## Abstract

The global-mean surface air temperature response of the climate system to a specific radiative forcing shows characteristic time scales. Identifying these time scales and their corresponding amplitudes (climate sensitivity) allows one to approximate the response to arbitrary radiative forcings. The authors estimate these time scales for a set of atmosphere–ocean general circulation models (AOGCMs) based on relatively short integrations of 100–300 yr for some idealized forcings. Two modes can be clearly distinguished but a large spread in time scales and climate sensitivities exists among the AOGCMs. The analysis herein also shows that different factors influence the mode estimates. The value and uncertainty of the smallest time scale estimate is significantly lower when based on step scenarios than gradual scenarios; the uncertainty on the climate sensitivity of the slow mode can only be reduced significantly by performing longer AOGCM simulations; and scenarios with only a monotonically increasing forcing do not easily permit the climate sensitivity and the response time for the slow mode to be disentangled. Finally, climate sensitivities can be estimated more accurately than response times.

## 1. Introduction

In the perspective of climate mitigation, comparing the climate impact of various anthropogenic climate forcers such as greenhouse gases and aerosols is important. The radiative forcing of climate forcers, integrated over a certain time period, is the most widely known basis for comparison (Houghton et al. 1990) and has been used within the Kyoto Protocol to the United Nations Framework Convention on Climate Change (UNFCCC) for weighting the climatic impact of emissions of different greenhouse gases. More recently, focus is also on the comparison of the temperature impact of climate forcers (Shine et al. 2005, 2007; Berntsen and Fuglestvedt 2008). To determine the temperature impact, the temperature response to a given forcing is required. Although atmosphere–ocean general circulation models (AOGCMs) give the best available and most accurate description of the global temperature response, their use is limited because of their high computational cost. Different studies have shown, however, that the global- and annual-mean surface temperature response of an AOGCM can be well described by a limited number of modes. For example, Hasselmann et al. (1993) derived mode estimates for two different AOGCMs (ECHAM-1/LSG and GFDL) and Boucher and Reddy (2008) did so for the Met Office HadCM3 AOGCM (see Table 1 for expansions of model names). Often the number of modes estimated is limited to one or two. Once the modes in the climate system are identified, one can efficiently compare the approximate global- and annual-mean surface air temperature impact of various climate forcers for a broad variety of scenarios.

As AOGCMs differ substantially in their response to a certain forcing (Solomon et al. 2007), the mode estimates vary strongly among AOGCMs. In addition, there might be a dependency of the mode estimates on the type and length of the scenario used in the AOGCM. Assuming two modes in the climate system, using a step scenario Hasselmann et al. (1993) found a very low value for the small time constant, while Boucher and Reddy (2008), using a gradual scenario, found considerably higher values. Also for the slow mode, both studies found very different values, probably related to the length of the scenarios. Olivié and Stuber (2010) used a very simple two-box model with two time constants that crudely represents the atmosphere–ocean system, and obtained smaller values for the time constants when using step scenarios than when using gradual scenarios.

Using rather low spatial resolutions, a few studies have presented results from very long AOGCM integrations, for example, 5000 yr with GFDL_R15a (Stouffer 2004), 3000 yr with CCSM3 (Danabasoglu and Gent 2009), or 2000 yr with FOAM (Yang and Zhu 2011). These studies show that very long time scales are present in the climate system, especially in the deep ocean with values larger than 1000 yr. On the other hand, a growing number of AOGCM simulations are available with a relatively high spatial resolution but a limited temporal extent of 100–300 yr (e.g., from CMIP3, but also from the more recent CMIP5). Although these simulations are much shorter than the longest time scales in the climate system, they possibly can give interesting information on the shorter time scales. In addition, for comparing the temperature impact of different climate forcers, attention is often limited to time horizons of 20, 50, 100, or 500 yr (Solomon et al. 2007; Fuglestvedt et al. 2010).

In this study we estimate the atmospheric response times for a set of AOGCMs that performed idealized forcing simulations with lengths on the order of 100–300 yr (mostly from CMIP3, but also some other simulations). In addition we try to identify some of the factors that influence these estimates. As our approach (Olivié and Stuber 2010) gives best estimates but also uncertainties, one can assess whether the mode estimates have been well constrained. Our analysis will try to answer the following questions:

How many modes appear to be present in the global-mean surface air temperature response?

How strongly do the mode estimates differ among the AOGCMs?

What is the impact of the type of scenario on the mode estimates?

Are some mode characteristics easier to estimate, and do all mode characteristics have to be known with the same accuracy?

The mode estimates we obtain are also compared with those of Hasselmann et al. (1993) and Boucher and Reddy (2008).

The structure of the paper is as follows. In section 2 we introduce the role of modes to characterize the surface temperature response to a radiative forcing. In section 3 we describe the method that we use to estimate the modes, and in section 4 we present our results. In section 5 we discuss our results, and in section 6 we present our conclusions.

## 2. Modes in the climate system

In Hasselmann et al. (1993), Sausen and Schumann (2000), Boucher and Reddy (2008), and Fuglestvedt et al. (2010), the global-mean temperature response *T*(*t*) to a radiative forcing *F*(*t*) is described by

where *E*(*t*) is the response to a *δ*-pulse forcing and is assumed to be written as the sum of *n* (≥1) decaying exponential functions, that is,

Putting explicitly the factor in front of the exponential function in Eq. (2) allows for a physical interpretation of *f _{i}*: the sum of

*f*equals the climate sensitivity

_{i}*λ*, that is,

and this can be shown with Eq. (1) when *E*(*t*) is the unit step forcing [see Eq. (A2)] and *t* → ∞. We will call *f _{i}* the climate sensitivity of mode

*i*and

*λ*the total climate sensitivity. The time constant

*τ*describes how fast the temperature responds to the radiative forcing [see Eq. (2)].

_{i}Here we investigate the characteristics of the AOGCMs assuming that one (*n* = 1) or two (*n* = 2) modes are present [in a sensitivity test we will also assume three modes (*n* = 3)]. The respective responses of an AOGCM to a *δ*-pulse forcing for *t* ≥ 0 are then approximated by

Some values of mode estimates used in the literature are shown in Table 2. Assuming one mode, Hasselmann et al. (1993) suggest a time constant of 36.8 yr, but assuming two modes they suggest that the small time constant varies between 1.2 and 2.86 yr and the large time constant varies between 23.5 and 41.67 yr. Boucher and Reddy (2008) also suggest the presence of two modes with time constants of 8.4 and 409.5 yr. One can see in Table 2 also a large spread in *λ*, which varies between 0.428 and 1.06 K W^{−1} m^{2}. Li and Jarvis (2009) suggest the presence of three modes and a total climate sensitivity around 1.25 K W^{−1} m^{2}.

## 3. Method

To estimate *τ _{i}* and

*f*we use an inverse modeling method described in Olivié and Stuber (2010) and based on Tarantola (2005). Starting with a priori best estimates and uncertainties on

_{i}*τ*and

_{i}*f*

_{i}, the method finds a posteriori best estimates and uncertainties (including correlations) for the modes in the AOGCM. It searches in an iterative way the extreme value of a likelihood function that takes into account how much the a posteriori mode estimates deviate from their a priori value, and how much the mode-based response [see Eq. (1)] deviates from the AOGCM time series for a given scenario.

As a priori estimates for *f*_{1} and *τ*_{1} we take 0.5 K W^{−1} m^{2} and 10 yr, respectively. When the presence of a second mode is assumed, we take 0.2 K W^{−1} m^{2} and 100 yr as a priori estimates for *f*_{2} and *τ*_{2}, respectively. For all parameters we assume a lognormal probability distribution with an a priori uncertainty of 100%. These a priori estimates and uncertainties are chosen to cover the variety in values presented in Table 2. (An overview of the a priori estimates and uncertainties of the mode characteristics can also be found at the top of Tables 4 and 5). In addition, we do sensitivity tests where we use 400 yr as a priori estimate for *τ*_{2} in the two-mode case, and 400% as a priori uncertainty for all parameters.

From a general perspective, we use two different types of scenarios: step scenarios with a sudden important change in the forcing and more gradual scenarios. The scenarios differ also in their length and in the amplitude of the forcing. An overview of the different scenarios can be found in Table 3; the same scenarios have also been used in Olivié and Stuber (2010). The step scenarios C20 and C2 show a sudden increase in the CO_{2} concentration by a factor of 6.5 corresponding to an increase in the radiative forcing by 5.35 W m^{−2} × ln(6.5) = 10 W m^{−2}, and scenario Cx2 shows a sudden doubling in the CO_{2} concentration corresponding with an increase in the radiative forcing by 5.35 W m^{−2} × ln(2) = 3.71 W m^{−2}. After this sudden increase, the scenarios C20 and C2 show a decay of the CO_{2} concentration with a relaxation time of 20 and 2 yr, respectively, while the forcing is kept constant for scenario Cx2 (as if the relaxation time is infinite). The gradual scenarios Cg2 and Cg4 show a 1% yr^{−1} increase in the CO_{2} concentration (and thus a linear increase in the forcing) over a certain period, after which the concentration (and thus the forcing) is kept constant. For scenario Cg2 this point is reached after 70 yr and the forcing is then kept constant at 5.35 W m^{−2} × ln(2) = 3.71 W m^{−2} (CO_{2} doubling), whereas for Cg4 it is reached after 140 yr for a value of 5.35 W m^{−2} × ln(4) = 7.42 W m^{−2} (CO_{2} quadrupling).

We will estimate the mode characteristics for two AOGCMs based on step scenarios [these simulations are described in more detail in Olivié and Stuber (2010)] and for 15 AOGCMs based on gradual scenarios. The latter simulations have been taken from the World Climate Research Programme’s CMIP3 database, which has been used for the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (Solomon et al. 2007).

## 4. Results

### a. Using step scenarios

In this section we describe the results from deriving mode estimates for the two AOGCMs forced by the step scenarios C20, C2, and Cx2. Figure 1 shows the results from using simultaneously these three step scenarios for UKMO-HadCM3 or CNRM-CM3. The dashed lines show the mode-based results [see Eq. (1)] using the a priori mode estimates, the full lines show the results using the a posteriori estimates, and the separate symbols are the results from the AOGCMs. In general, with one mode we can reproduce rather well the results from the AOGCMs. However, for scenario C20, it slightly overestimates the response just after the peak (years 10–20) and it underestimates the response during the last 40 yr. Similarly, for Cx2, we are unable to correctly represent the small increase in the stabilization phase [which leads to overestimating the initial response (years 10–30) and underestimating the later response (years 60–100)]. Although the results using the presence of two modes are rather similar to the results using one, with two modes we can better reproduce the peak for scenario C20 and the long-term rise in the response to scenario Cx2. It is the presence of the slow mode with *τ*_{2} and *f*_{2} that allows this long-term rise. The improvement can be clearly seen by comparing the root-mean-square error (RMSE), based on the difference between mode-based results and AOGCM results. When using all three scenarios, the RMSE is reduced when going from one mode to two modes from 0.30 to 0.20 K for UKMO-HadCM3 and from 0.29 to 0.21 K for CNRM-CM3. It is also worth noting that both with one and two modes we can reproduce simultaneously the three step scenarios that have different relaxation times for the CO_{2} perturbation (2, 20, and ∞ yr).

Tables 4 and 5 show the a posteriori values of the mode parameters for UKMO-HadCM3 and CNRM-CM3 based on the step scenarios. The lines with C20, Cx2, or C2 mentioned in brackets indicate that only one of the three scenarios has been used for deriving the coefficients (if nothing is mentioned, the three scenarios have been used simultaneously). When using one mode, the single time constant has values of 5.0 and 3.4 yr for UKMO-HadCM3 and CNRM-CM3 respectively, and the total climate sensitivity represented by *f*_{1} has a value of 0.58 or 0.45 K W^{−1} m^{2}. When assuming two modes, the lowest time constant has values of 3.1 and 2.5 yr, which is less than when assuming one mode. The second time constant takes values between 81.0 and 132.3 yr. These quite different values for *τ*_{2} are mainly a consequence of the experiment C20. Using only C20 leads to considerable differences in *τ*_{2}, while using only Cx2 or C2 gives very similar results (see Table 5). UKMO-HadCM3 and CNRM-CM3 differ quite strongly in their behavior the last 50 yr for C20: the temperature in UKMO-HadCM3 decreases gradually in this period, while in CNRM-CM3 it remains almost constant on a value different from 0 K. This behavior of CNRM-CM3 can only be modeled by a large time constant. Concerning *τ*_{1} in both the one- and two-mode assumption, we see that using Cx2 alone leads to slightly higher values for *τ*_{1}. This might be related to the first part of the scenario, where a step with relaxation time *τ _{p}* = ∞ is less exacting than a step with a smaller value for

*τ*. However, the last part of the scenario might play a role, where the observed linear increase requires a larger value of the time constant. For C2, we obtain the smallest values of

_{p}*λ*.

The total climate sensitivity for two modes (determined by *f*_{1} + *f*_{2}) is 0.72 and 0.61 K W^{−1} m^{2} for UKMO-HadCM3 and CNRM-CM3, respectively, and is considerably larger than when assuming one mode. An important part (more than one-third) of the total climate sensitivity is caused by the slow (i.e., *τ*_{2}) component.

Tables 4 and 5 also contain the derived time scale *τ _{m}* [see Eq. (A1)], which is an

*f*-weighted average over

_{i}*τ*, and whose physical interpretation is illustrated in the appendix. This mean response time

_{i}*τ*is the delay time that can be observed between a linearly increasing forcing and its response (see sections b and c of the appendix), or in general between a slowly varying forcing and its response (see section d of the appendix).

_{m}For the a posteriori uncertainties, we see that for *τ*_{1} and *f*_{1} they are considerably reduced with respect to the a priori ones. For example, when using one mode for UKMO-HadCM3, we find that the uncertainties are reduced from 100% down to 3% for *f*_{1} and to 15% for *τ*_{1} (see Table 4). Similar reductions are found for CNRM-CM3: down to 4% for *f*_{1} and to 22% for *τ*_{1}. This reduction is stronger when using one mode compared to two modes: when using all UKMO-HadCM3 scenarios for the two-mode case, we find that the uncertainties are reduced down to 9% for *f*_{1} and to 25% for *τ*_{1} (see Table 5). This illustrates that when more parameters have to be estimated, using the same information does not reduce uncertainties on the parameters as much. In the two-mode case, the uncertainty reduction for *τ*_{2} and *f*_{2} is much smaller than for *τ*_{1} and *f*_{1}. Especially for *τ*_{2} the uncertainty remains very high. This shows that determining time constants of the order of 100 yr using simulations with a length of only 100 yr is difficult. It also means that considerably different values of *τ*_{2} might still give an equally good fit. One can also see that the uncertainty reduction is in general stronger for the *f _{i}* than for the

*τ*

_{i}. When using only one scenario at the time (C20, Cx2, or C2), the uncertainty reduction for most of the parameters is strongest for C20 and weakest for C2. Only for

*τ*

_{1}in the two-mode case, the uncertainty reduction for C2 is slightly stronger than for Cx2.

Tables 6 and 7 show the a posteriori correlations between the parameters. These values represent the dependencies that exist between the a posteriori values of mode characteristics. When using one mode, the a posteriori correlation between *f*_{1} and *τ*_{1} is quite low. When using two modes, we find a strong a posteriori correlation between *f*_{2} and *τ*_{2} except when using only scenario C2. As a consequence of the little information available in C2, uncertainties are reduced minimally (see Table 5) and correlations stay close to their a priori value (which is 0).

### b. Using gradual scenarios

In this section, we estimate the modes for 15 different AOGCMs using gradual scenarios (scenarios Cg2 and Cg4 in Table 3). Figure 2 shows the results when using simultaneously the gradual scenarios Cg2 and Cg4 to estimate the modes. The dotted lines show the mode-based results using the a priori parameter values and the full lines show the mode-based results using the a posteriori values. With the a posteriori estimates, the mode-based results correspond very well with the AOGCM data. With two modes, we can better represent the small rise during the stabilization phase present for some AOGCMs and the RMSE is slightly smaller in the two-mode case.

Tables 4 and 5 give the a priori and a posteriori values of the parameters of the mode characteristics. Using one mode, we obtain values for *τ*_{1} between 5.9 and 24.3 yr. The lowest values are obtained for AOGCMs with only short time series available, namely MIROC3.2(hires), UKMO-HadCM3, and UKMO-HadGEM1. They have only simulated the first part of the Cg2 scenario (where the CO_{2} concentration rises by 1% yr^{−1}), but not the stabilization phase (see Fig. 2). For most of the other models we find values of *τ*_{1} larger than 10 yr. Using two modes, we find values for *τ*_{1} that are in general smaller than 10 yr. The *f*_{2} value is small for ECHO-G, FGOALS-g1.0, INM-CM3.0, and MRI-CGCM2.3.2 because they have a very flat stabilization phase. In general the total climate sensitivity when using two modes (*f*_{1} + *f*_{2}) is slightly higher than using one mode (*f*_{1}).

In the one-mode case, we see that the uncertainties of *τ*_{1} and *f*_{1} are comparable with the values obtained using step scenarios. Only for the short simulations [MIROC3.2(hires), UKMO-HadCM3, and UKMO-HadGEM1] do we find much larger uncertainties. These simulations can only clearly determine the fraction *f*_{1}/*τ*_{1}, corresponding with a strong correlation between both (see Table 6), implying that the actual a posteriori values are very uncertain. For the two-mode case, as expected, the uncertainties for *τ*_{1} and *f*_{1} are larger than in the one-mode case. The uncertainty on *τ*_{1} in the one-mode case is clearly also much higher for gradual than for step scenarios. The uncertainties on *τ*_{2} and *f*_{2} remain important; however, for *f*_{2} they are in general slightly smaller than when using the step scenarios (the relatively long Cg2 and Cg4 scenarios allow us to constrain the slow mode better). For the three AOGCMs with only short time series available, we find extremely high uncertainties on *f*_{2} and *τ*_{2}, as these short simulations contain almost no information on these parameters.

Tables 6 and 7 show the correlations between the characteristics of the modes. For the one-mode case, much higher correlations between *f*_{1} and *τ*_{1} are found than using step scenarios. For the two-mode case, we see a small correlation between *f*_{2} and *τ*_{2} for the three short simulations (the same explication as for C2 is valid here; see section 4a). There is also a relatively high anticorrelation between *f*_{1} and *f*_{2} in the two-mode case for the gradual scenarios from ECHO-G, GFDL-CM2.0, GISS-EH, INM-CM3.0, and MRI-CGCM2.3.2. This high anticorrelation is reflected in a low a posteriori uncertainty on the total climate sensitivity *λ* (i.e., considerably lower than its components *f*_{1} and *f*_{2}). The physical explication for this is that these five simulations all show a very flat stabilization phase and that *λ* can thus be estimated with high accuracy, but that the partitioning of *λ* between *f*_{1} and *f*_{2} is more uncertain.

### c. Averaged values

At the bottom of Tables 4 and 5 we show values for the a posteriori estimates and uncertainties, averaged over the different AOGCMs. We show five different values: “step” is the average over the two AOGCMs that performed step scenarios, namely UKMO-HadCM3 and CNRM-CM3; “gradual” is the average over all AOGCMs that performed gradual scenarios; “gradual+” is the average over all AOGCMs that performed long enough scenarios, including a considerable stabilization period; “gradual−” is the average over the three AOGCMs with only short simulations, namely MIROC3.2(medres), UKMO-HadCM3, and UKMO-HadGEM1; and “all” is the average over all AOGCMs. We calculate the averages using the a posteriori uncertainties as weights. These averaged values allow an easy comparison between the effect of using step scenarios (step), short gradual scenarios (gradual−), and long gradual scenarios with a stabilization phase (gradual+).

The top panels in Fig. 3 show histograms of the a posteriori parameter estimates in the one-mode and two-mode case for all AOGCMs. The histograms contain the same data as Tables 4 and 5 and are based on the two pulse scenarios and the 15 gradual scenarios. We find spreads on the order of 0.2–0.4 K W^{−1} m^{2} for climate sensitivities, and around 30%–50% for time constants. The difference between the blue and black distribution indicates the impact of the one- or two-mode assumption on *λ* and *τ _{m}*. It shows that in the two-mode case total climate sensitivities are around 0.05–0.10 K W

^{−1}m

^{2}higher than in the one-mode case, and that mean response times are more than a factor of 2 larger.

### d. Sensitivity tests

In this section we describe the results of some sensitivity tests on the impact of the a priori estimate and uncertainty of the mode characteristics, and on the impact of assuming three modes. We first test using a higher a priori *λ* in the one-mode case. Until now we used an a priori value of 0.5 K W^{−1} m^{2} for *λ* in that case while we used 0.7 K W^{−1} m^{2} in the two-mode case. Assuming one mode and using 0.7 K W^{−1} m^{2} as a priori value for *f*_{1} (and thus for *λ*), there is almost no impact on the results: very similar a posteriori estimates, uncertainties, and RMSE are found.

In a second test, we use 100 yr as a priori estimate for *τ*_{1} in the one-mode case (instead of 10 yr), as if testing the presence of only a slow mode. We find that the a posteriori estimates of *τ*_{1} are very close to the ones in Table 4 (on average only 0.8 yr higher for the step scenarios and around 2.0 yr higher for the gradual scenarios). Only the three AOGCM experiments that lack a stabilization phase lead to much larger values for *τ*_{1} (increases by 13.1, 15.2, and 24.0 yr) but with still a good agreement with the AOGCM data. This could have been expected because of the much larger uncertainty on *τ*_{1} and *f*_{1} (and a high correlation between them) for these AOGCMs.

In a third test we use 400 yr as a priori estimate for *τ*_{2} in the two-mode case (instead of 100 yr). The results for the a posteriori values of the mode characteristics are given in Table 8. It shows that we obtain considerably larger values for *τ*_{2}, with a value of 300.4 yr averaged over all AOGCMs. These different estimates are also visualized in the histograms in Fig. 3 (middle panels). The mode-based results (not shown) are almost equally good with similar values for the RMSE. The large volatility of the a posteriori estimate of *τ*_{2} could be expected because of the large uncertainty of 86% already present in Table 5.

In a fourth experiment, we redo the estimation in both the one- and two-mode case assuming an a priori uncertainty on all parameters of 400% (instead of 100%). By doing this, a posteriori values more easily deviate from their a priori values, leading to a larger spread, while the RMSE improves only marginally. A broader distribution for the a posteriori values of the mode parameters can clearly be seen in the histograms of Fig. 3 (bottom panels).

Finally, we investigate whether a third mode with a large time constant can be estimated. Assuming the presence of a third mode with a priori best estimates for *f*_{3} and *τ*_{3} of 0.1 K W^{−1} m^{2} and 400 yr, the mode-based results almost do not change with respect to the two-mode case: a posteriori RMSE, estimates, uncertainties, and correlations for *f*_{1}, *τ*_{1}, *f*_{2}, and *τ*_{2} are almost equal for the two- and three-mode case, while the a posteriori uncertainties on *f*_{3} and *τ*_{3} remain large.

## 5. Discussion

### a. Number of modes and type of scenario

We found that two modes better reproduce the AOGCM results, and the lower RMSE indicates that there are at least two important time scales in the response of the atmosphere–ocean system to radiative forcings: *τ*_{1} represents the fast response of the atmosphere and the ocean mixed layer, and *τ*_{2} represents the time scale of the slow response of the deeper ocean. In addition to the lack of an ocean-like time scale when assuming only one mode, it also shows lower a posteriori total climate sensitivities with differences up to 30% (a sensitivity test mentioned in section 4a shows that this difference is not caused by the lower a priori value for *λ* in the one-mode case). For these two reasons, an approximation based on one mode might deviate strongly from one using two modes when used for long scenarios. A glimpse of this can already be seen in the Cg2 and Cg4 results where significant differences between the one- and two-mode approaches are visible in the temperature tendency at the end of the simulations. For the scenarios used here, a sensitivity test with the presence of a third mode improves only marginally the behavior of the mode-based results. This, however, does not exclude the possibility that a third mode might show up if much longer scenarios were used. Li and Jarvis (2009), who used the same 1000-yr-long AOGCM data series on which the values in Boucher and Reddy (2008) are based, obtained a lower RMSE for the second half of the 1000-yr period when using three modes.

Our results show also that the type of scenario influences the estimates for the modes. For the two-mode case, the type of scenario significantly affects the value of the small time constant *τ*_{1} (i.e., step scenarios tend to favor small values of *τ*_{1}). The values of *f*_{1}, *τ*_{2}, and *f*_{2} are comparable for step and gradual scenarios. Assuming one mode, where the only time constant is a compromise between the *τ*_{1} and *τ*_{2} from the two-mode case, we also see that step scenarios favor a smaller value for *τ*_{1}.

### b. Parameter uncertainties

Figure 4 shows for the different parameters the ratio of the a priori and a posteriori uncertainties as a function of the number of data used. The results based on the step scenarios C20, C2, and Cx2 are shown as squares, both for the cases where these scenarios are used separately (*N* = 50 or *N* = 100) and all together (*N* = 250), while the results based on gradual scenarios are shown as diamonds. We also indicate a linear regression line that is based only on the gradual scenarios. One sees that in general uncertainties reduce more when using more data, and that this reduction is stronger for climate sensitivities than for time constants. Using step scenarios leads for most cases to a stronger uncertainty reduction than using gradual scenarios (most squares lie below the linear regression line). For example, for the determination of *τ*_{1} in both the one- and two-mode case, C20 alone (with only *N* = 100) obtains a similar uncertainty reduction as the gradual scenarios with *N* ≃ 500. Only for *f*_{1} does the use of C2 alone lead to a weaker uncertainty reduction than for the gradual scenarios. For the two-mode case, it is clear that longer time series will further reduce the uncertainty on *f*_{2}, whereas this is less straightforward for *f*_{1}.

Uncertainties are smaller in the one-mode case than in the two-mode case, but this should be interpreted with care. It solely indicates that the a posteriori parameter values give the best possible agreement between the one-mode-based results and AOGCM data (other values for the parameters would give worse fits). The low uncertainty does not guarantee, however, that the correspondence is good (e.g., using two modes gives a lower RMSE than using one mode).

In the two-mode case, rather large a posteriori uncertainties are found for *τ*_{2} (and to a lesser extent for *f*_{2}). Limited uncertainty reduction on a parameter (as for *τ*_{2}) indicates that significantly different values for the parameters (but taking into account the correlations) give equally good fits. This has been explicitly shown by one of the sensitivity experiments for the two-mode case: an averaged standard a posteriori value for *τ*_{2} of 93.4 yr and an averaged a posteriori value of 300.4 yr (based on the a priori estimate of 400 yr) give both equally good fits. However, the strongly different functional dependence (due to differences in *τ*_{2}) will create big differences if using these estimates for scenarios considerably longer than the ones used to derive the estimates. As mentioned above, using scenario simulations that are extended over some more centuries might reduce the uncertainty on *τ*_{2} although this reduction is expected to be slow (see green line in bottom panel of Fig. 4). Another way to reduce the uncertainty on *τ*_{2} might be the extension of the modes to describe also the temperature in the deep ocean (Berntsen and Fuglestvedt 2008; Olivié and Stuber 2010) and the use of additional deep ocean data from the AOGCMs for the estimation.

### c. Comparison with other studies

In Hasselmann et al. (1993), three sets of parameters were presented assuming one or two modes. Their values for the parameters in the one-mode case [also used in Sausen and Schumann (2000)] are 36.8 yr for *τ*_{1} and 0.607 K W^{−1} m^{2} for *f*_{1}. They were derived based on a 100-yr-long simulation with an AOGCM for a gradual CO_{2} scenario without stabilization (Cubasch et al. 1992; Hasselmann et al. 1993, 1997). Looking to our results in the one-mode case, we see that the mean over all the gradual scenarios gives a value of 0.61 K W^{−1} m^{2} for *f*_{1} and a value of 16.2 yr for *τ*_{1}. While our *f*_{1} estimates correspond well with Hasselmann et al. (1993), our *τ*_{1} values are considerably smaller. However, the sensitivity experiment using an a priori estimate for *τ*_{1} in the one-mode case of 100 yr (instead of 10 yr) shows that three AOGCM experiments that lack a stabilization phase give much larger values for *τ*_{1}. As the scenario for these three models is actually very similar to the one used in Hasselmann et al. (1993), one can conclude that scenarios that have only an increasing phase allow a large value for *τ*_{1}.

Hasselmann et al. (1993) also present estimates in a two-mode case based on a comparison with a 100-yr-long Cx2 scenario from two AOGCMs (ECHAM-1/LSG and GFDL). Their values for *f*_{1} and *λ* are in general smaller than what we obtain. Their values for *τ*_{1} (2.86 and 1.2 yr) are relatively close to the values we obtain when using only the Cx2 scenario (i.e., 4.3 and 4.4 yr). Their values for *τ*_{2} (41.67 and 23.5 yr) are considerably lower than the values we obtain (i.e., 103.7 and 107.8 yr). However, there are very large uncertainties of 93% on these values (see Cx2 in Table 5), indicating that smaller values of *τ*_{2} might give equally good fits.

Boucher and Reddy (2008) also assumed the presence of two modes and found time constants of 8.4 and 409.5 yr and a total climate sensitivity of 1.06 K W^{−1} m^{2}, with a contribution of 40% from the slow mode. These values were derived using a 1000-yr-long simulation with the UKMO-HadCM3 model (using a scenario similar to Cg4, but reaching the quadrupling in CO_{2} concentration already after 70 yr). Their total climate sensitivity is to the high end of the sensitivities we derive, and, similarly, their contribution from the slow mode is to the high end of what we find. Their value for *τ*_{1} corresponds very well with the values we obtain using the gradual scenarios (6.8 yr; see Table 5), but their value of *τ*_{2} is considerably higher than ours. As argued before, *τ*_{2} is poorly constrained and very different values still give very good correspondence with the AOGCM results. In the sensitivity experiment represented in Table 8 we found an average a posteriori value of 300.4 yr for *τ*_{2}, which is comparable to the value of 409.5 yr in Boucher and Reddy (2008). Also, Hooss et al. (2001), based on a Cg4 experiment extended up to 850 yr with the ECHAM-3/LSG AOGCM, found a value of 400 yr for the time constant of the slow mode. It is possible that the AOGCM simulations we have used here would give clear indications for larger *τ*_{2} if continued up to 1000 yr.

For the AOGCMs that performed gradual scenarios, Solomon et al. (2007) derived climate sensitivities based on the atmosphere part of the AOGCMs (AGCMs) coupled with a slab ocean model (except for CNRM-CM3) [see Tables 8.2 and S8.1 of Solomon et al. (2007)]. Figure 5 shows the relation between the values for *λ* obtained using the AGCMs coupled with a slab-ocean model and the values obtained by us assuming one (black) or two (blue) modes. In general, when using both the one- and two-mode approach we find considerably smaller values. This might indicate that there are conceptual difficulties in trying to deduce the climate sensitivity of an AOGCM from an equilibrium experiment where the AGCM is coupled to a slab-ocean model. However, better agreement is found in the two-mode case when using larger values of *τ*_{2} (see the sensitivity experiment using 400 yr as a priori estimate for *τ*_{2}), indicating that the AOGCMs have not reached a new equilibrium at the end of the Cg2 and Cg4 simulations.

### d. Importance of mode parameters

Until now, we concentrated on the estimation of the mode characteristics using specific scenarios. However, an important application of the identification of the modes is their use to calculate the response to a forcing scenario that is of interest for a particular research question and that can differ significantly from the scenarios used until now. For example, an approximately exponential type of increase in the concentration of CO_{2} has been observed since the start of the industrial era. But different scenarios for CO_{2} can be expected in the future if mitigation of CO_{2} emissions occurs. Also, people often study the impact of short pulse emissions when comparing the climate impact of different species. These very different scenarios might impose different constraints on the accuracy of the mode parameters, and it might be relevant to know which mode parameters should be known with high accuracy, and for which high precision is less important. We shortly look here into the role of the time scale and climate sensitivity of the different modes in the response to a general forcing. In section d of the appendix, we show that the temperature response *T*(*t*) assuming *n* modes to a general forcing *F*(*t*) can be written as

The first term on the right-hand side of Eq. (6) indicates that should be well estimated. The second term in Eq. (6) contains the derivative of the forcing and therefore is a bit more complicated. However, if the forcing is changing slower than a characteristic time scale *τ _{f}*, that is,

then the absolute contribution of mode *i* to the second term can be approximated; that is,

This means that for the contribution of the fast modes for which *τ _{i}* ≪

*τ*the following approximation in Eq. (7) is valid:

_{f}indicating that the knowledge of the exact value of small *τ _{i}* is not important. However, for rapidly varying forcings, the response will contain parts that will be very similar to the expression in Eq. (2). The factor

*f*/

_{i}*τ*in front of the exponential makes that a good estimate for the value of the small time constants becomes very important.

_{i}## 6. Conclusions

We estimated the modes present in the global- and annual-mean temperature response of different AOGCMs to radiative forcing perturbations, based on relatively short simulations on the order of 100–300 yr. The analysis also discussed the origin of the large spread in mode characteristics found in the literature and obtained in our study. We summarize our results by answering the questions mentioned in the introduction.

How many modes appear to be present in the global-mean surface air temperature response?

Based on the results we obtained, we found that two modes can be clearly identified for simulations shorter than 300 yr: a short time scale of around 3.5–3.9 yr (representing the response time of the atmosphere and the ocean mixed layer) and a long time scale of around 93.4–300.4 yr (representing the response of the deep ocean). Using two modes, one is able to well reproduce AOGCM results both for gradual and step scenarios, allowing in general a smaller RMSE and a better representation of the temperature tendency than when using one time constant. We found that the total climate sensitivity *λ* and the mean response time *τ _{m}* using one mode are smaller than when using two modes. Introducing a third mode did not improve the behavior of the mode-based approach for the scenarios investigated here (length up to 300 yr).

How strongly do the mode estimates differ among AOGCMs?

Differences in the behavior of the AOGCMs are strongly reflected in the mode characteristics estimates. The total climate sensitivity varies (excluding the results from the three AOGCMs with only short gradual scenarios) between 0.45 and 0.76 K W^{−1} m^{2} when assuming one mode. When assuming two modes it varies between 0.49 and 0.83 K W^{−1} m^{2} or between 0.56 and 1.01 K W^{−1} m^{2} in case of using a higher a priori value for *τ*_{2}. The values of *f*_{2} and *τ*_{2} in the two-mode case reflect the behavior at longer time scales (which can be very different among the AOGCMs) and their values considerably vary depending on the AOGCM. The contribution from the slow component to the total climate sensitivity is on average 30%–35%, but values around 15% and 40% have also been found. The averaged values presented in Tables 5 and 8 represent our best estimates for the mode characteristics. These are a possible set of parameters that may be used for studying the temperature response to forcings on time scales up to 300 yr. However, one should be aware of the large spread in mode characteristics among the different AOGCMs.

What is the impact of the type of scenario on the mode estimates?

The type of scenario used to estimate the mode characteristics has a significant impact on the a posteriori estimates. Step scenarios tend to attribute lower values to the smallest time constant and are more appropriate to constrain its value. The smallest time constant in the two-mode case is on average 2.8–3.0 yr using step scenarios, whereas it is 6.8–8.4 yr using gradual scenarios. When using only one mode, the time constant is a compromise between the two time constants of the two-mode case, but we see again that it is clearly smaller for the step than for the gradual scenarios with average values of 4.2 and 16.2 yr, respectively. We also see that the scenario C2 (very rapidly decaying pulse) and a short gradual scenario without a stabilization phase are not very well suited to estimate the mode characteristics. The uncertainty on the climate sensitivity of the slow mode can be reduced by using longer simulations.

Are some mode characteristics easier to estimate, and do all mode characteristics have to be known with the same accuracy?

Our analysis of the a posteriori uncertainties shows that the uncertainty reduction is stronger for climate sensitivities than for time constants. Assuming two modes, the uncertainty reduction is stronger for the fast mode than for the slow mode. It is mainly the climate sensitivity of the slow mode that benefits from performing longer simulations. A large uncertainty on the parameters of the slow mode in the two-mode case does not necessarily imply large uncertainties on the mode-based temperature evolution due to the correlations among the parameters. However, one should be careful when using the modes on time scales longer than the length of the simulations with which they have been estimated. Also, the required accuracy on climate sensitivities and time constants for the different modes depends on the characteristics of the scenarios for which the modal approximation will be used. For example, it is important to accurately estimate the time constants of the fast mode when having rapidly changing forcings, whereas their exact values are not very important when calculating the climate response to a slowly varying forcing.

## Acknowledgments

We thank Nicola Stuber for performing the UKMO-HadCM3 step scenario experiments and David Salas for the help with running the CNRM-CM3 model. We also thank Jan Fuglestvedt and Terje Berntsen for the interesting discussions, and Aurore Voldoire for comments on the manuscript. We acknowledge the modeling groups, the Programme for Climate Model Diagnosis and Intercomparison (PCMDI), and the WCRP’s Working Group on Coupled Modelling (WGCM) for their role in making available the WCRP CMIP3 multi-model dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy. This work was partly funded by the European Union FP6 Integrated Project QUANTIFY (http://www.ip-quantify.eu/) under Contract 003893 (GOCE).

### APPENDIX

#### The Mean Response Time

We define the mean response time in the *n*-mode case as

where *τ _{i}* and

*f*are the response time and climate sensitivity of mode

_{i}*i*. By the following examples we try to indicate the relevance of the mean response time and its physical interpretation. These examples are also illustrated in Fig. 6.

##### a. Step forcing

When adopting for the radiative forcing the step function

where *F _{c}* is a constant, the contribution from each mode

*i*to the convolution integral in Eq. (1) can be calculated analytically and gives

##### b. Linearly increasing forcing

When adopting for the radiative forcing the linearly increasing function

with *t _{c}* and

*F*constants, the contribution from each mode

_{c}*i*to the convolution integral in Eq. (1) can be calculated analytically and gives

For very small *τ _{i}*, this response approaches

suggesting that the response follows almost perfectly the shape of the forcing. For the general case (where *τ _{i}* does not have to be small), the response to the linear increasing forcing can be rewritten in the case of

*t*≫

*τ*as

_{i}This means that for a linear increasing forcing, the response has an asymptotic delay of *τ _{i}* with respect to the instantaneous response in Eq. (A6) and thus also with respect to the forcing. When multiple time scales

*τ*are present, the response for

_{i}*t*≥ 0, where

*t*≫

*τ*for all

_{i}*i*, can be written as

So the delay of the response becomes *τ _{m}* (i.e., the average of the time scales weighted by

*f*).

_{i}##### c. Scenarios Cg2 and Cg4

When adopting the radiative forcing

(this forcing can represent the Cg2 or Cg4 scenarios), the contribution from each mode *i* to the convolution integral in Eq. (1) can be calculated analytically and gives

For very small *τ _{i}*, this response approaches

This means that the response follows almost perfectly the shape of the forcing. Similar to the linear increasing case, one can easily derive approximative expressions.

##### d. General scenario: Comment on the cold start problem

Here we show that *τ _{m}* appears in general in the responses to all type of forcings (not only linear increasing forcings). We will prove that the delay between forcing and response is

*τ*, and that

_{m}*T*(

*t*) from Eq. (1) can be written as

A system with one time constant can be described by the differential equation

and we first show that its solution can be written as

Taking the time derivative of Eq. (A14)

If we then use the Taylor series expansions

and

and substitute them into Eq. (A16), one can easily see that the equality is fulfilled.

For a general system with various time constants, the response can therefore be written as

where we used the earlier definition of *τ _{m}*, the Taylor series expansion for

*F*(

*t*−

*τ*), and the shorthand notation .

_{m}We think that this *τ _{m}* explains large parts of the phenomena described in Hasselmann et al. (1993) and can be easily determined based on the characteristics of the system. Eventually, one can also calculate the cold start effect as

## REFERENCES

_{2}emission paths using a simplified structural integrated assessment model (SIAM)

_{2}forcing experiment revisited

_{2}and NO

_{x}emissions scenarios

## Footnotes

Prior affiliation: Centre National de Recherches Météorologiques, Météo-France, Toulouse, France.