lower layer adjusts much slowly due to the larger heat capacity of the deeper ocean^{29}. We use an asymptotic solution, obtained using the twotiming method and verified to be accurate, to better show the dependence on these two timescales.
For the TCR experiment, the warming response to a switchedon radiative forcing
\({Q}_{{TCR}}\left(t\right)={q}_{{TCR}}{t}\) for \(t \, > \, 0\) and \({Q}_{{TCR}}\left(t\right)=0\) for \(t\le 0\) is found to be:
The instantaneous “equilibrium” response^{28}, \(\frac{Q\left(t\right)}{\lambda }\), is the response obtained by ignoring ocean inertia. Due to ocean inertia, there is a lag of \(\Delta \left(t\right)\) in the surface temperature response, starting at 0, reaching the asymptotic value of \({\tau }_{0} \, \approx \, 80\) yr in \(\tau \, \approx \, 240\) yr. The TCR experiment period is too short to reach that asymptotic value. The transient lag during the shorter TCR experiment period is linearly increasing, as \(\Delta \left(t\right) \, \approx \, (\frac{\gamma }{\lambda+\gamma })t\), with the end value: \(\Delta \left(70\,{{{{{\rm{yr}}}}}}\right) \, \approx \, 20\,{{{{{\rm{yr}}}}}}\). The thermal parameters used in the estimates here are from the mean of the emulator model ensemble^{28}.
Regression of response to forcing yields:
The normalized TCR can be found as:
The usual climate gain factor estimated using an energybalance (one layer) model is \(\frac{1}{\lambda }\), which is larger because ocean heat uptake is not taken into account.
Because the solarcycle response does not penetrate through the whole depth of the mixed layer^{31}, it does not experience the fulldepth heat capacity. We let \(\eta\) be the fraction of the mixed layer depth that it penetrates to (to be determined from observation or constrained by the lag in the response). Since the signal does not penetrate the deeper ocean layer, \({T}_{d}\) is not affected by it, and can be set to be zero^{30} in the approximate solution. For a sinusoidal solarcycle forcing: \({Q}_{{SOLAR}}={q}_{{SOLAR}}\sin (\omega \, {t})\), the approximate solution is:
The analytical result in Eq. (7) is found to be very accurate compared to the full twolayer numerical solution (See Fig. 5). The lag is between 0 to 2 years for \(\eta\) between 0 and 1. Especially accurate are the approximate expressions for the normalized TCR and also for the solar response regressed against forcing, which are being used in the expected linear relationship (see Fig. 5c, d). These relationships are independent of model parameters used in Fig. 5a, b.
Although the two phenomena are forced by separate forcings and the responses can be extracted separately, their sensitivity metrics are related by a linear relationship because they each are proportional to \(\frac{1}{\lambda+\gamma }\); see Fig. 5c, d. Therefore:
When the regressed form of the model climate sensitivity metric is not available, we use the normalized TCR:
The slope a depends only on the lag of the solar response. It is close to 1 because the lag in the solarcycle responses in observations is close to 0, subject to a maximum of 6month uncertainty because annualmean data are used.
Linear relationship in CMIP6 models
54 CMIP6 models are available on esgfnode.llnl.gov. Meehl et al^{1}. computed TCR for 37 models consistently using the ESMValTool. Zelinka et al.^{32} computed the ERF in a set of 27 models; one (ECEarth3) of these was not used in Meehl et al. Among the 26 common models, 6 (CAMSCSM10, CESM2WACCM, CNRMCM61HR, FGOALSF3L, INMCM48, and SAM0UNICON) did not follow CMIP6’s protocol, which requires at least 500 years of spinup of their oceans before TCRs are calculated (see Supplementary Table S1, and the discussion on the effect of not having sufficient spinup on TCR in Methods, which can give rise to 30% uncertainty); they are objectively excluded. In addition, NORESM2LM did not provide information on branching off time or preindustrial control, and so is also excluded. Information on MPIESM12LR, which was not included in ref. ^{32}, became available recently, and is added to our list. Sensitivity to including all 27 CMIP6 models is discussed in a later section and Fig. 10. CMIP5 models did not adopt a consistent spinup protocol^{33}, thus introducing an additional but artificial decadal variability, and so are not used.
CMIP6 models seem to be consistent with Eq. (9). We use linear leastsquares fit of the model pairs, and MonteCarlo bootstrapping to find the spread of the slopes. The CMIP6 ensemble yields a regressed slope of 1.02 [0.97,1.16] for a best fit from the origin, consistent with Eq. (9) with slope 1 obtained above from a simpler model. This consistency test is not a proof; for that purpose, a CMIP6 model with many versions each with a different TCR is needed, which is however not available.
Emergent constraint
Suppose we do not know the functional form \(G\left(x\right)\) that relates the quantity y, to be constrained, to a quantity \(x\), which could be observed: \(y=G\left(x\right)\), but would like to discover it. The discovered relationship that “emerges” by exploring model pairs \(\left(x,y\right)\) empirically is referred to as an “emergent constraint”. Often because of the limited model points, the best one can do is through a local approximation around a typical point \(\left({x}_{0},{y}_{0}\right)\) in a cluster of model data points: \(y{y}_{0}\cong {G}^{{\prime} }\left({x}_{0}\right)\left(x{x}_{0}\right)\), that is, a linear relationship of the form: \(y={A\,x}+B\). Model points are then regressed onto this line. The constant term \(B={y}_{0}{G}^{{\prime} }\left({x}_{0}\right){x}_{0}\) is sometimes referred to as the “intercept”. It is often nonzero in practice regardless of whether the true functional relationship passes through the origin. Both slope and “intercept” of the local approximation are affected by the scatter from the centroid of data points.
In Fig. 6 we empirically fit the model pairs using leastsquares to this line. Bootstrapping with replacement generates 10,000 random sets of data with 20 points each. The 10,000 linear leastsquares fits form a probability density function, from which 5%–95% band is defined. The result obtained through robust regression remains the same. Projecting the observed (blue band) onto this 5–95% range of slopes determined from model scatters yields
In the language of regression analysis, x is the predictor, while y is the predictand. The first value, 0.65, is the predicted mean value (or sometimes called the “best estimate”). The brackets enclose the 90% confidence level of that predicted mean; the maroon dashed lines in Fig. 6 indicate the prediction limits of the mean. This interval does not necessarily include 90% of the predicted model y given model x. The latter is called^{34} the 90% predicted y interval given x, and could be much wider. It is useful for a different purpose: to predict where individual model’s TCR lies in 90% of the cases given a value of x. We are asking instead where the “best estimate” lies in 90% of the cases.
The centroid of the model data points happens to lie within the observed range (the blue vertical band). This is the narrowest part of 90% interval bounded by the dashed curves. The uncertainty range would have been larger if the observed range had occurred to the right or left of this narrow point. Also, the derived range is not sensitive to the slope of the regression lines since the various straight lines pivot near this point.
For comparison, recent emergent constraints on TCR using CMIP6 models based on simulated historical warming post1975 gave 1.68 [1.0 to 2.3] °C by Nijsse et al.^{11}, and 1.60 [0.90 to 2.27] °C by Tokarska et al.^{35}. AR6^{9} combined these to claim 1.7 [1.1 to 2.3] °C. These have wider spreads of 77%, 86% and 71%, respectively. Ours, at 23%, is the narrowest, about \(1/3\) of existing constraints.
However, all three, including AR6’s and ours, may be subject to a criticism of modelfitting bias, which is often raised in machine learning literature, but has so far not been considered in Emergent Constraint research. The twoparameter fit to the straight line used in the Emergent Constraints includes an “intercept”, which is not physically justified but is employed solely for the purpose of yielding a better fit to the “training data”. The “training data” here is the CMIP6 models. This may be a bias toward CMIP6 models.
Eliminating this bias, Emergent Constraint without an intercept should yield a wider range:
obtained for fits to lines through the origin (Fig. 7). The uncertainty range of our oneparameter fit is still \(\sim 1/3\) of existing constraints: 120%^{7} and 139%^{8}, when compared to other oneparameter fits of historical warming.
A still more conservative approach^{34,36} makes no assumption about the existence of any relationship between \(x\) and \(y\), and suggests not leveraging models whose “observables” (\(x\)) are inconsistent with the current observed to “discover” a linear relationship. So only the model points within the blue band are fully weighted. Weighting of the points away from the blue band less using KullbackLeibler divergence in information theory, a new probability density distribution is then constructed, from which it yields: \(\overline{TCR}=0.62 \, \left[ 0.39 \;{\rm to }\; 0.86\right] \, {{\,\!}^\circ}{\rm C / W \, m^{2}} \, ; \; \; r_{TCR}=76\%\). See Fig. 8. This wider uncertainty range is not adopted here since we have established that a linear relationship does exist.
Another “line of evidence” uses direct observational constraint of warming in the instrumental record (without using the Emerging Constraint) to yield a \({r}_{{TCR}}\) of 120% by Lewis and Curry^{7}, and 139% by Richardson^{8} (see “Methods”). Though their uncertainty ranges are much larger, these direct constraints were obtained without having to use an adjustable “intercept”, and are therefore not subject to criticisms often leveled at some Emergent Constraints about not including “structural uncertainty”^{37}.
Processbased estimate of TCR
AR6^{9} uses an energybalance model for its processbased estimate. Here we use a twolayer model emulator, with its parameter calibrated against observations instead of models for the estimate. This approach does not use CMIP6 climate model pairs to empirically determine the relationship and thus avoids the possible criticism that we used some CMIP6 models which may “run too hot”. We find almost the same result on the TCR constraint as above. It shows consistency with the Emergent Constraint approach but is probably not independent enough to be considered an additional line of evidence.
Using Eq. (9), which is independent of CMIP6 models, we have
The proportionality constant, \(b=\frac{\partial {TSI}}{\partial {F}_{{SOLAR}}}\), is the ratio of TSI and the ERF after stratospheric and tropospheric adjustment, where \(\frac{\partial {F}_{{SOLAR}}}{\partial {TSI}}=\frac{1}{4}\left(1\alpha \right)\xi\), \(\alpha\) being the planetary albedo and \(\xi\) the ratio of solar radiation at the top of the troposphere and top of the atmosphere.
The planetary albedo has been measured very accurately by satellites, with an accuracy of ±1% for the 2.5–97.5% range or, equivalently, ±0.86 for the 5–95% range, with very little interannual variability and has a symmetry between the two hemispheres^{38}. Gray et al. performed a wavelength dependent calculation of the albedo for the solarcycle specific irradiation and concluded that it is almost the same as the wavelengthaveraged planetary albedo. In contrast, the modeled planetary albedo is more variable, and perhaps overtly sensitive to the surface temperature. This explains some of the scatter of the slope in Fig. 6. Another cause for the scatter may be due to the slight differences in the solarcycle response lags in models. We use observed values for these quantities to calibrate the emulator.
The simulated value of \(\xi\) is sensitive to the spectral resolution of the input solar spectrum due to the stratospheric O_{3} absorption in the Ultra Violet (UV) range. Gray et al.^{39} estimated the stratosphere adjusted radiative forcing (RF) for the 11year solar cycle based on fixed dynamical heating approach and obtained a value of \(\xi\) to be 0.78 using a solar UV spectrum with 1nm resolution. ~0.15 of the reduction from 1.00 is due to the stratospheric O_{3} absorption of the solar variation at wavelength below 300 nm; the rest of the reduction is due to the combination of stratospheric O_{3} absorption of the solar variation above 300 nm and the stratospheric temperature adjustment^{40}. Earlier studies using coarser spectral UV resolutions obtained higher values: Larkin et al.^{41} first obtained a value of 0.88 using a twostream model with six spectral bands; Hansen et al.^{42} reduced it to 0.83 using a solar spectrum at 5nm resolution. We adopt Gray et al.’s^{39} value with 1nm resolution. AR6 also adopted this value for stratosphereadjusted RF, to which it added –0.06 for tropospheric adjustment to yield \(\xi=0.72\) for use with ERF.
By imposing 5 satelliteobserved solarinduced O_{3} changes and using 2 radiative transfer schemes, Isaksen et al.^{43} showed that the O_{3} solar RF varied from –0.005 W m^{–2} to 0.008 W m^{–2} relative to the mean. These values are close to zero since an increase in ozone in the stratosphere absorbs more of the ultraviolet part of the solar radiation, but the increase in ozone heating emits more downward longwave radiation that almost compensates the decrease in shortwave radiation. Larkin et al.^{41}. also tested their solar RF by replacing the simulated O_{3} solar response with the observed, and the values remain the same as 0.23 W m^{–2} in both cases due to the abovementioned compensation. Because of their broadband calculations, they reported the resulting solar radiative forcing to only the second decimal place. We adopt Isaksen et al.’s values for the uncertainty in \(\xi\). The standard deviation of the 10 net O_{3} solar radiative forcings listed in Isaksen et al.’s^{43} Table 1 is 0.0047. For a mean radiative forcing of \(0.26 \times 0.72=0.187 \, {{{\rm{W}}}} \, {{{\rm{m}}}}^{2}\), we estimate a percentage uncertainty in \(\xi\) to be \(0.0047/0.187\times 1.65\), which is ±4.1% (5–95% range). For observational data analysis we use: \(\frac{\partial {F}_{{SOLAR}}}{\partial {TSI}}=\frac{1}{4}\left(1\alpha \right)\xi=0.127\pm 4.2\%\), where \(\xi=0.72 \; (\pm 4.1\%)\) (5–95% range), \(\alpha=0.29 \; (\pm 0.86\%)\) (5–95% range), and the total uncertainty for this quantity is \(\sqrt{{\left(4.1\%\right)}^{2}+{\left(0.86\%\right)}^{2}}=4.2\%\) (5–95% range). The reciprocal is: \(b=\frac{1}{0.127}( \pm\! 4.2\%)=7.87 \pm 0.33\) (5–95% range).
The uncertainty in the reconstructed detrended TSI is 0.049 Wm^{−2} for the period 1950–2014 (see “Methods” and Fig. 9), about 5%.
The solar response determined from HadCRUT5^{17} yields, for the “very likely” range:
Because some of the errors are not Gaussian, the above estimate may not be accurate. By performing MonteCarlo bootstrapping of the slope, along with error in variable, we obtain the more accurate result of 34.4%, which is close to (13); both should be rounded to 34%. The best estimate for the normalized TCR is 0.66 °C/W m^{–2}.
For comparison, Padilla et al.^{30} obtained estimates of TCR of 1.6 [1.3 to 2.6]\({\!\,}^\circ {{{{{\rm{C}}}}}}\), giving \({r}_{{TCR}}=81\%\). They also used a twolayer emulator as here but with \({T}_{d}\) set to zero, and applied historical forcing and observed warming to constrain model parameters.
TCR in conventional units
Our central value in terms of normalized TCR, based on the ProcessBased Estimate and the oneparameter Emergent Constraint, is 0.66–0.67 °C /Wm^{−2}. We take the average as the best estimate of the central value. For comparison with the TCR in conventional units, we need to multiply it by \({F}_{2\times C{O}_{2}}\). The value of \({F}_{2\times C{O}_{2}}\) to use, for consistency, should be what was divided into model TCR to normalize it as the vertical coordinate in Fig. 7. For the 5 models that are consistent with the observational constraints of solar amplitude and slope in these figures, the mean value is 3.38 W m^{–2}. Using this mean value, our best estimate (the average of processbased estimate and the emergent constraint) is TCR = 2.2 °C. Using AR6’s mean value of 3.93 W m^{–2} for doubling CO_{2} forcing would have increased the value of estimated TCR in °C beyond AR6 TCR’s “likely” and “very likely” ranges. However, this higher ERF value is not constrained by observations.
Using the common procedure of Emergent Constraint (with an intercept), a narrower range of:
in conventional units is obtained here, which is \(1/3\) that of AR6’s estimate using historical warming and Emergent Constraint. However, given our reservation about the 2parameter fit used in the common Emergent Constraint procedure, a more conservative estimate is
based on our best estimate of the central value and the range in Eq. (11).
Sensitivity to including models that do not follow protocol
There are 7 CMIP6 models that were excluded in Fig. 7 because they do not follow the CMIP6 protocol, which requires that models have at least 500 years of spinup of their oceans before the calculation of TCR. Including them (see Fig. 10) widens the uncertainty range slightly, from
to
The upper range remains unchanged. It lowers the lower uncertainty range slightly from 0.58 to 0.56. We argue that these models should not be included.
Discussion
Our observational constraint of Earth’s Transient Climate Response based on the solarcycle phenomenon is \(\sim 1/3\) as narrow as existing estimates, which were previously all based on the observed historical warming. This reduction of uncertainty is achieved through a number of factors: (i) the solar phenomenon has more accurately measured forcing, compared to the large aerosol uncertainty in historical warming; (ii) the TCR from models are calculated using a consistent protocol; (iii) we use a climate sensitivity metric that involves normalizing model TCR by their individual radiative forcing, thus avoiding the including the latter’s 12% uncertainty in TCR itself; (iii) only instrumental temperature record since 1950s is used.
With improvements in the modeling of physical processes, climate models have become more sensitive. CMIP6^{1} models now generally have a higher TCR than previous generations, with a 5–95% intermodel range of 1.3 °C–3.0 °C. Our central estimate of 2.2 °C is near the midpoint of the intermodal spread of CMIP6 model TCRs, implying that some sensitive CMIP6 models (compared with previous generation) may be more consistent with the proposed observational constraint. In particular, climate models, such as GFDLCM4, CANESM5, UKESM10LL, BCCESM1 and IPSLCM6ALR, satisfy both of our observational constraints for TCR and for solar cycle response. We obtained essentially the same estimate without using CMIP6 models in a ProcessBased Estimate. Our central estimate is within AR6’s^{9} “very likely range” of 1.2 °C–2.4 °C. Our narrowing of the “very likely” range is mainly from a lifting, by more than 60%, of the lowend estimate.
Having an independent line of evidence in addition to the existing constraint based on historical warming potentially allows further reduction of the uncertainty by combining them. We leave this task to a future project.
Just as successive Assessment Reports yield more and more accurate estimates of the TCR as observational datasets improve, we expect that future estimates of TCR using the solarcycle approach may need to be revised as surface temperature datasets change. We already saw changes when HadCRUT4 was replaced by HadCRUT5, which widened the uncertainty range and lowered slightly the mean. Also, as future generations of CMIP models include more solar cycles in the satellite era, accuracy of the extracted solarcycle response is expected to improve; at this time we have to stop the comparison with observation at 2014. What we report here is our current best estimate.
Methods
Linear regression
Given two timedependent datasets \(x\left(t\right)\) and \(y\left(t\right)\), we construct a linear regression model as: \(y\left(t\right)={a\; x}\left(t\right)+{residue}\). The linear regression coefficient a is obtained from
which may be also viewed as the functional derivative \(\frac{\partial y\left(t\right)}{\partial x\left(t\right)}\). The bars denote the temporal averages. The integration is over the length of the record. For sinusoidal forcing (e.g., the solarcycle forcing), the integration is over a full period. For the TCR experiment it is over the 70 years of the experiment.
Climate sensitivity and its uncertainties
The TCR is commonly defined by the difference between year 70 and year 1 global mean surface temperature. We use \(\Delta\) to denote difference. So the normalized TCR is defined as the ratio of differences of response temperature to forcing:
Note that \(\Delta T\) and \(\Delta F\) are not functions of t.
For transient climate sensitivity, the forcing is timedependent, and so it is not appropriate to use the ratio of response to the forcing to define this sensitivity metric. A more appropriate measure of sensitivity to external forcing should be given as the regression of the response against the forcing. In the case of climate sensitivity, the response is taken to be the global mean surface temperature change (\(T\)) (after the removal of unforced variability), and the forcing is taken to be the global mean radiative forcing (\(F\)): \(\mu=\left\langle T\left(t\right)F\left(t\right)\right\rangle\), where \(\mu\) is the climate sensitivity of interest. To obtain \({\mu }_{{SOLAR}}\), we constructed another regression coefficient: \({\kappa }_{{SOLAR}}=\left\langle {T}_{{SOLAR}}\left(t\right){TSI}\left(t\right)\right\rangle\), where \({\kappa }_{{SOLAR}}\) is the solarcycle response we obtained using LDA (supplementary materials). We then obtained \({\mu }_{{SOLAR}}\) through the relation \({\mu}_{SOLAR}=b \, {\kappa}_{SOLAR}\), where \({b}^{1}=\left\langle {F}_{{SOLAR}}\left(t\right){TSI}\left(t\right)\right\rangle=\frac{0.72 \, \left(1\alpha \right)}{4}\) and \(\alpha\) is the planetary albedo.
It is difficult, though not impossible, to determine the equilibrium climate sensitivity from instrumental measurements because the current climate is far from equilibrium. The situation is different for TCR. Previously, the historical record of global warming change (\(\Delta T\)) in response to the change in radiative forcing (\(\Delta F\)) from the beginning of the industrial period to recent decades has been used, which is referred to as historical warming method or the Otto et al. method^{6,8}:
This ratio of differences is strictly speaking not a regression of response against forcing—perhaps we should have used a different symbol for it—but is close since the forcing and response are both approximately linear. The difference used in Otto et al. is the decade of 2000s minus 1860–1879. The best estimate for TCR is 1.3 °C, with a 5–95% range [0.9 °C to 2.0 °C]. Lewis and Curry^{7}, using the same method but different periods, also obtain 1.33 °C as the best estimate, but a wider spread of 0.9 °C to 2.5 °C, 120% of its central value, due mostly to their adopting AR5’s \(\Delta F\) with a wider uncertainty range. Their best estimate falls below the multimodel means of CMIP5, suggesting seemingly that some models overestimate future warming. This has generated much debate.
There are four major uncertainties in the historical warming method:

(i)
As pointed out by Richardson et al.^{8}, “the single largest contribution to formal error in calculated TCR is, however, due to uncertainty in \(\Delta F\)”. And the largest component of the uncertainty is due to aerosol forcing:^{7} “Without a reduction in aerosol ERF uncertainty, additional observational data and extended time series may not lead to a major reduction in ECS and TCR estimation uncertainty”. The spread of Otto et al.’s TCR estimates was mainly due to this large uncertainty in radiative forcing. Richardson et al.’s TCR has a spread of 1.0 °C to 3.3 °C, or 139% of its central value. This spread is even larger than that of Otto et al.’s TCR, not because of uncertainties in surface temperature records stated in (ii) below, but mostly because an updated, larger uncertainty for \(\Delta F\) was used. This is where using the solarcycle data has a marked advantage, as its forcing is much better measured and does not involve aerosol forcing.

(ii)
A second source of error is in the historical data of surface temperature over such a long time span required in Otto et al.’s method, when earlier data are geographically less complete, and that data involved the blending of air temperature and sea surface temperature, while models use surface air temperature in their TCR calculation. The adjustment in temperature record by Richardson et al. led to 24% higher warming: 15 of the 24 percentage points arise from masking models to HadCRUT4.4 geographical coverage and 9 from watertoair adjustment in the model. This modelderived scaling factor was responsible for moving the “best estimate” from 1.33 °C to 1.66 °C, now within the CMIP5 model mean. For the solar cycle, we use only the modern record since 1950, aided also by the recent availability of a geographically complete dataset, GISTEMP Version 4, based on NOAA Global Historical Climatology Network (GHCN) Version 4 (meteorological stations) and ERSSTv5 (ocean)^{44}. These geographically complete datasets yield a solarcycle response very close to the central value of HadCRUT5.
For an assessment of the watertoair temperature difference we use the 2m temperature from European Center for MediumRange Weather Forecasts (ECMWF) ReAnalysis (ERA). The 2m temperature is obtained by interpolating the air temperature at the lowest level of the model (~10 m) and the skin temperature (SST), which in recent decades is remotely sensed. The latest ERA version 5 (ERA5) starts from 1979^{45}. To extend the data before 1979, we combine the previous version, ERA40, from 1960 to 1978 with ERA5 from 1979 to 2004^{46}. The watertoair temperature difference obtained from ERA is –10%. ERA uses a model to assimilate observed data, and so this adjustment can also be criticized as being model dependent. However, since this error is within the range of HadCRUT5’s spread, we chose not to let it affect our reported result on TCR. In NCEP reanalysis the 2mtemperature is not a standard analyzed product.

(iii)
A third source of uncertainty is related to the fact that the observed warming is the sum of forced response and unforced response. The latter includes the Atlantic Multidecadal Oscillation (AMO), believed to be caused by the Atlantic Meridional Overturning Circulation (AMOC)^{47}. Tung et al.^{48} and Chen and Tung^{49} found that the large multidecadal variation in the observed globalmean surface temperature is mostly contributed by the AMO, and it can double the observed warming during its positive phase. Hu et al.^{50} found that while AMO does not affect ECS, a weaker AMOC enhances surface warming and increases TCR (and vice versa). van der Werf and Dolman^{51} found that the calculated TCR are affected substantially with different choice of AMO indices. Recent studies using post1975 observed warming to constrain TCR^{10,11} is even more prone to AMO contamination than using the centennial warming in Otto et al.’s method. In our analysis method, spatialtemporal information is used to extract the forced solarcycle response, which greatly reduces the influence of AMO, which has a different spatial structure and much longer period.
Sensitivity to aerosols from volcano eruptions
Volcanic aerosols tend to cool the surface and could affect our extracted solarcycle signal, especially when the two large eruptions were spaced a decade apart. Two large volcanic eruptions, El Chichón in March 1982 and Pinatubo in June 1991, coincidentally erupted during solar max. So the response to the solarcycle forcing could possibly be underestimated. However, the effect should be temporary and lasted two to three years. We tested the sensitivity of our method to the volcanic aerosols by excluding the two years after the El Chichón and also after Pinatubo eruptions. The resulting observed mean response and the 5–95% range are only 0.003 higher: 0.087 [0.073 to 0.100] respectively, in units of °C/W m^{–2}. The difference is well within the uncertainty of the HadCRUT5 data. The distribution of the HadCRUT5 ensemble responses with the Lblock resampling test shown in Fig. 3 shows that the mean responses is still 99% confident.
Extraction of solarcycle signal
We use a spatialtime filter to extract the solarcycle signal from the surface temperature dataset to minimize the contamination of the signal by other climate noises, such as El Niño. The same method is applied to both model and observations. The spatial part is the Linear Discriminant Analysis (LDA). The surface temperature data at each grid point is first linearly detrended before the application of LDA, to eliminate the possibility of global warming contaminating the solar signal, although this effect is small. We define a solar plus (minus) year as the year when the TSI is above (below) the local mean for the period under consideration. The local mean is found using the method of Empirical Mode Decomposition. Collectively the solar plus years are referred to as belonging to the “solar max group”, and the solar minus years as belonging in the “solar min group”. As described in more detail in the Supplementary Information, LDA finds the spatial pattern that best separates the solar plus years from the solar minus years given the criterion for defining these two groups. We do not use a threshold in defining the two groups and therefore no data are excluded for this purpose. The observed (or modeled) temperature field is projected onto this LDA pattern to produce a time series, denoted by \({T}_{{LDA}}\left(t\right)\). The second step of the procedure is to regress this time series onto the TSI, denoted by \(S\left(t\right)\). This step further ensures that global warming signal is eliminated. It is the regressed solarcycle response that is required in the definition of climate sensitivity, \({\mu }_{{SOLAR}}\).
ERF for solar radiative forcing and its uncertainties
AR6 uses ERF (Effective Radiative Forcing) while earlier assessment reports used RF (Radiative Forcing). RF is defined as the radiative forcing at the tropopause after stratospheric ozone and circulation have adjusted, while ERF including fast tropospheric adjustment except surface temperature. The intent is to measure the effectiveness of radiative forcing in its effect in warming the surface in the same way as CO_{2}. ERF for solar radiative forcing is the TSI variation from solar min to solar max, divided by 4 (so that it is per unit area of a spherical earth), multiplied by the ratio \(\xi\) of the downward radiation at the tropopause vs that at the top of the atmosphere, and multiplied by \(\left(1{albedo}\right)\). The simulated value of \(\xi\) is sensitive to the spectral resolution of the input solar spectrum due to the stratospheric O_{3} absorption in the UV range. Gray et al.^{39} estimated the adjusted RF for the 11year solar cycle based on fixed dynamical heating approach and obtained a value of \(\xi\) to be 0.78 using a solar UV spectrum with 1nm resolution. ~0.15 of the reduction from 1.00 is due to the stratospheric O_{3} absorption of the solar variation at wavelength below 300 nm; the rest of the reduction is due to the combination of stratospheric O_{3} absorption of the solar variation above 300 nm and the stratospheric temperature adjustment^{40}. Earlier studies using coarser spectral UV resolutions obtained higher values: Larkin et al.^{41} first obtained a value of 0.88 using a twostream model with six spectral bands; Hansen et al.^{42} reduced a value to 0.83 using a solar spectrum at 5nm resolution. We adopt Gray et al’s^{39} value with 1nm resolution. AR6 also adopted this value for Stratosphereadjusted RF, to which it added –0.06 for tropospheric adjustment to yield \(\xi=0.72\) for use with ERF.
By imposing 5 satelliteobserved solarinduced O_{3} changes and using 2 radiative transfer schemes, Isaksen et al.^{43} showed that the O_{3} solar radiative forcing varied from –0.005 W m^{–2} to 0.008 W m^{–2} relative to the mean. These values are close to zero due to the fact that an increase in ozone in the stratosphere absorbs more of the ultraviolet part of the solar radiation, but the increase in ozone heating emits more downward longwave radiation that almost compensates the decrease in shortwave radiation. Larkin et al.^{41} also tested their solar radiative forcing by replacing the simulated O_{3} solar response with the observed, and the values remain the same as 0.23 W m^{–2} in both cases due to the abovementioned compensation. Because of their broadband calculations, they reported the resulting solar radiative forcing to only the second decimal place. We adopt Isaksen et al.’s values for the uncertainty in \(\xi\). The standard deviation of the 10 net O_{3} solar radiative forcings listed in Isaksen et al.’s^{43} Table 1 is 0.0047. For a mean radiative forcing of \(0.26\times 0.72=0.187 \, {\rm W \, m^{2}}\), we estimate a percentage uncertainty in \(\xi\) to be \(0.0047/0.187\times 1.65\), which is ±4.1% (5–95% range).
The planetary albedo has been measured very accurately by satellites, with an accuracy of ±1% for the 2.5–97.5% range or, equivalently, ±0.86 for the 5–95% range, with very little interannual variability and has a symmetry between the two hemispheres^{38}. Gray et al.^{39} performed a wavelength dependent calculation of the albedo for the solarcycle specific irradiation and concluded that it is almost the same as the wavelengthaveraged planetary albedo. In contrast, the modeled planetary albedo is more variable, and perhaps overtly sensitive to the surface temperature. This explains some of the scatter of the slope in Figs. 6, 7.
Thus, for observational data analysis we use:
where \(\xi=0.72\pm 4.1\%\) (5–95% range), \(\alpha=0.29 \; (\pm 0.86\%)\) (5–95% range), and the total uncertainty for \({\hat{F}}_{{SOLAR}}\) is \(\sqrt{{\left(4.1\%\right)}^{2}+{\left(0.86\%\right)}^{2}}=4.2\%\) (5–95% range). The reciprocal is used in Eq. (5): \(b=\frac{1}{0.127} \pm 4.2\%=7.87 \pm 0.33\) (5–95% range).
Uncertainty in TSI reconstructions
Reconstruction of longterm variation of TSI is based on proxies of solar magnetic activity, mostly sunspot records and chromospheric indices on centennial time scale. A commonly accepted reconstruction is that by Naval Research Laboratory (NRL)^{52}. Most CMIP6 used the recommendation by Mattes et al.^{53}. It is difficult to provide an overall estimated of the uncertainties of these two reconstructions. We have decided to estimate the overall uncertainty of the TSI reconstruction by calculating the difference of the two datasets. The raw data are shown in Fig. 9a, giving a mean absolute difference of 0.075 W m^{–2} for the period 1950–2004 and 0.087 W m^{–2} for the period 1950–2014. Much of the uncertainty is related to the secular trend, while our calculation uses only the difference between the solar max and solar min years. In Fig. 9b we show the mean absolute difference of the detrended time series. Because the trend is nonlinear, linear detrending is not appropriate. We use the Empirical Mode Decomposition^{18,19}. The mean uncertainty is 0.055 W m^{–2} for the period 1950–2004. The mean difference is even smaller, at 0.049 W m^{–2} for the period ending 1950–2014 because there are satellite measurements for the added 10 years. Since the TSI varies by about 1 W m^{–2} from solar min to solar max, the uncertainty of 0.049 W m^{–2} is about 5%. As can be seen in Fig. 9b, the grouping of the solar plus years and solar minus years are identical with either of the solar reconstructions after nonlinear detrending, yielding identical solar temperature response using the LDA method.
Uncertainty in method
The LDA spatial filter is very efficient in the sense that the projected time series \({T}_{{LDA}}\left(t\right)\) after the first step is already very close to the TSI time series^{22}, with a small residue of 0.02 °C–0.04 °C. This small residue is eliminated by the second, regression step. This residue in the intermediate step is not counted as an error in our twostep procedure.
The major uncertainty in the LDA method is related to the determination of the truncation parameter \(r\) for the data regularization. r should be large enough to include the target signal (the solar response in our case) but small enough to prevent overfitting with noise to generate an artificially high separation of the two groups. For HadCRUT5 over 1956–2014, the value of κ reaches a plateau when r = 30, implying that most of the solar response is included when r ≥ 30. Further increasing r leads to overfitting (Supplementary Information). Based on two different values of r (30 and 38), we estimate an additional 1% to the overall uncertainty in \({r}_{{TCR}}\) in Eq. (13) for the 5–95% range for κ.
Sensitivity to period considered
Solar forcing is not sinusoidal. Its TSI has variable amplitudes and slightly variable period. Different response to different forcing period is to be expected. This makes it more important that models and observations are analyzed using the same period when we want to use observations to constrain the models. When there is a secular trend in TSI, the nonlinear trend needs to be first removed. Here we use the Empirical Mode Decomposition^{19,54} for that purpose. For the period 1950–2004, the secular solar trend is small; multicycle mean can be used instead. This is not always the case. Solar Cycle 24, from December 2008 to December 2019, is highly unusual. It is the lowest on record since Solar Cycles 12–15 during 1878–1923, leading many to speculate about the possibility of a “Grand Solar Minimum”. We do not expect that our conclusion about model’s TCR would be affected if both models and observations cover the same period, since we extract the response regressed against the forcing. However, including this extra cycle would include a secular trend, complicating the methods for defining solar plus years or solar minus years. This problem is solved using EMD to remove the nonlinear trend, and the two groups are defined relative to that nonlinear trend locally.
Amdur et al.^{55} considered the period 1959–2019, including Solar Cycle 24, without removing the nonlinear trend. The year 1959 is a solar max and 2019 a min, which by themselves include an imbedded trend from solar max to solar min. They obtained \({\kappa }_{{SOLAR}}=0.07 \pm 0.12 \, {{\,\!}^\circ} {\rm C / W \, m^{2}}\) (2.5–97.5% range) for HadCRUT4 when using the same LDA method as ours. (There may be a typo in their text, where they listed 0.10 ± 0.12, while their Table listed 0.07 ± 0.12). Their central value of 0.07 is very close to our 0.08, but their uncertainty range is more than twice that of their central value and includes 0, partly due to the secular trend in the solar forcing, and partly due to their phase randomization test, which likely led to an overestimated uncertainty, as discussed in Camp and Tung^{25}. They further obtained different values when they slide the 60year window to include different periods with unequal number of solar plus years and solar minus years. Having unequal number of the two solar groups introduces bias when we do not have sufficiently long record to average out the bias of having too many solar maxes or too many solar mins. There was also an error in their code when they were supposed to have used the Composite Mean Difference method to extract the solar cycle response: They actually used Composite “Sum” Difference, which amplifies the problem related to unequal number of samples in each group.
The signal extracted by other authors using purely time series analysis, such as multiple linear regression^{56}, tended to be smaller than what we found using spacetime methods^{20,57}. One should always use the most discriminating method known at the time. The difference between a more discriminating method and a less discriminating method should not be counted as an uncertainty in the method and has not been included in our uncertainty quantification. If this were counted, the uncertainty range can be arbitrarily large because there is always a worse method.
A method easier to implement than LDA is the Composite Mean Difference (CMD) method of Camp and Tung^{25}, and the associated statistical tests as described in detail in Zhou and Tung^{20}. CMD uses the difference in spatial structure between the mean of solar plus years and solar minus years and projecting the spacetime data onto that spatial pattern. CMD is more intuitive and yields the solar cycle signal that is very close to that obtained by LDA but has a larger uncertainty. For this reason, CMD is not employed in this study.
Uncertainty in RF for CO_{2}
Myhre et al.^{58}, obtained by fitting to the results of a broadband model, the following canonical formula for the RF of 2×CO_{2}:
where \(C\) is the atmospheric concentration of CO_{2} and \({C}_{0}\) is its initial value. For doubling CO_{2}, this formula yields \({F}_{2\times C{O}_{2}}=\) 3.71 W m^{–2}. The constant coefficient has an uncertainty of 1%. The broadband model has a reported error of –2.4% compared to the linebyline model. Updating the linebyline radiative transfer calculations and after reviewing the uncertainties involved, Etminan et al.^{59} concluded that the total uncertainty for \({F}_{2\times C{O}_{2}}\) is ±10% (5–95% range), retaining what was previously adopted by IPCC AR5. IPCC AR6 found ERF to be 0.2 W m^{–2} higher than Stratosphere adjusted RF, with a ±12% uncertainty. We suggest that this uncertainty in model diagnosed ERF, which is not an observable quantity, can be avoided by constraining the normalized TCR. In climate model runs, this ERF does not need to be specified. The doubling CO_{2} atmospheric concentration can be specified without uncertainty.
Uncertainty in surface temperature data
HadCRUT5^{17} is a combination of a global landsurface temperature dataset, CRUTEM5, and the global SST dataset, HadSST4. There is no interpolation (infilling) performed and so there are some coverage gaps. Although poor in the 1850–1880 period, the data coverage is good after 1950, at over 75%. We use only the data after 1950. Two hundred ensemble members, an increase from the 100 ensemble members in the previous version, are used to provide an assessment of observational uncertainties, such as those arising from changing instrumental and observational practices, changes in station locations, and changes in local land use. These uncertainties are shared by other datasets. We apply the LDA method to each ensemble member to extract a solarcycle response. The larger number of ensemble members yields a larger uncertainty range (about twice) compared from that obtained from the previous version, HadCRUT4.6. The central value is obtained as the ensemble mean.
Richardson et al^{8}. pointed out the importance of using a geographically complete dataset. They found a significant 15% difference in the inferred TCR between a masked version and the complete model dataset and adjusted their mean value of TCR by this modeled ratio to account for the effect if the dataset were complete. However, this adjustment is model dependent. Currently one geographically complete observational dataset is available, provided by GISTEMP, which builds upon the latest NOAA Merged Land–Ocean Global Surface Temperature Analysis. The NOAA seasurface temperature (ERSSTv5) is complete, based on satellite observations since 1979, but some missing data exist over land. Unlike HadCRUT5, GISTEMP does not provide a way for us to assess their uncertainty. Its solarcycle responses however fall within the uncertainty range of HadCRUT5 (see Fig. 1).
In 2008, because geographically complete datasets were not available, Tung et al.^{22} included two reanalyses, ERA40 and NCEP. Solarcycle response from ERA40 and NCEP differ significantly. While ERA40’s solarcycle response is 0.12 °C, NCEP is an outlier at 0.17 °C. While ERA40 produced 2m temperature by interpolating from SST, NCEP’s “surface” temperature (at 0.995 sigma level) was produced by the model using upper air observations and surface pressure. It does not assimilate the observed 2m temperature over land^{60}. Tung and Camp^{61} found the solar signal obtained from NCEP’s “surface” temperature to be the same as that found from its 925hPa temperature, which tends to be higher in magnitude than that from the surface temperature. NCEP’s interannual variance is generally higher than ERA’s^{61}. It should be pointed out that the NCEP’s higher value was not used in constraining the TCR in Tung et al.^{22}. At the time, the linear relationship in Eq. (12) was not available because the reported radiative transfer calculations available (from the top of the atmosphere to the tropopause) were to us ambiguous. That study resorted to an inequality that the TCR should be larger than a minimum value corresponding to the lowest solarcycle response. NCEP’s higher value was therefore not used as a constraint. It was used only to indicate the uncertainty of this lower bound and did not affect the conclusion. HadCRUT3 used then has now been updated to HadCRUT5 and GISS has gone through a major update incorporating NOAA’s ERSSTv5. The newer datasets are geographically more complete, and lead to lower solarcycle responses in surface temperature.
CMIP6 models
Previously, it was thought that climate models either cannot generate solarcycle signals at the surface or, if they could, the amplitude would be too small to be detected over climate noise^{62}. Consequently, the solarcycle response has not been used to constrain model climate sensitivities. Some still think the surface response to the small radiative forcing is too small as calculated from a backofenvelope estimate, without realizing that, like greenhouse gas warming, the response should be amplified by the various climate feedbacks in the troposphere. It was also argued by some studies that exotic mechanisms not included in CMIP models, such as cosmic rays forming condensation nuclei for clouds, need to be invoked. Solarcycle responses from the climate models are extracted from each model’s surface temperature using the method of LDA^{61,63}, which utilizes the spatialtemporal information of the phenomenon. The models studied here include some with interactive ozone chemistry and some without, and yet there is no systematic difference between them in their solarcycle response. This is noteworthy as one of the proposed mechanisms, the socalled “stratospheric pathway”^{64}, involves interactive ozone in the stratosphere: The UV portion of the solar irradiance produces ozone in the stratosphere, and the additional ozone heating generates a stratosphere temperature signal. Although it is known that very little of the ultraviolet portion of the 11year solar cycle forcing gets through the stratosphere into the troposphere because of the effective ozone absorption in the stratosphere, there have been some proposals for an indirect influence of the stratosphere on the troposphere^{64}. The proposed pathway may affect regional variability for which an interactive ozone model is important. Here we study the global surface signal.
There have been proposals of the galactic cosmic rays (GCR) affecting the troposphere via their forming condensation nuclei for clouds. In fact, it was suggested that the early twentieth century global warming was caused by the cosmic rays. The second author, Tung, was in a National Academy of Sciences team that examined this issue in 2012^{62} and found no evidence supporting the proposal and evidence contradicting it. Later in 2017 a satellite called “CLOUD” was launched to study this issue. The results were that the cosmic rays are “unlikely to be comparable to the effect of large variations in natural primary aerosol emissions”.
Fluxes of energetic particles consisting of corona mass ejection of energetic particles (mostly ionized hydrogen) and the effect of solar wind on cosmic rays can potentially affect atmospheric composition, but mostly in the upper atmosphere. Most CMIP6 models are unable to account for these particle fluxes with the exception of a few, such as WACCM. Notable recent studies focusing on the impacts of GCRs, solar proton events and energetic electron precipitation on chemical composition of the atmosphere include Calisto et al.^{65} and Rozanov et al.^{66}. They showed using the SOCOL model that the GCR induced changes in the chemical species, such as NO_{x}, HO_{x}, O_{3}, are largest as measured in percentage (~10%) in the middle troposphere. However, since these chemical tracers are most abundant in the stratosphere and less abundant in the troposphere by at least an order of magnitude, the contribution of the tropospheric change to the total NO_{x} abundance is very small.
In our recent publications^{67,68} we estimated the solarcycle induced changes in NO_{2} using groundbased observations, WACCM, and our own 1D photochemical model. The changes are largely in the stratosphere, near the 1hPa level, with an exception during rare nonlinear events such as tropopause folding in the midlatitude^{68}. Thus, the overall solarcycle (or GCR) induced changes in the tropospheric NO_{x} is much less than other variabilities, e.g. anthropogenic pollution, and therefore we do not focus on its climate impact in the current study.
The aforementioned studies also suggested that the GCRinduced changes of stratospheric species could lead to a reduction in stratospheric O_{3}, which in turn could strengthen the polar night jets and potentially indirectly impact surface climate [e.g. in northern Europe]. Matthes et al.^{53} used one of the CMIP6 model, WACCM and obtained a weakened polar night jet in the solar maximum, consistent with those studies, potentially impacting regional climate. We are interested however in globally averaged surface response.
In summary, these and the studies mentioned earlier suggest that while these particle fluxes may affect chemical composition in the upper atmosphere, their impact on surface global temperature response is likely small. Furthermore, even if the effect is not small, it does not affect our results as long as these variations follow the time variation of the \({TSI}\left(t\right)\), though some may be anticorrelated. This is because we extract the total solar cycle response from all solar forcing and regressed against the \({TSI}\left(t\right)\) for both models and observation. In observation, obviously the effect is included. In the few models which include particle flux forcing and SSI, the response should include it. Though if such particle forcing and or SSI is important, the denominator of the regression could be larger or smaller. But since both the models and observations used the same regression, the relationship between them does not change.
CMIP6 models and their TCR
Model TCRs were taken from Meehl et al.^{1} for CMIP6. Uncertainty in how TCR is calculated is usually not adequately included in previous uncertainty analyses of climate sensitivity. As pointed out in our previous work^{69}, one major uncertainty is the internal variability in the control run affecting the baseline temperature which was subtracted from the 70^{th} year temperature to calculate TCR. This variability yields TCR values that could be 30% different for the same model depending on which year is chosen for the baseline. We previously recommended taking a 100year mean of the preindustrial control run for the baseline value, and that the control run be at least 1,000 years long. We also showed that the subtraction of the climate drift of a parallel control run is no substitute for adequately long control runs. These protocols are partially adopted for the CMIP6 protocol: The control runs were recommended to be at least 500 years, and the ESMValTool in Meehl et al. calculated TCR using 20year averaging of the 60^{th}–79^{th} years of the run minus the PreIndustrial baseline value. For some models the control runs are still too short, and substantial uncertainty may remain^{69,70}. In AR4, the model with the highest TCR was MIROC(hires), at 2.6 °C. It had only 109 years of coupled spinup. MIROC6 now has 1000 years of ocean spin up. The MIROCES2L model has 3,000 years of ocean spinup and 1000 years of coupling to the atmosphere. Their TCR of 1.6 is now among the lowest in CMIP6. The uncertainty associated with inadequate time for the spinup and control runs remains unquantified and unquantifiable by us. Therefore, we exclude from our study models with such problems (See Supplementary Table S1).
Solar responses simulated in all ensemble members of the same model are averaged to give a single value that is listed in Supplementary Table S1, except for MIROC6. MIROC6 has 50 ensemble members starting with different branchoff times of the preindustrial simulation. Some members branched off too early, not giving adequate spinup time before branching. The average solar response of MIROC6 is here obtained using the 20 ensemble members that have branchoff times of at least 600 years.
Lag in solarcycle response relative to forcing
The lag extracted from observations is close to zero. Since we used annual mean data to avoid the rather large seasonal variability, a lag of a few months cannot be ruled out, but such a lag cannot be more than six months. We assign an uncertainty in this slope using the two extreme values of lag, 0 and 6 months.
Data availability
All data used in this study are publicly available. The CMIP6 data were downloaded from https://esgfnode.llnl.gov/. The ERA40 and ERA5 data were downloaded from https://www.ecmwf.int/. The HadCRUT5 data were downloaded from https://www.metoffice.gov.uk/hadobs/hadcrut5/. The GISTEMP4 data were downloaded from https://data.giss.nasa.gov/gistemp/. The solar flux used in CMIP6 models can be obtained at https://solarisheppa.geomar.de/cmip6. This is also the solar forcing used here in our analyses. For comparison purposes we also used the Naval Research Lab’s solar forcing: https://data.giss.nasa.gov/modelforce/solar.irradiance/
Code availability
The algorithm of the linear discriminant analysis (LDA) used to extract the solar signal in this study is contained in the Supplementary Information. The work presented in the main text was obtained using the LDA algorithm implemented in IDL.
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Acknowledgements
KFL has been supported in part by the National Science Foundation’s Coupling, Energetics, and Dynamics of Atmospheric Regions Program under Grant 2230265. KKT’s research was supported in part by National Science Foundation under Grant 1536175, and by the Frederic and Julia Wan Endowed Professorship.
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K.F.L. and K.K.T. conceived the project and designed the experiments. K.F.L. carried out the experiments, performed the analysis and statistical tests. K..F.L. developed the tools and wrote the Supplementary Information. K.K.T. interpreted the results and wrote the main parts of the paper.
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Li, KF., Tung, KK. Solar cycle as a distinct line of evidence constraining Earth’s transient climate response. Nat Commun 14, 8430 (2023). https://doi.org/10.1038/s41467023435837
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DOI: https://doi.org/10.1038/s41467023435837
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