Emergent constraint on equilibrium climate sensitivity from global temperature variability

Transcription

1 Letter doi: /nture5450 Emergent constrint on equilirium climte sensitivity from glol temperture vriility Peter M. Cox 1, Chris Huntingford & Mrk S. Willimson 1 Equilirium climte sensitivity (ECS) remins one of the most importnt unknowns in climte chnge science. ECS is defined s the glol men wrming tht would occur if the tmospheric cron dioxide (CO ) concentrtion were instntly douled nd the climte were then rought to equilirium with tht new level of CO. Despite its rther idelized definition, ECS hs continuing relevnce for interntionl climte chnge greements, which re often frmed in terms of stiliztion of glol wrming reltive to the preindustril climte. However, the likely rnge of ECS s stted y the Intergovernmentl Pnel on Climte Chnge (IPCC) hs remined t degrees Celsius for more thn 5 yers 1. The possiility of vlue of ECS towrds the upper end of this rnge reduces the fesiility of voiding degrees Celsius of glol wrming, s required y the Pris Agreement. Here we present new emergent constrint on ECS tht yields centrl estimte of.8 degrees Celsius with 66 per cent confidence limits (equivlent to the IPCC likely rnge) of. 3.4 degrees Celsius. Our pproch is to focus on the vriility of temperture out long-term historicl wrming, rther thn on the wrming trend itself. We use n ensemle of climte models to define n emergent reltionship etween ECS nd theoreticlly informed metric of glol temperture vriility. This metric of vriility cn lso e clculted from oservtionl records of glol wrming 3, which enles tighter constrints to e plced on ECS, reducing the proility of ECS eing less thn 1.5 degrees Celsius to less thn 3 per cent, nd the proility of ECS exceeding 4.5 degrees Celsius to less thn 1 per cent. Mny ttempts hve een mde to constrin ECS, typiclly using either the record of historicl wrming or reconstructions of pst climtes 4. Methods sed on historicl wrming re ffected y uncertinties in ocen het uptke nd the contriution of erosols to net rditive forcing 5,6. These methods lso dignose the effective climte sensitivity over the historicl period, which my e different to ECS, owing to the strength of climte feedcks vrying with the evolving pttern of surfce temperture chnge 4,7 9. Although methods sed on pst climtic periods, such s the Lst Glcil Mximum 10, re more closely relted to the concept of equilirium, they suffer insted from even lrger uncertinties in the reconstruction of net rditive forcing. As n lterntive, the emergent constrint pproch uses n ensemle of complex Erth system models to estimte the reltionship etween modelled ut oservle vrition in the Erth system nd predicted future chnge,11. The model-derived emergent reltionship cn then e comined with the quntifiction of the oserved vrition to produce n emergent constrint on the predicted future chnge,11,1. Here we present n emergent constrint on ECS tht is sed on the vriility of glol-men temperture. To inform our serch for n emergent constrint, we consider the simple Hsselmnn model 13 for the vrition in glol men temperture Δ T in response to rditive forcing Q: dδ T C = Q λ Δ T = N dt (1) The constnt het cpcity C in this model is simplifiction tht is known to e poor representtion of ocen het uptke on longer timescles However, we find tht it still offers very useful guidnce out glol temperture vriility on shorter timescles. The climte Temperture nomly (K) ECS (K) Simultion of glol wrming record < 1.0 W m K 1 > 1.0 W m K 1 Oservtions Yer ECS versus root-men-squre error in simultion h m o j l d f i g e n Root-men-squre error in ΔT (K) Figure 1 Historicl glol wrming., Simulted chnge in glol temperture from 16 CMIP5 models (coloured lines), compred to the glol temperture nomly from the HdCRUT4 dtset (lck dots). The nomlies re reltive to seline period of The model lines re colour-coded, with lower-sensitivity models (λ > 1 W m K 1 ) shown y green lines nd higher-sensitivity models (λ < 1 W m K 1 ) shown y mgent lines., Sctter plot of ech model s ECS ginst the root-men-squre error in the fit of ech model to the oservtionl record. Individul CMIP5 model runs re denoted y the letters listed in Extended Dt Tle 1. k c p 1 College of Engineering, Mthemtics nd Physicl Science, University of Exeter, Exeter EX4 4QF, UK. Centre for Ecology nd Hydrology, Wllingford OX10 8BB, UK. 18 jnury 018 VOL 553 NATURE Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

2 RESEARCH Letter Metric of vriility versus time PDF of emergent constrint 0.5 < 1.0 W m K 1 > 1.0 W m K Oservtions 0.5 (K) Prolity density End of window ECS (K) ECS (K) Emergent reltionship fit p k d m l i f h o c n e j Liner regression Oservtionl constrint g CDF CDF of emergent constrint Emergent constrint CMIP5 models feedck fctor λ determines how the net top-of-tmosphere plnetry energy lnce N vries with temperture chnge Δ T in response to rditive forcing chnge Q. ECS nd λ re inversely relted, with constnt of proportionlity tht is the rditive forcing due to douling of tmospheric CO, Q CO so tht ECS = Q CO /λ. Although the dignosed Q CO vries cross the model ensemle 17, the uncertinty in ECS is predominntly due to uncertinty in λ, which vries from 0.6 W m K 1 to 1.8 W m K 1, s shown in Extended Dt Tle 1. If Q cn e pproximted s white-noise forcing with vrince Q, the Hsselmnn model cn e solved to give expressions for the vrince of glol temperture T nd the one-yer-lg utocorreltion of the glol temperture α 1T, which cn e comined to yield n eqution for ECS (see Methods): ECS= Q CO (K) Figure Metric of glol men temperture vriility., Ψ metric of vriility versus time, from the CMIP5 models (coloured lines), nd the HdCRUT4 oservtionl dt (lck circles). The Ψ vlues re clculted for windows of width 55 yr, fter liner de-trending in ech window. These 55-yr windows re shown for different end times. As in Fig. 1, lower-sensitivity models (λ > 1 W m K 1 ) re shown y green lines nd higher-sensitivity models (λ < 1 W m K 1 ) re shown y mgent lines., Emergent reltionship etween ECS nd the Ψ metric. The lck dotdshed line shows the est-fit liner regression cross the model ensemle, with the prediction error for the fit given y the lck dshed lines (see Methods). The verticl lue lines show the oservtionl constrint from the HdCRUT4 oservtions: the men (dot-dshed line) nd the men plus nd minus one stndrd devition (dshed lines). T 1 Q CO Ψ α = () Q loge 1T Q ECS (K) Figure 3 Emergent constrint on ECS., The PDF for ECS., The relted CDF. The horizontl dot-dshed lines show the 66% confidence limits on the CDF plot. The ornge histogrms (oth pnels) show the prior distriutions tht rise from equl weighting of the CMIP5 models in 0.5 K ins. where Ψ = T/ logeα1 T is our key metric of glol temperture vriility. This eqution is essentilly fluctution dissiption reltionship 18 relting the vriility of the climte (Q, T, α1 T) to its sensitivity to externl forcing (ECS). Oservtionl records of glol men temperture chnge 3 enle Ψ to e estimted for the rel world. The vrince of the net rditive forcing is pproximtely equl to the vrince of the top-of-thetmosphere flux N, which cn in principle e estimted from stellite mesurements. However, the ville stellite records re currently too short to provide relile estimtes of N. In ddition, the rditive forcing due to douling CO (Q CO ) is not oservle in the rel world. This mens tht the right-hnd side of eqution () cnnot e directly estimted from oservtions. Fortuntely, we find tht the vrition in ECS is wekly correlted with Q CO / N cross the model ensemle (see Extended Dt Tle 1). We cn therefore pproximte the predicted grdient of the ECS versus Ψ emergent reltionship using the ensemle men vlue of Q CO / N (= 8.7). Our theory therefore predicts grdient of the ECS versus Ψ emergent reltionship of 87. = 1.. Figure 1 shows the simultion of glol wrming in the historicl simultions with the 16 models in the CMIP5 ensemle 19,0 used here (see list in Extended Dt Tle 1). Here nd throughout, highersensitivity models (λ < 1.0 W m K 1 ) re shown in mgent nd 30 NATURE VOL jnury Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

3 Letter RESEARCH lower-sensitivity models (λ > 1.0 W m K 1 ) re shown in green. Oservtions from the HdCRUT4 dtset 3 re shown y the lck line mrked with dots. Figure 1 illustrtes tht oth high- nd lowsensitivity models re le to fit the historicl record with resonle fidelity, despite implying very different future climtes. Models with higher ECS vlues lso hve longer response times, nd there re vritions cross the models in net rditive forcing nd in ocen het uptke llowing models with oth high nd low sensitivities to reproduce historicl glol wrming 1. As result, the fit to the glol temperture record does not provide direct constrint on ECS, s shown in Fig. 1. To test whether vriility is etter constrint on ECS, we de-trend the glol men temperture records from the models nd the oservtions. Our pproch to de-trending is informed y techniques designed to detect precursors of potentil tipping points such s criticl slowing down 3. The method pplied in tht cse is to use moving window, to linerly de-trend within tht window, nd then to clculte sttistics of the de-trended residuls. For tipping point detection, the fvoured vrile is often the utocorreltion, which mesures the memory in fluctutions of the nlysed vrile 3. We use similr pproch, lthough here we pply it to nlyse the reltionship etween Ψ nd ECS cross the ensemle of models, rther thn to detect declining system resilience in single reliztion of the system. We nlyse the nnul-men glol-men temperture time series from 16 CMIP5 historicl simultions nd compre to the HdCRUT4 oservtionl dtset. Although there were nother 3 historicl runs ville in the CMIP5 rchive, we chose to use just one model vrint from ech climte centre, to void ising the emergent constrint towrds the centres with the most model runs in the rchive. Where there ws more thn one model vrint from modelling centre, we took the model vrint from tht centre tht hd the smllest rootmen-squre (r.m.s.) error in the fit to the record of oserved glol wrming from 1880 to 016. The remining 3 model runs (which included some initil condition ensemles) were susequently used to test the roustness of the emergent constrint (see Extended Dt Fig. 1). Figure shows the resulting vrition in Ψ for ech of the models nd the oservtions, using window width of 55 yr, nd dt from 1880 to 016 to mtch the ville oservtionl dtsets. Although Ψ vries in time, the different models re clerly distinguished, in contrst to the simultions of historicl glol wrming (Fig. 1). In prticulr, the Ψ vlues seprte higher-sensitivity models (mgent lines) from lower-sensitivity models (green lines), with higher-sensitivity models producing lrger Ψ vlues. It is lso worth noting tht Ψ from the oservtionl dt re within the rnge of the lower-sensitivity models ut clerly outside the rnge of the higher-sensitivity models. Figure shows the emergent reltionship etween ECS nd the time-men Ψ vlues cross the model ensemle, with est-fit grdient tht is very close to our theoreticl vlue. The verticl lue lines show the oservtionl constrint on Ψ from the HdCRUT4 dtset, ut similr oservtionl constrints re lso derived from other dtsets of glol men temperture (see Extended Dt Tle ). As in previous studies 11,1 the emergent reltionship from the historicl runs nd oservtionl constrint cn e comined to provide n emergent constrint on ECS. This involves convolving the prediction error implied y the fit of the sctter plot to the emergent reltionship, with the uncertinty in the oservtions, to produce proility density function (PDF) for the y-xis vrile (see Methods). Figure 3 shows the resulting PDF for ECS (lck curve). For comprison, the prior PDF implied y the equl-weighted model ensemle is shown y the ornge histogrm. The emergent constrint PDF is shrply peked round est estimte of ECS =.8 K, which is slightly smller thn the centre of the IPCC rnge of K. Our est estimte of ECS is considerly lrger thn the vlues derived from rw energy udget constrints 8,4,5 ut similr to some recent ECS (K) Proility (%) Emergent constrint versus window width Best estimte 66% confidence limits Window width (yr) Proility of high/low ECS versus window width Proility of ECS < 1.5 K Proility of ECS > 4 K Window width (yr) Figure 4 Sensitivity of the emergent constrint on ECS to window width., Centrl estimte nd 66% confidence limits. The thick lck r shows the minimum uncertinty t window width of 55 yr nd the red r shows the equivlent likely IPCC rnge of K., Proilities of ECS > 4 K (red line nd symols) nd ECS < 1.5 K (lue line nd symols). estimtes tht ccount for time-dependent nd forcing-dependent feedcks 9,6. Figure 3 shows the resulting cumultive density function (CDF), which gives the proility of ECS tking vlue lower thn the vlue shown on the x xis. The lck horizontl lines in Fig. 3 show the 66% confidence limits (. K to 3.4 K), or pproximtely.8 ± 0.6 K. Reltive to the IPCC rnge of K, this constrint on ECS therefore reduces the uncertinty y out 60%. Indeed, even the 95% confidence limits from the emergent constrint (1.6 K to 4.0 K) fit well within the IPCC likely rnge for ECS. Our constrint is therefore t odds with suggestion tht the lower 66% confidence limit for ECS could e s high s 3 K (ref. 7). If we insted use ll 39 historicl runs in the CMIP5 rchive, we find slightly weker emergent reltionship, ut derive very similr emergent constrint on ECS (Extended Dt Tle ). The constrint is lso roust to the choice of oservtionl dtset, nd to whether or not the model glol temperture is clculted just cross the points where there were oservtions 8 (Extended Dt Tle nd Extended Dt Fig. ). Our choice of window width ws informed y sensitivity studies in which the emergent constrint ws clculted for rnge of this prmeter. Figure 4 shows the est estimte nd 66% confidence limits on ECS s function of the width of the de-trending window. Our est estimte is 18 jnury 018 VOL 553 NATURE Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

4 RESEARCH Letter reltively insensitive to the chosen window width, ut the 66% confidence limits show greter sensitivity, with the minimum in uncertinty t window width of out 55 yr (s used in the nlysis ove). As Extended Dt Fig. 3 shows, t this optimum window width the est-fit grdient of the emergent reltionship etween ECS nd Ψ (= 1.1) is lso very close to our theory-predicted vlue of Q CO / Q (= 1.). This might e expected if this window length optimlly seprtes forced trend from vriility. Figure 4 shows the proility of ECS > 4 K nd ECS < 1.5 K s function of window width. For comprison, the IPCC likely rnge of K implies 5% proility of ECS > 4 K, nd 16% proility of ECS < 1.5 K. At the optimum window width of 55 yr, oth proilities re close to their minimum vlues of less thn.5%. Our emergent constrint therefore gretly reduces the uncertinty in the ECS vlue of Erth s climte, implying less thn 1 in 40 chnce of ECS > 4 K, nd renewing hope tht we my yet e le to void glol wrming exceeding K. Online Content Methods, long with ny dditionl Extended Dt disply items nd Source Dt, re ville in the online version of the pper; references unique to these sections pper only in the online pper. received 7 July; ccepted 13 Decemer Collins, M. et l. Long-term climte chnge: projections, commitments nd irreversiility. In Climte Chnge 013: The Physicl Science Bsis. Contriution of Working Group I to the Fifth Assessment Report of the Intergovernmentl Pnel on Climte Chnge (eds Stocker, T. F. et l.) Ch. 1 (Cmridge Univ. Press, 013).. Hll, A. & Qu, X. Using the current sesonl cycle to constrin snow ledo feedck in future climte chnge. Geophys. Res. Lett. 33, L0350 (006). 3. Morice, C. P. et l. Quntifying uncertinties in glol nd regionl temperture chnge using n ensemle of oservtionl estimtes: the HdCRUT4 dtset. J. Geophys. Res. 117, D08101 (01). 4. Knutti, R. et l. Beyond equilirium climte sensitivity. Nt. Geosci. 10, (017). 5. Gregory, J. M. et l. An oservtionlly sed estimte of the climte sensitivity. J. Clim. 15, (00). 6. Forster, P. M. et l. Evluting djusted forcing nd model spred for historicl nd future scenrios in the CMIP5 genertion of climte models. J. Geophys. Res. Atmos. 118, (013). 7. Gregory, J. M. & Andrews, T. Vrition in climte sensitivity nd feedck prmeters during the historicl period. Geophys. Res. Lett. 43, (016). 8. Forster, P. M. Inference of climte sensitivity from nlysis of Erth s rdition udget. Ann. Rev. Erth Plnet. Sci. 44, (016). 9. Armour, K. C. Energy udget constrints on climte sensitivity in light of inconstnt climte feedcks. Nt. Clim. Chng. 7, (017). 10. Annn, J. D. & Hrgreves, J. C. Using multiple oservtionlly-sed constrints to estimte climte sensitivity. Geophys. Res. Lett. 33, L06704 (006). 11. Cox, P. M. et l. Sensitivity of tropicl cron to climte chnge constrined y cron dioxide vriility. Nture 494, (013). 1. Wenzel, S. et l. Projected lnd photosynthesis constrined y chnges in the sesonl cycle of tmospheric CO. Nture 538, (016). 13. Hsselmnn, K. Stochstic climte models. I. Theory. Tellus 8, (1976). 14. McMynowski, D. G. et l. The frequency response of temperture nd precipittion in climte model. Geophys. Res. Lett. 38, L16711 (011). 15. Cldeir, K. & Myhrvold, N. P. Projections of the pce of wrming following n rupt increse in tmospheric cron dioxide concentrtion. Environ. Res. Lett. 8, (013). 16. Geoffroy, O. et l. Trnsient climte response in two-lyer energy-lnce model. Prt I: Anlyticl solution nd prmeter clirtion using CMIP5 AOGCM experiments. J. Clim. 6, (013). 17. Flto, G. et l. Evlution of climte models. In Climte Chnge 013: The Physicl Science Bsis. Contriution of Working Group I to the Fifth Assessment Report of the Intergovernmentl Pnel on Climte Chnge (eds Stocker, T. F. et l.) Ch. 9 (Cmridge Univ. Press, 013). 18. Leith, C. E. Climte response nd fluctution dissiption. J. Atmos. Sci. 3, 0 06 (1975). 19. Tylor, K. E., Stouffer, R. J. & Meehl, G. A. An overview of CMIP5 nd the experiment design. Bull. Am. Meteorol. Soc. 93, (01). 0. Andrews, T. et l. Forcing, feedcks nd climte sensitivity in CMIP5 coupled tmosphere-ocen models. Geophys. Res. Lett. 39, L0971 (01). 1. Kiehl, J. T. Twentieth century climte model response nd climte sensitivity. Geophys. Res. Lett. 34, L710 (007).. Lenton, T. M. et l. Tipping elements in the Erth s climte system. Proc. Ntl Acd. Sci. USA 105, (008). 3. Scheffer, M. et l. Erly-wrning signls for criticl trnsitions. Nture 461, (009). 4. Otto, A. et l. Energy udget constrints on climte response. Nt. Geosci. 6, (013). 5. Lewis, N. & Curry, J. A. The implictions for climte sensitivity of AR5 forcing nd het uptke estimtes. Clim. Dyn. 45, (015). 6. Mrvel, K. et l. Implictions for climte sensitivity from the response to individul forcings. Nt. Clim. Chng. 6, (015). 7. Sherwood, S. C., Bony, S. & Dufresne, J.-L. Spred in model climte sensitivity trced to tmospheric convective mixing. Nture 505, 37 4 (014). 8. Cowtn, K. & Wy, R. G. Coverge is in the HdCRUT4 temperture series nd its impct on recent temperture trends. Q. J. R. Meteorol. Soc. 140, (014). Acknowledgements This work ws supported y the Europen Reserch Council (ERC) ECCLES project, grnt greement numer 7447 (P.M.C.); the EU Horizon 00 Reserch Progrmme CRESCENDO project, grnt greement numer (P.M.C. nd M.S.W.); the EPSRC-funded ReCoVER project (M.S.W.); nd the NERC CEH Ntionl Cpility fund (C.H.). We lso cknowledge the World Climte Reserch Progrmme s Working Group on Coupled Modelling, which is responsile for CMIP, nd we thnk the climte modelling groups (listed in Extended Dt Tle 1 of this pper) for producing nd mking ville their model output. Author Contriutions All uthors collortively designed the study nd contriuted to the mnuscript. P.M.C. led the study nd drfted the mnuscript. C.H. ws the led on the time-series dt for the CMIP5 models. M.S.W. led on the theoreticl nlysis. Author Informtion Reprints nd permissions informtion is ville t The uthors declre no competing finncil interests. Reders re welcome to comment on the online version of the pper. Pulisher s note: Springer Nture remins neutrl with regrd to jurisdictionl clims in pulished mps nd institutionl ffilitions. Correspondence nd requests for mterils should e ddressed to P.M.C. (p.m.cox@exeter.c.uk). Reviewer Informtion Nture thnks P. Forster nd T. Muritsen for their contriution to the peer review of this work. 3 NATURE VOL jnury Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

5 Letter RESEARCH Methods Theoreticl sis for the emergent reltionship. We hypothesize tht eqution (1) (the Hsselmnn model ) is resonle pproximtion to the short-term vriility of the glol men temperture nomly Δ T: dδ T C + λ Δ T = Q dt If trends rising from net rditive forcing nd ocen het uptke cn e successfully removed, the net rditive forcing term Q cn e pproximted y white noise. Under these circumstnces, eqution (1) is essentilly the Ornstein Uhleneck eqution, which descries Brownin motion, nd hs stndrd solutions (for exmple, see Uhleneck_process) for the lg-one-yer utocorreltion of the temperture: nd the rtio of the vrinces of T nd Q: α 1T (3) λ = exp (4) C T Q = 1 λ C These two equtions cn e comined to eliminte the unknown het cpcity C nd therefore to provide n expression for the climte feedck fctor λ: λ = Q T (5) α 1 log e 1T (6) The ECS nd λ re inversely relted y constnt of proportionlity, which is the rditive forcing due to douling of tmospheric CO (Q CO ), so tht ECS = Q CO /λ. Thus, we cn lso derive n expression for ECS in terms of the vriility of T nd Q: ECS = Q CO T Q log α e 1T Lest-squres liner regression. Lest-squres liner regressions were clculted using well estlished formule (see for exmple LestSquresFitting.html). The liner regression f n etween time series given y y n nd time series given y x n is defined y grdient nd intercept : (7) f = + x (8) Minimizing the lest-squres error for y n involves minimizing: n N n= 1 n 1 s = { yn fn} (9) N where N is the numer of dt points in ech time series. In this cse, the est-fit grdient is given y: xy = (10) x Here x = n N = { x x } 1 n / N is the vrince of x n nd xy = n N = 1 { x n x} { yn y} / N is the covrince of the x n nd y n time series, with mens of x nd y, respectively. The stndrd error of is given y: = s N which defines Gussin proility density for : P ( ) = 1 π x ( ) exp Finlly, the prediction error of the regression is the following function of x: 1 { x x = s + + x} f( ) 1 N N x (11) (1) (13) This expression defines contours of equl proility density round the est-fit liner regression, which represent the proility density of y given x: Pyx { } = 1 f π ( y f( x)) exp f (14) where f = f ( x), s descried ove. Clcultion of the PDF for ECS. The emergent constrint derived in this study is liner regression cross the CMIP5 models etween ECS nd the Ψ sttistic of the de-trended glol temperture. In the context of the lest-squres liner regression presented ove, ECS is equivlent to y, nd Ψ is equivlent to x. The liner regression therefore provides n eqution for the proility of ECS given Ψ (tht is, the eqution for P{y x} ove). In ddition, the Ψ sttistic clculted from the de-trended oservtionl dtset provides n oservtion-sed PDF for Ψ. Given these two PDFs, P{ECS Ψ} nd P(Ψ), the PDF for ECS is clculted y numericlly integrting: P(ECS) = P{ECS Ψ} P( Ψ)d Ψ (15) Dt vilility. The dtsets generted during the current study re ville from the corresponding uthor on resonle request. Code vilility. The Python code used to produce the figures in this pper is ville from the corresponding uthor on resonle request. 018 Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

6 RESEARCH Letter Extended Dt Figure 1 Test of emergent reltionship ginst models not used in the clirtion. The test set includes dditionl models from some climte centres (lelled f x, f y nd so on), nd initil condition ensemles with prticulr models (lelled c, c 3 nd so on). The lck dot-dshed line shows the est-fit liner regression cross the model ensemle, with the prediction error for the fit given y the lck dshed lines (see Methods). The verticl lue lines show the oservtionl constrint from the HdCRUT4 oservtions: the men (dot-dshed line) nd the men plus nd minus one stndrd devition (dshed lines). Individul CMIP5 model runs re denoted y the letters listed in Extended Dt Tle Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

7 Letter RESEARCH Extended Dt Figure Comprison of Ψ sttistics for the 16 CMIP5 models from filtered-men temperture nd glol-men temperture. The filtered model output clcultes re-men vlues of temperture using only the points where there re oservtions in the HdCRUT4 dtset. All cses nlyse nd use 55-yr window width. The dotted line is the 1:1 line. 018 Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

8 RESEARCH Letter Extended Dt Figure 3 Grdient of emergent reltionship etween ECS nd Ψ s function of window width. The dotted line shows the grdient predicted with eqution () using the ensemle-men vlue of Q CO / N. Note tht the theory (dot-dshed line) fits est t the optiml window width of 55 yr. All cses here nlyse nd use the 16-model ensemle. 018 Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

9 Letter RESEARCH Extended Dt Tle 1 Erth system models used in this study, s provided y the CMIP5 project 19 The first column shows the symol used for ech model in Figs 1 nd. The third nd fourth columns list λ nd the ECS vlues s given in IPCC AR5 tle 9.5 (ref. 17). The fifth nd sixth columns show sttistics clculted in this study for the period nd using window width of 55 yr. The fifth column shows the rtio of the rditive forcing due to douling CO (Q CO) to the stndrd devition of the net top-of-tmosphere flux N; nd the sixth column shows the time-men Ψ sttistic for ech model. 018 Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.

10 RESEARCH Letter Extended Dt Tle Roustness of the emergent constrint to the choice of oservtionl dtset nd model ensemle The ALL dtset tkes the men nd stndrd devition of the Ψ vlues for ll four glol-men temperture dtsets (y conctenting the individul Ψ time series). The filtered model output clcultes re-men vlues of temperture just using the points where there re oservtions in the HdCRUT4 dtset 7. All cses nlyse nd use 55-yr window width. 018 Mcmilln Pulishers Limited, prt of Springer Nture. All rights reserved.