| 31 Aug 2026
Emergence of a non-linear global surface air temperature response to rapidly increasing effective radiative forcing
Abstract. A time series of global surface air temperature anomalies (GSAT) and effective radiative forcing (ERF) obtained from IPCC Sixth Assessment Report (AR6) for 1977 to 2025 provides evidence that GSAT is a non-linear, power-law function of ERF when ERF is rapidly increasing. With ERF > 0 W m−2 (n = 48), the GSAT-ERF relationship is described by a power-law, GSAT = 0.41 + 0.11 × ERF1.90 (R² = 0.896; likely range = ERF1.74-2.24). A linear relationship was rejected (F(1,45) = 15.41, p < 0.0003). GSAT, ERF1.90, and the Niño 3.4 sea surface temperature anomaly index (ONI) were used in an autoregressive distributed lag model [ARDL(1,0,0)] yielding GSATt = 0.32 + 0.23 GSATt-1 + 0.08 ERF1.90 + 0.11 ONI. This model is equivalent to a two-layer energy-balance model with the deep-ocean temperature held constant. The ARDL model explains 96 % of GSAT variability from 1977 to 2025 (R2 = 0.959). Residual diagnostics indicate no evidence of residual autocorrelation or heteroskedasticity and bounds testing (F = 40.51) supports cointegration. The model was validated by expanding-window recursive out-of-sample prediction from 1994 to 2025 (Theil’s U = 0.53). Earth’s climate system has entered a new regime in which the radiative response to the energy imbalance at the top of the atmosphere is progressively decreasing. ARDL projections for GSAT in 2041–2060 with ONI = 0 exceed the AR6 best estimate by 0.6 °C. If ERF continues to rise 0.7 W m−2 decade−1, GSAT by mid-century will likely be 2.8 ± 0.3 °C.
Review of "Emergence of a non-linear global surface air temperature response to rapidly increasing effective radiative forcing" by Craig R. Smith
This paper applies statistical techniques from econometrics to find a relationship between observed global mean surface air temperature change T and effective radiative forcing F (estimated by AR6 methods) over recent decades. The main conclusion is that T = 0.41 + 0.11 F^n, where n=1.7--2.2 i.e. F^2 (roughly). The author suggests that this is evidence for a strongly non-linear climate feedback, which substantially raises the projections of T by the middle of the 21st century.
As I advised the editor when accepting this assignment, I am not familiar with these statistical techniques. Consequently I cannot follow the methods and most of the results, and am unable to comment on them, so the majority of my detailed comments are on the introduction. I suspect that I am typical of most climate scientists in this regard. This paper is attempting a difficult and potentially valuable step of importing ideas from one discipline into another, but I think that, in order to succeed, a lot of technical explanation needs to be included - if I'm right that most potentially interested readers would find it opaque. Perhaps my comments listing the many things I'm ignorant of will be helpful!
My responses of "Fair" to the questions in the form aren't proper judgements. I would rather not answer the questions, because I don't consider myself in a position to do so competently.
Because I can't follow the reasoning, I can't judge the validity of the conclusion. I find it surprising. In order to accept it, I need some physical interpretation of why T should have this form. The author is right that at high temperatures we expect feedbacks to become non-linear. To the next order of approximation, we would get R = F - N = lambda T + q T^2, with lambda>0 and q<0 (cf. e.g. 10.1029/2020GL089074). Assuming that constant ocean heat uptake efficiency kappa>0 is an adequate approximation, that gives F = kappa T + lambda T + q T^2 = rho T + qT^2, where rho = lambda + kappa is the climate resistance. Solving the quadratic equation, T = (-rho + sqrt(rho^2 + 4qF)/2q. While |qF|<<rho, sqrt(rho^2 + 4qF) = rho (1 + 4qF/rho)^(1/2) \approx rho (1 + 2qF/rho - (qF/rho)^2), so T = F/rho - qF^2/2rho^2. There is an F^2 term, but it doesn't dominate the linear term. Also, there isn't a constant term, as in the author's model.
In general, I am uneasy about an approach which isn't motivated by physical understanding. I am inclined to think that if the relationship between F and T is non-linear, it may be evidence of other degrees of freedom in the system having an effect, through unforced variability. In reality, and in an AOGCM, the system is obviously far more complex than a global-mean relationship between the two numbers (F and T). Simple climate models like FAIR are useful because they are formulated on a physical basis, and then calibrated to agree with more complex models (AOGCMs) that explicitly model (some of) the physical processes involved.
In order to give confidence, I think that an essential step is the one the author hints at right at the end, namely to test the ARDL model's predictive capacity on AOGCM data. AOGCMs aren't the real world, but they're somewhat like it and satisfy physical essentials. If the model is fitted to 1977-2025 in a historical AOGCM simulation, does it match that AOGCM's SSP projections to 2060 or beyond?
48. Meyssignac et al. evaluated lambda(t) from obs and compared it with AGCMs forced by observed SSTs. As you say, they conclude that lambda(t) varied more in reality than in the AGCM experiments. However, the variation of lambda(t) over the historical period by the "SST pattern effect", which is implied at line 43, was not noticed first by them, but several years earlier, in those AGCM experiments, and has subsequently been confirmed in many AGCMs (10.1002/2016GL068406, 10.1038/NGEO2828, 10.1029/2022JD036675). Another important fact (also discussed by 10.1029/2022JD036675, for instance, and in many other papers) is that AOGCM historical experiments do *not* reproduce these variations in lambda. The AGCMs included in the AOGCMs are the same ones, so the discrepancy means that the AOGCMs do not produce the same kind of SST pattern variation as occurs in reality. Because both AGCMs and AOGCMs could be called "climate models", I think that this phrase isn't sufficiently precise in the present context.
49. It's helpful to note that they used the negative-stable convention, but it would be clearer to give their numbers in your positive-stable convention i.e. omit the minus.
52. Constant ocean heat uptake efficiency kappa, in DeltaN = kappa DeltaT, is not an assumption in those models in general i.e. it's not built in; it's an approximation to the emergent behaviour, which holds for the first several decades under scenarios of steadily increasing forcing. Constant climate feedback lambda, in DeltaF - DeltaN = R = lambda DeltaT, is indeed an assumption they make, following AOGCMs, as you say at 53. That is, only *part* of the linearity F propto DeltaT is an assumption (the dominant part, via lambda). This is discussed by 10.1007/s00382-023-06989-z.
59. The relationship between DeltaT and DeltaN is linear, according to the eq at line 39, if lambda is constant, V is negligible and DeltaF is linearly related to DeltaT. The latter approximately applies while kappa can be treated as constant i.e. under those scenarios of constantly increasing DeltaF, for some decades. It also applies if DeltaF is constant, which is a special kind of linear relationship to DeltaT. It doesn't work in stabilisation scenarios, for example.
62-63. Yes, that's true, but the Planck feedback, governed by the Stefan-Boltzmann law, is only one of the phenomena involved. In that case you can show that first order is good enough. This isn't obvious for water vapor and surface albedo effects, and especially not for changes in cloud, which dominate the AOGCM spread in lambda. The adequacy of the first-order approx is an emergent result from AOGCMs, rather than expected a priori.
69. It also must assume constant kappa, which will not hold in reality and does not hold in AOGCMs very well - not because of nonlinearity in the physical processes, but because the system is more complicated than a global mean.
70-72. It's unclear what scenarios you have in mind, with what models. DeltaF and DeltaT are not linearly related in general.
74-76. In the figures of Forster et al 2026 I don't see a sharp increase specifically in the last decade, and I can't find such a remark in the paper. I agree there that DeltaF, DeltaN and DeltaT are rapidly increasing, and now have values and rates which haven't occurred before in the historical record. The graphs suggest that you might say this began in about 1980. "New regime" sounds like like a discontinuity, but I can't see evidence for that.
96. I suggest that "Econometric models" should start a new paragraph.
102, 105. The "climate physics" (in a global-mean sense) and the energy balance equation are emergent in AOGCMs and reality, rather than underlying. The energy balance is a consequence of energy conservation in all the physical processes. We can't infer that a statistical description is sufficient for the real climate system on the basis of its similarity to the behaviour of a simple climate model, or that it will apply in a different climate state.
122. As the forcing takes the climate system into states we haven't observed before, new behaviour may emerge which can't be captured in a statistical model based on what has been observed up to now. For that reason I would not be confident in the predictions of such a model.
141-143. I don't know what a "cointegrated system" or a "cointegrating relation" is. These terms and concepts aren't used in climate and Earth system science (as far as I know), so I think that some tutorial explanation is needed for this journal, to avoid readings getting lost at this point.
145. I suppose that "enters" and "specification" might be technical terms, whose meaning I don't know.
149. I don't know what "weakly exogenous" means. I guess that "exogenous forcing" means what would usually be called "external forcing" in climate science. In fact "forcing" of the climate system is always understood to be external. On what criterion is it "weak"?
151. What does "orthogonal to the ERF regressor" mean?
152. I don't follow what A4 means. Are you saying that the ERF is raised to some power? What would be the physical meaning of that?
160. I don't know what an ARDL equation is. I've just looked up what it stands for, and I found later than its functional form is stated at line 240. However, it would take some exposition for me to appreciate what it does.
162. What does "ONI" stand for?
165. By "lag coefficient" do you mean the lag-one autocorrelation coefficient of GSAT, or the lagged correlation coefficient of GSAT with some other quantity, or something else?
167-187. I have skipped this bit because of not knowing the concepts and terminology.
188. The climate sensitivity parameter is the reciprocal of the climate feedback parameter (ERF-DeltaN)/DeltaT. The quantity DeltaT/ERF is the reciprocal of the climate resistance ERF/DeltaT (the term used by Gregory and Forster 2008). The difference between them is the ocean heat uptake efficiency DeltaN/DeltaT. All these quantities can have differential counterparts, but only the differential climate feedback parameter d(ERF-DeltaN)/dDeltaT has generally been considered; that's the quantity studied for its historical time-dependence, discussed in the introduction. This quantity is interesting because (ERF-DeltaN)/DeltaT was previously assumed to be time-constant and scenario-independent. However, DeltaN/DeltaT was never expected to be constant; its main use is as an interpretative device.
208-209. Since the linear specification is a special case of the power law, with nu=1, why it is considered separately? What is the physical motivation for considering nu different from unity, or the choice of a power law, if not linear?
226-356. I have skipped most of the methods because of not being aware of concepts and techniques. There are many terms which are unknown to me, such as "attenuation bias", "simulation-extrapolation correction", "I(0) and I(1) regressors", "Dickey--Fuller test", "Kwiatkowski--Phillips--Schmidt--Shin test", "Pesaran--Shin--Smith test", "Breusch--Godfrey Lagrange Multiplier test", "Breusch--Pagan and White tests", "heteroskedasticity", "expanding-window recursive validation", "Diebold--Mariano" test.
390-391. Some authors argue that the unusual SST trend in the Pacific, related to the PDO pattern, ended in the 2010s e.g. Loeb et al. (2020, 10.1029/2019GL086705).
426. What does "admitted an interpretation within a two-layer energy-balance framework" mean? As far as I know, the linear form is an approximate solution of the two-layer model for linearly increasing forcing, and the solution couldn't take any of the other forms with usual simple assumptions about the time-profile of forcing.
734-735. I agree that applying the procedure to output from AOGCMs (or ESMs - but I think this is about the physical climate system) and two-layer model output is an essential step. That would be a "perfect-model" test, in which a model is a substitute for reality, whose future we do not know. The test is whether the ADRL specification, fitted to the recent past of the historical simulation, is able to match the future projection.