On the diagnosis of climate sensitivity using observations of ﬂuctuations

It has been shown that lag-covariance based statistical measures, suggested by the Fluctuation Dissipation Theorem (FDT), may allow estimation of climate sensitivity in a climate model. Recently Schwartz (2007) has used measures of the decay of autocorrelation in a global surface temperature time series to estimate the real world climate 5 sensitivity. Here we use a simple climate model, and analysis of archived coupled climate model output from the IPCC AR runs, for which the climate sensitivity is known, to test the utility of this approach. Our analysis of archived data show that estimates of climate sensitivity derived from century-long time scales typically grossly underestimate the models’ true climate sensitivity. We analyze the behavior of the simple model with 10 adjustable heat capacity in two surface layers, subject to various stochastic forcings and for various climate sensitivities, modulated by albedo and water vapor feedbacks. We use our simple climate model to demonstrate:


Introduction
and Bell (1980) introduced the idea that the climate sensitivity of the earth or of a general circulation model (GCM) could be predicted using the Fluctuation Dissipation Theorem (FDT) (Callen and Green, 1952). The FDT states that for systems near equilibrium, the rate at which anomalies dissipate is related the sensitivity of the 5 system to a finite change in forcing. Bell (1980) and Cionni et al. (2004) were able to show that global climate sensitivity could in fact be predicted to useful accuracy for a simplified climate model and for a highly complex chemical GCM, respectively using measures based on the FDT, depending on the lag-autocorrelation matrix of the model temperature. Langen and Alexeev (2005) used the FDT to predict the zonal mean 10 response of a GCM to climate forcing, using 500 yr of model output data. North et al. (1993) andvon Storch (2004) considered the dissipation of fluctuations in a GCM and discussed the applicability of the FDT to climate models. Great hopes have been staked on the use of the FDT to diagnose sensitivity using satellite infrared radiance data (Goody et al., 1998). 15 In all of these discussions, two aspects of the potential use of the FDT for climate sensitivity diagnoses have been neglected. First, there has been no explicit discussion of the use of the FDT in the case where the heat capacity of the system is uncertain. Second, there has been no thorough investigation of how the length of the time series needed for an accurate diagnosis of climate sensitivity depends on the parameters 20 (climate sensitivity, heat capacity, number of independent variables) of the system. In this paper, we investigate the practical usefulness of the FDT for the purpose of diagnosing climate sensitivity. To do so, we use an extremely simple climate model that is nevertheless includes parameters controlling climate sensitivity, heat capacity and internal heat transports. Introduction

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Printer-friendly Version Interactive Discussion capacity and climate sensitivity. Schwartz (2007) diagnosed a value of the climate sensitivity (≈0.3 K W −1 m 2 ) that was quite low compared to other estimates of climate sensitivity (≈0.75 K W −1 m 2 ). Schwartz argued that the discrepancy was due to the models' over-estimate of the climate system's decorrelation time scale, arguing that while his results showed a decorrelation time scale of about 5 yr, climate models typ-5 ically required some 30 yr to fully adjust to a step change in climate forcing. We will show that actual climate model time series typically show decorrelation time scales similar to that found by Schwartz (2007) for observed global average surface air temperature. Thus, application of Schwartz's technique to climate model output results in a substantial underestimate of the climate models' climate sensitivity. We will use a 10 simple heuristic climate model to show how a model with a relatively long adjustment time may nevertheless exhibit a short decorrelation time-scale. In Sect. 2 we review the use of lag-correlation analysis to estimate climate sensitivity. In Sect. 3, we apply Schwartz (2007)'s method to the IPCC AR4 model runs, to test the method on modeled climate systems whose climate sensitivity is known. In Sects. 4 15 and 5 we introduce our simple climate model and apply it to explain the systematic underestimate of climate sensitivity by the lag-correlation method. These errors are due both to the inadequate length of the time series involved, and to the assumption of a single heat capacity for the climate system. "Note added in proof: the author has recently become of aware of three comments 20 on Schwartz (2007), some of which overlap with material in this paper (Foster et al., 2008;Knutti et al., 2008;Scafetta, 2008), and a reply (Schwartz, 2008). In particular, Knutti et al. (2008)  Interactive Discussion response of the climate system to a step change in climate forcing. We assume that the unforced fluctuations, though in reality driven by internal instabilities, can be treated as though they were driven by random fluctuations of the forcing. The system tends to return to its equilibrium temperature because of the dominant blackbody feedback, whereby an increase in temperature leads to an increase in outgoing radiation, 5 and thus a return to equilibrium. Thus, in a system with large climate sensitivity, a small fluctuation in forcing leads to feedbacks that tend to slow the return of the system to equilibrium. On the other hand, in a system with small climate sensitivity the same fluctuation in forcing causes a smaller temperature change, and thus a more rapid return to equilibrium. Thus a time series of surface temperature ought to show larger lag-autocorrelation in the system with larger climate sensitivity, and smaller lagautocorrelation in the system with smaller climate sensitivity.
There are a number of reasons to suspect that this relationship might not be so simple in the real world. The relative importance of external and internal drivers of global temperature is different at different time scales, and it is reasonable to suppose 15 that feedbacks might therefore act differently at different time scales. For instance, interannual variations of global mean surface temperature in the real climate system are largely driven by internal oscillations of the system (e.g. ENSO), where the mean temperature change is a small residual of large and nearly compensating patterns of heating and cooling. In this case it is clear that any geographic dependence in the 20 strength of climate feedbacks might make the global feedback have a different relationship to the global temperature for these internally forced variations than for externally forced climate change (e.g. by a secular trend in carbon dioxide concentration).
Keeping these caveats in mind, we now formalize our treatment of the relationship between the lag-correlation and climate sensitivity for a purely linear system. Our 25 discussion follows that of Penland and Sardeshmukh (1995). Consider a linear system forced by white noise (ξ) and a fixed forcing f: A change in f from f 0 to f 1 will result in a change in the long-term mean value of x of (x 1 −x 0 )=−B −1 (f 1 −f 0 ) −1 . Thus we call −B the sensitivity matrix. −B can be determined by observation of the lag-covariances of the system, as follows: where C xx (τ)=<x(t+τ)x T (t)>, and τ is an arbitrary time lag. By taking the exponent of 5 both sides of Eq.
(2), and then integrating over all time lags τ, one obtains which is a form of the FDT. Note that no use is made in this estimate of the white 10 noise forcing, ξ, and that the sensitivity is given in units of time: it is the time scale of the equivalent restoring force of the system. In any practical application of the FDT to climate sensitivity diagnosis, we must assert that the climate system can be approximated by a system like Eq. (1), and we must be able the calculate the forcing in units of Kelvins per unit time. This requires independent knowledge of the heat capacity of 15 the climate system. One can use either Eqs.
(2) or (4) to obtain B. Schwartz (2007) Cionni et al. (2004) employed a method inspired by the FDT, where instead of calculating the integral of the lag covariance matrix of the system x, they calculated the integral of the lag covariance of the system x and the forcing ξ: τ L , the upper limit of integration in Eq. (5) is the lag beyond which no significant correlation is observed. One way to choose a value for this variable is to integrate Eq. (5) for a sufficiently large τ L that B est reaches a maximum, and then begins to decrease. Cionni et al. (2004) accurately estimated climate sensitivity based on a 10 yr time series of 10 model output data from a GCM run with fixed surface temperature and known forcings.
In their study, diagnosis of effective heat capacity was not needed because only the small land surface heat capacity contributed to the decorrelation time. In the real world the task becomes more difficult, due to short time series and a lack of accurate forcing information.

Application to IPCC AR4 model output
Here we apply the tools described above to test the ability of analysis based on lagcorrelation of temperature timeseries, combined with estimation of global heat capacity by the methods of Schwartz (2007) to predict climate sensitivity in the IPCC AR4 model runs. Figure 1a shows the autocorrelation of the detrended global mean surface 20 air temperature as function of lag (in years) for 18 coupled model simulations of the 20th century (the "20c3m" runs). The three curves shown in blue are three models arbitrarily selected for having longer decorrelation time scales, while the three red curves  (Hansen et al., 1996) is shown in light blue, and is the same curve shown in Schwartz (2007), Fig. 2c. Note the good agreement between the GISS curve and the IPCC model curves. Figure 1b shows the relaxation time constant, − 1 τ ln c(τ) for the same global mean surface air temperature time series. Again, the light blue 5 curve showing the GISS observations falls well within the range of the IPCC model lag autocorrelation curves. We calculate the model relaxation time scale by finding the lag for which the autocorrelation is less than zero, and averaging the relaxation time scale either over all lags up to the lag with zero correlation, or the relaxation over the three years prior to that lag, whichever is greater. Using this method, the average relaxation time scale for the models is 8.3 yr, with a standard deviation of 4.5 yr, and a range of 2.8-18.5 yr. Thus, Schwartz (2007)'s estimate of 5 yr falls in the range of the model relaxation time scales.
Following Schwartz (2007) we calculate the models' effective heat capacity using two methods: by regressing the global heat storage against the global temperature, and by 15 taking the ratio of the trends of heat storage and temperature. Heat storage is obtained by integrating the global downward flux of heat at the atmosphere's lower boundary, which is the sum of the latent and sensible heat fluxes, and the net downward long and shortwave fluxes. Scatter plots of annual and global mean heat storage anomaly versus annual and global mean temperature anomaly are shown in Fig. 2, offset from 20 zero mean for clarity. The light blue points refer the observed ocean heat storage, and were taken by digitizing the lower right panel of Fig. 4 in Schwartz (2007). That figure plots GISS surface temperatures against ocean heat storage from the surface to 3000 m from the Levitus data set (Levitus et al., 2005), scaled upwards to account for the heat capacity of land. It is apparent that the slope of the dependence of heat 25 storage on surface temperature (shown in light blue) for the observed data is small (12 W yr m −2 ) compared to the range of values found for the models. With these two quantities calculated, we can now, following Schwartz (2007), make an estimate of the equilibrium climate sensitivity, and see how well it predicts the actual model sensitivity. One difficulty with this procedure concerns the definition of climate sensitivity. Since the theory underlying the use of the decorrelation time scale to predict climate sensitivity assumes a linear model of climate dynamics, the estimate ought to 5 work best for a very small climate forcing perturbation. However, for reasons of practical interest, climate models are generally run for finite periods using finite forcing. For the IPCC AR4 coupled model runs, we can define climate sensitivity as either: 1. the full response to a doubling of carbon dioxide, where the doubling occurs over a period of 150 yr, and the models are then run for an additional 100 yr allowing In Fig. 3, we show that these sensitivities are related but different. Here, transient sensitivity of the models (in K/100 yr) derived from a linear fit to the trend of global mean temperature for the first 50 yr of runs with carbon dioxide increasing 1% yr −1 is compared with climate sensitivity derived from the temperature difference for the years 20 2000 to 2100 in the "SRESa1b" climate forcing scenario. Among the nineteen models with output for both experiments stored in the PCMDI archive, the correlation coefficient of these two measures of climate sensitivity is 0.57, indicating that important aspects of the climate system (whether the effective climate forcing due to the imposed changes in greenhouse gas forcing, the climate feedbacks or the effective heat capacities of 25 different components of the climate system) respond differently to these two forcing scenarios. In the following discussion we will present results comparing the climate Interactive Discussion sensitivity derived using the procedure of Schwartz (2007) with the transient climate response of the models in the 1% yr −1 forcing scenario. The reader should note however, that similar comparisons have been done with the SRESa1b-derived sensitivities, with broadly similar results (not shown). In Fig. 4a, we confirm that the estimates of global effective heat capacity derived 5 from regression of global heat storage against global heat temperature in the 20c3m model runs are reasonably well correlated (r=0.70) with the estimates derived from the ratio of the trends of the two numbers. Heat capacity is shown in units of equivalent mixed layer depth (m), calculated by dividing the heat storage per unit area by the heat capacity of 1 m 3 of sea water. Having established this, we proceed using the 10 average of the two methods. Figure 4b shows the relationship between the decorrelation time scale and the effective global heat capacity. Contrary to the prediction of a linear, single heat reservoir theory of climate dynamics, the relationship is weak, and anticorrelated (r=−0.48): models with large decorrelation time scales tend to have small heat storage. When we compare decorrelation time with model sensitivity derived 15 from the first 100 yr of the SRESa1b runs (Fig. 4c), the correlation is very weak and marginally negative (r= − 0.20). Finally we compare the estimate of climate sensitivity calculated by dividing the decorrelation timescale by the effective heat global heat capacity, as calculated by Schwartz (2007). We find that the estimate of climate sensitivity made by Schwartz (0.30 K W −1 m 2 ) fall squarely within the range of climate sensitivity 20 derived from the IPCC AR4 models (mean=0.22 K W −1 m 2 , min=9.9×10 −4 K W −1 m 2 , max=1.8 K W −1 m 2 ). When the estimated climate sensitivity is compared with the actual climate sensitivity derived from the SRESa1b runs (Fig. 4d) heat uptake combined with a realistic decorrelation timescale should lead the models to underestimate climate sensitivity. However, the fact that the models themselves do not obey this relationship (i.e. the ratio of decorrelation time scale over effective heat capacity is slightly negatively correlated with model climate sensitivity) is a strong argument against the use of the simple linear theory to estimate the earth's climate 5 sensitivity. In the next two sections we use a very simple model of the climate system to explore why the theory might fail for climate system models, and for the real climate system.

Model description
For simplicity and ease of integration, we make use of a global mean climate model 10 with a single atmospheric layer and purely radiative interactions between the surface and the atmosphere. The features needed include rapid integration, an ability to simulate arbitrary climate sensitivity, and simulation of both radiative and dynamical energy fluxes. The model includes two ocean mixed layers (upper and lower) with an adjustable linear coupling, to simulate heat storage in the ocean, a single atmospheric 15 layer to account for greenhouse warming, and gray-body radiation. Feedbacks include a temperature-dependent atmospheric infrared emissivity (intended to represent the water vapor feedback), and a temperature-dependent surface albedo (intended to represent ice-albedo and low cloud feedbacks). The model is forced with modified red noise applied to either the solar variability or the infrared emissivity.

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The thermodynamic equations for the lower and upper mixed layers, and for the atmosphere are: where C s and C s2 are the heat capacities of the upper and lower mixed layers, T s and T s2 are their respective temperatures, C a and T a are the heat capacity and temperature of the atmosphere, γ is the coupling time constant between the upper and lower mixed layers, S is the solar constant, S is the solar absorption of the atmosphere, α is the 5 surface albedo, σ is the Stefan-Boltzmann constant, and is the infrared emissivity of the atmosphere. The atmospheric emissivity and the surface albedo are subject to linear feedbacks on atmospheric and surface temperature, respectively: where 0 and α 0 are the basic state emissivity and surface albedo, respectively, T and T α are the basic state atmosphere and surface temperatures, respectively, and f and f α are the feedback parameters. A is a red-noise forcing, and c α is a constant that controls its magnitude. The solar constant can also be forced with red noise: where c S is a constant controlling the magnitude of the forcing. A and A S are generated using: where r is a random variable with a flat distribution over the range [0 1], β is the parameter that determines the autocorrelation of the red-noise forcing, and the scaling is performed after x has been calculated for as many time steps as necessary. Solving Eqs. (7)-(9) for equilibrium conditions, with feedbacks and stochastic forcing assumed to be zero yields  Figure 5b shows that as the feedback parameters increase, the time required for the temperature to reach 90% of its equilibrium value is increased. This increase appears at first glance to be very similar to that of the climate sensitivity. However, as shown in Fig. 5c, the ratio of the response time to the climate sensitivity is not constant, but increases with increasing emissivity feedback. This lack of proportionality of the climate 15 sensitivity and the adjustment timescale is due to the fact that the emissivity feedback parameter acts on T a , rather than on T s . For a lag-autocorrelation based methodology to accurately predict the climate sensitivity, the fluctuations of both T a and T s would have to be observed. Nevertheless, for a modest range of possible climate sensitivities (e.g. over a factor of 3), the ratio of the climate sensitivity to the adjustment time varies

Simple model results and sensitivity experiments
We now turn to the results of a series of experiments designed to show how statistics like those used by Schwartz (2007) can give misleading results about climate sensitivity. Figure 6 shows the simple model's response to changes in the solar constant. In Fig. 6a, the solar constant is increased by step change of 10 W m −2 . The feedback This behavior suggests that a diagnostic method based on decorrelation time scale 15 will have trouble distinguishing between a system with low heat capacity on short time scales and one with low heat capacity on all time scales. Pursuing this thread of argument, we now apply Schwartz (2007)'s methodology to much longer synthetic time series, in order to test its ability to extract the model's true dynamics from finite time series. Figure 7 shows the lag autocorrelation r(∆t) for 20 two model parameter settings. Figure 7a shows results for a single 25 m mixed layer. Figure 7b shows results for a 25 m mixed upper layer, and a 1000 m mixed lower layer. In each case, the model was run for 10 000 yr, forced by white noise solar variability, and the time series of annual mean T s was calculated. Red curves show r(∆t) calculated separately for 100 100-year chunks of data. The black curve shows r(∆t) calculated 25 for the full 10 000 yr record. These results demonstrate two key points.
First, the uncertainty with which one can establish the value of the autocorrelation of a given 100 yr time series is not the relevant uncertainty for the calculation of climate  Fig. 7a that a single curve of autocorrelation versus lag derived from a 100 yr time series can give only a poor estimate of the true curve. For instance, while the autocorrelation curve derived from the 10 000 yr time series crosses zero at 10 yr lag, the mean of the 100 yr autocorrelation curves' zero crossing is 8.4 yaers, with a standard deviation of 5.7 yr. Even for a model with a single heat capacity, 5 analysis of the lag-autocorrelation of hundred year temperature time series results in both large negative bias in estimating the true decorrelation time scale (−∆t/ ln(r(∆t), averaged over lags from 8 to 12 yr), and large variance of the decorrelation time scale of randomly selected hundred year segments. Thus, our simple model run with a single heat capacity, and with parameters yielding real decorrelation time scales ranging from 10 5 yr up to 25 yr, produces 100 yr segments within a standard deviation of the mean that have decorrelation time scales of 5 yr or less. In particular, for a 25 yr decorrelation time scale, the expected mean decorrelation time scale of a randomly selected 100 yr segment is 15 yr, with a standard deviation of 10 yr. Thus, the fact that the observed temperature time series of the last 50 yr of global temperature exhibits a decorrelation 15 time scale of 5 yr (Schwartz, 2007), means only that the true value could be anywhere from 3 yr to 25 yr, and have the observed value within a single standard deviation of the expected value. Second, when the model has more than a single heat reservoir, autocorrelations based on a hundred year sample can substantially underestimate both the true decor-20 relation time scale and the adjustment timescale. This is clear from panel Fig. 7b where the lag autocorrelation curve based on 10 000 yr of data exceeds 95% of the curves based on 100 yr segments out to 30 yr lag.
Finally we demonstrate how our simple model with a fast surface heat reservoir coupled to a large deep heat reservoir can fool the method of 25 Schwartz (2007) into falsely ranking the climate sensitivities of the model run for various parameter choices.
In Fig. 8 Results in red are for parameter choices that give low climate sensitivity (f α =f =−0.01, Sensitivity: 0.1 K W −1 m 2 ) and a single deep mixed layer (C s =1.3×10 9 J m −2 K −1 (333 m), γ=0 W m −2 K −1 ). Panel a shows the response of each model to a step change in climate forcing. The high sensitivity, weakly coupled model 5 jumps up rapidly, but after the temperature difference between the upper and lower mixed layers increases, the flux of heat from the upper to the lower layer slows the temperature rise until equilibrium is reached. The low sensitivity, single mixed layer model approaches equilibrium at the same constant e-folding rate. Panel b shows the lag autocorrelation curves for 100 100-yr segments of model output forced by noise.

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The lag autocorrelation curves show higher autocorrelation for the low climate sensitivity model, just the opposite of the result predicted by Schwartz (2007), even though the climate sensitivities vary by a factor of 4.

Discussion and conclusions
Some clear conclusions may be drawn from our work. 15 1. The FDT is poorly suited to the evaluation of model sensitivity in practice. Although it can produce precise, accurate estimates, given sufficient time, the time required for accurate evaluation of model sensitivity is in general several times longer than a model would require to come to equilibrium with a step-change in forcing.

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2. Estimates of climate sensitivity using real global mean temperature variations are consistent with those using output data from the 20c3m IPCC AR4 model runs. There is no inconsistency between the model climate sensitivities and the observed global mean temperature decorrelation timescales, as claimed by Schwartz (2007). Introduction

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