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Transient Climate Response in a Two-Layer Energy-Balance Model. Part I: Analytical Solution and Parameter Calibration Using CMIP5 AOGCM Experiments [Journal of Climate](Journal of Climate Via Acquire Media NewsEdge) ABSTRACT This is the first part of a series of two articles analyzing the global thermal properties of atmosphere-ocean coupled general circulation models (AOGCMs) within the framework of a two-layer energy-balance model (EBM). In this part, the general analytical solution of the system is given and two idealized climate change scenarios, one with a step forcing and one with a linear forcing, are discussed. These solutions give a didactic description of the contributions from the equilibrium response and of the fast and slow transient responses during a climate transition. Based on these analytical solutions, a simple and physically based procedure to calibrate the two-layer model parameters using anAOGCMstep-forcing experiment is introduced. Using this procedure, the global thermal properties of 16 AOGCMs participating in phase×of the Coupled Model Intercomparison Project (CMIP=) are determined. It is shown that, for a given AOGCM,theEBMtuned with only the abrupt 4&×CO2 experiment is able to reproduce with a very good accuracy the temperature evolution in both a step-forcing and a linear-forcing experiment. The role of the upper-ocean and deep-ocean heat uptakes in the fast and slow responses is also discussed. One of the main weaknesses of the simple EBM discussed in this part is its ability to represent the evolution of the top-of-the-atmosphere radiative imbalance in the transient regime. This issue is addressed in Part II by taking into account the efficacy factor of deepocean heat uptake. (ProQuest: ... denotes formulae omitted.) 1. Introduction Determining the response of the climate system to an imposed external perturbation is a major challenge in climate science. The global and annual mean surface temperature response is a useful metric to determine the magnitude of a climate change induced by an externally imposed radiative perturbation. Indeed, many studies suggest that most of the climate variables are related to the global mean surface temperature response. Coupled atmosphere-ocean general circulation models (AOGCMs) are themost comprehensive tool to study climate changes and perform climate projections. They can be used to assess the changes in global temperature but they are computationally expensive. Alternatively, simple climate models (SCMs), which estimate approximately the global mean surface temperature change for a given externally imposed perturbation in the earth's radiation balance (Meinshausen et al. 2011; Good et al. 2011; Friend 2011; van Hateren 2013), can be used to emulate the AOGCM responses in order to cover a wide range of scenarios at a negligible computational cost. Energy-balance models (EBMs) are physically based SCMs. They are useful to summarize AOGCM global thermal properties and to compare and analyze AOGCM responses (Raper et al. 2002; Soden and Held 2006; Gregory and Forster 2008; Dufresne and Bony 2008). In the case of a small perturbation, some EBMs assume that the thermal energy balance of the climate system is only expressed as a linear function of temperature perturbation (Budyko 1969; Sellers 1969). The net radiative imbalance caused by an external forcing and a temperature change can be expressed as N&×F - ?T. The radiative feedback parameter ? with respect to the global mean surface air temperature T depends on the type of forcing (Hansen et al. 200=). The imposed radiative forcing F includes the effects of fast (few months) stratospheric and tropospheric adjustments (Gregory and Webb 2008). In this formulation of the radiative imbalance N, the assumption of linear dependency in T suffers from some limitations (Gregory et al. 2004; Williams et al. 2008; Winton et al. 2010; Held et al. 2010) that are addressed in Part II of this study (Geoffroy et al. 2013, hereafter Part II). In equilibrium, N×0 and the steady-state temperature is equal to Teq×F/?. Equilibrium climate sensitivity (ECS), which is defined as the equilibrium mean surface air temperature perturbation resulting from a doubling carbon dioxide radiative forcing, is commonly used as a metric of anthropogenic climate change. However, this metric is not sufficient to study the transient regime because of the climate system's thermal inertia. Indeed, the rate of change in the heat content of the climate system is equal to the earth's radiative imbalance and this change occurs on large time scales because of the large thermal inertia of the deep ocean (Dickinson 1981; Hasselmann et al. 1993; Murphy 199=; Gregory 2000; Held et al. 2010). Based on empirical relationships, Gregory and Mitchell (1997) and Raper et al. (2002) propose a formulation for the deep-ocean heat uptake, which is proportional to the surface temperature perturbation, H×kT. However, this formulation is not able to represent the equilibrium temperature response in the case of a step-forcing or a stabilization scenario because the deep-ocean temperature response is by definition neglected. The solution to circumvent this shortcoming is to introduce a second layer that represents the deep ocean. Splitting of the climate system into two thermal reservoirs with different heat capacities allows one to account for the ocean thermal saturation along a transient regime until equilibrium and to represent the two distinct time scales of the global mean climate system response (Hasselmann et al. 1993; Held et al. 2010). This system is similar to the three-layer EBM presented in Dickinson (1981), the atmosphere and the upper-ocean layers being considered as one single layer characterized by the surface air temperature. In this study, we derive the analytical solution of this two-layer energy-balance model and propose a calibration method to determine the equivalent thermal parameters of a givenAOGCM.We then assess the validity of this simple framework to represent the behavior of the complex coupled models in response to an idealized forcing scenario by analyzing the results of 16 AOGCMs participating in the fifth phase of the Coupled Model Intercomparison Project (CMIP=). The role of each layer's heat uptake in the fast and slow components of the transient response is also discussed. The structure of the paper is as follows: after introducing the theoretical framework and describing the analytical solutions for different forcing scenarios in section 2, the methodology used to adjust the two-layer EBM response to AOGCMs results is presented and applied to CMIP= AOGCMs in section 3. 2. Theoretical framework a. Two-layer energy-balance model We consider the linear two-layer energy-balance model described in Held et al. (2010). Held et al. (2010) also proposed an alternative model with an additional parameter, an efficacy factor for deep-ocean heat uptake, which will be discussed in Part II. The climate system is split into two layers (Gregory andMitchell 1997;Gregory 2000). The first one corresponds to the atmosphere, the land surface, and the upper ocean, and the second one represents the deep ocean. The state of each layer is described by the temperature perturbations T and T0. Note that T is usually taken as the global mean surface air temperature perturbation fromthe control climate, while T0 is a characteristic temperature perturbation of the deep ocean. Here, T and T0 satisfy the following system of equations: ... (1) ... (2) This system has two prognostic variables T and T0 and five free parameters: ?, ?, C, C0, and a radiative forcing amplitude parameter F. Whatever the radiative forcing agent, the radiative forcing formulation requires at least onemodel-dependent reference radiative parameter Fref caused by stratospheric and tropospheric adjustments. In the case of a CO2 perturbation, the radiative forcing can be expressed as a function of the CO2 concentration and a radiative parameter following (Houghton et al. 1990) ... (3) where [CO2]t is the time-dependent carbon dioxide concentration, [CO2]0 is the preindustrial CO2 concentration, and F23CO2 is the net radiative forcing associated with a doubling of the atmospheric CO2 concentration. Here C(dT/dt) and C0(dT0/dt) are the tendencies of heat contents respectively of the upper and the lower layer; C and C0 are effective heat capacities (per unit area) of the upper ocean (we neglect the heat capacities of the atmosphere and land surfaces) and the deep ocean, respectively. The parameter g is a heat exchange coefficient. The heat flux exchange between the two layers is thus assumed to be proportional to the difference between the two temperature perturbations. In the limit of an infinite deep-ocean heat capacity C0 [arrow right] 8, T0 is zero and the expression of the heat flux exchange is the one proposed by Gregory and Mitchell (1997) with k×?. In this one-layer model (the deep-ocean layer is an external infinite reservoir), the temperature perturbation follows the equation ... (4) Such systems can be advantageously described in terms of equivalent electrical circuits (Schwartz 2008; van Hateren 2013). In appendix B, the equivalent electrical circuits of the one- and two-layer EBMs are presented and differences are discussed. A main difference between the one- and two-layer EBM is that the one-layer EBM does not allow the equilibrium temperature to be reached. Indeed, the temperature response in equilibrium is equal to F/(?1k). The zero-layer EBM [C = 0 in Eq. (4)] is an approximate version of the one-layer EBM commonly used (Raper et al. 2002; Dufresne and Bony 2008). Such a model does not allow the representation of the transition for step-forcing cases because the temperature response is a linear function of the forcing. Finally, the zero- and one-layer EBM are per definition not able to represent all kinds of climate change scenarios. The temperature TH associated with the climatesystem heat uptake is defined as the disequilibrium temperature difference between T and the instantaneous equilibrium temperature Teq(t)×F(t)/? (Winton et al. 2010). The latter is the equilibrium temperature associated with the instantaneous forcing applied at time t. The heat-uptake temperature represents the instantaneous rate of heat storage in the climate system and follows the equation ... (5) Contrary to Winton et al. (2010), a negative heat-uptake temperature corresponds here to a positive heat storage in the climate system. b. Analytical solutions In appendix B, the general solutions of both the mean surface temperature and the deep-ocean temperature responses are derived for any forcing function t/F(t). With an integration by parts of Eqs. (B8) and (B9), the temperature perturbations T and T0 can be written as the sum of the balanced temperature Teq(t) and two modes characterized by two distinct time scales, tf (fast) and ts (slow), as follows: ... (6) ... (7) where ti, ai, and fi are parameters depending on C, C0, ?, and ?. Their expressions are given in Table 1. Note, in particular, that af + as×1 and ff af + fsas×1, and that ff < 0 while af, as, and fs are all positive. The sum term in Eq. (6) is the heat-uptake temperature TH. It is the sum of two modes that can be decomposed in two terms depending on the forcing function. The first contribution is an instantaneous deviation associated with a discontinuity of the forcing at t×0. The second term is due to the time evolution of the forcing. In the following paragraphs, we briefly discuss the analytical solution for two idealized forcings, step and linear. In appendixes C and D, we present solutions for stabilization, abrupt return to zero, and periodic forcings. 1) STEP FORCING For a step forcing, ... (8) the analytical solution of the two-layer energy-balance model is given by ... (9) ... (10) By decomposing the response as the sum of the equilibrium temperature response and the two modes characterized by the two distinct time scales, tf (fast) and ts (slow) following Eqs. (6) and (7), the temperature perturbations T and T0 are ... (11) ... (12) The sums of the last two terms of Eqs. (11) and (12) are respectively the perturbations TH(t)×T(t) - Teq and T0H(t)=T0(t)-Teq from the new equilibrium T=T0= Teq. Thus, ai is the partial contribution of the mode i to the TH initial amplitude in the case of a step forcing. Initially, both the slow and the fast terms of TH are negative with respective amplitudes -afTeq and -asTeq. During the transition, they increase exponentially toward zero with their respective relaxation times tf and ts as illustrated in Figs. 1a and 1b. Since fs > 0, the slow contributions of TH and T0H have the same sign, and the slow mode corresponds to a joint adjustment of the upper and lower layers. On the other hand, since ff< 0, the fast mode of T0H is of opposite sign to the slow mode of TH (in the fast mode, TH < 0 and T0H > 0). The perturbation heat flux from the lower layer to the upper layer is 2H×2g(TH 2 T0H) and its fast mode is of opposite sign to TH. The fast mode thus corresponds to an adjustment of the upper layer by both the radiation imbalance and the deepocean heat uptake. The two physical processes at play interact positively to adjust the smallest energy reservoir. This explains why the characteristic time scale tf is shorter than the characteristic time scale of a one-layer model of the upper layer without deep-ocean heat uptake (i.e., the limit of tf when C0 tends toward zero; tf , C/?). Still, tf is longer than the characteristic time scale of the one-layer model with the deep-ocean heat-uptake formulation of Gregory and Mitchell (1997) and Raper et al. (2002) that is the limit of tf when C0 tends toward infinity tf > C/(? + ?). In that model, the deep-ocean heat uptake damps TH more efficiently than in the twolayer model because of its infinite heat capacity. For a system with a vanishingly small upper-ocean heat capacity, the adjustment of the first layer is immediate leading to a slightly faster adjustment of the deep ocean. Indeed, the slow time scale is longer than its limit when C tends toward zero ts > C0(1/? + 1/?). 2) LINEAR FORCING To derive the analytical solution of the system for a linear forcing ... (13) Eqs. (6) and (7) require the knowledge of the integral ... which is ... Then the general solution can be written as ... (14) ... (15) As in the step-forcing case, the surface temperature perturbation is the sum of an instantaneous equilibrium temperature response Teq(t)×F(t)/? and an imbalance term that can be decomposed into a fast and a slow mode response as illustrated in Figs. 1c and 1d. Contrary to the abrupt case, the system is initially in equilibrium and deviates from its instantaneous equilibrium afterward. The fast and slow responses decrease with time and asymptotically tend toward negative limits. Their amplitudes are proportional to their respective relaxation times, resulting in a small amplitude of the fast response. Assuming a logarithmic relationship between the radiative forcing and the carbon dioxide concentration [Eq. (3)], the 1% yr-1 CO2 experiment corresponds to a linear forcing ... (16) 3. Multimodel analysis In this section, a method to tune the two-layer model parameters described above to fit the behavior of each AOGCM using only the idealized step-forcing experiments is proposed. The calibration method is then applied to 16 available AOGCMs participating in the CMIP= (Taylor et al. 2011) and is validated by using theAOGCMresponses to the linear forcing 1%yr-1 CO2 experiments. a. Method for parameter calibration The method uses only the transient response of an AOGCM step-forcing experiment. We assume that the top of the climate system corresponds to the model top of the atmosphere (TOA). Both radiative net flux change at TOA and surface temperature change T are used to adjust the two radiative parameters Fref (the adjusted radiative forcing amplitude) and ?. Only T is used to adjust the thermal inertia parameters C, C0, and g. The method can thus be decomposed in two steps. 1) FIRST STEP The first step consists of estimating the radiative parameters Fref and l by performing a linear regression of the radiative imbalance as a function of the mean surface air temperature perturbation (Gregory et al. 2004). Using this method, the computation of Fref and ? takes into account both stratospheric and tropospheric adjustments. However it assumes a linear dependence between the earth's radiative imbalance and the surface temperature perturbation such that N×F - ?T. The limitations of this assumption are discussed in Part II. 2) SECOND STEP The second step consists first of the calibration of the four mode parameters tf, ts, af, and as by fitting the global mean surface air temperature response. For t?tf, Eq. (11) can be approximated as follows: ... (17) Assuming that tf ? 30 yr, the linear regression of log(Teq - T) against t over the period 30-1=0 yr gives estimates of ts and as. Then af×1 - as is known and tf can be expressed from Eq. (11) as a function of these three parameters and the surface temperature response ... (18) Its value is estimated by averaging over the first 10 years of the step-forcing experiment. Finally, the remaining physical parameters of the model (the heat capacities C and C0 and the heat exchange coefficient g) are computed from the other parameters using the following analytical relationships (see Table 1): ... (19) ... (20) ... (21) This methodology is applied to instantaneous carbon dioxide quadrupling (abrupt 4×CO2) experiments (with a typical integration time of 1=0 yr) performed by an ensemble of 16 AOGCMs participating in CMIP=. Note that we use annual mean values over a period of 1=0 years even when longer simulations are provided. The name of the AOGCMs used and their expansions are provided in Table 2. b. Results 1) RADIATIVE PARAMETERS For the 16 AOGCMs considered here, the radiative parameters and the 43CO2 equilibrium temperature response are reported in Table 3. The results are in good agreement with the estimates of Andrews et al. (2012). The multimodel average of the net radiative forcing (6.9 W m-2) is very close to previous CMIP3 analysis results (Williams et al. 2008), and the relative intermodel standard deviation is about 13%. The estimates for the model's feedback parameters are consistent with previous results with older AOGCMs (Soden and Held 2006). The multimodel mean (1.13 W m-2 K-1) and standard deviation (0.31 W m-2 K-1) of the total feedback parameters (Table 3) are close to previous values obtained for CMIP3 models and for different types of scenarios. The 43CO2 equilibriumtemperature response ranges from4.1 to 9.1 K. The spread among the responses is as large as those of CMIP3 simulations. 2) CLIMATE SYSTEM INERTIA PARAMETERS In Table 4, we summarize the corresponding thermal parameters for each of the 16 models. We first note that the deep-ocean heat-capacity values are about an order of magnitude larger than the upperlayer heat-capacity values. The multimodel means of C (7.3 W yr m-2 K-1) and C0 (106 W yr m-2 K-1) are close to the estimates of Dickinson (1981) of 10 and 100 W yr m-2 K-1 for the ocean mixed layer and for the deep-ocean capacities respectively. The deepocean heat-capacity mean value is, however, larger than the estimate of Murphy (199=) for the ocean capacity of about =2 W yr m-2 K-1 (1.6= 109 J m-2 K-1). Considering that the ocean covers f0×70%of the Earth's surface, and using a constant water heat capacity of cp = 4180 J kg-1 K-1 and a constant density of saltwater r = 1030 kg m23, the AOGCM ensemble mean C0 value corresponds to an equivalent deep-ocean layer depth D0 equal to ... (22) Similarly, an upper ocean with an effective heat capacity equal to the AOGCM ensemble mean value is equivalent to a 77-m-thick mixed layer. The INM-CM4 model gives a very large value of C0 (317 W yr m-2 K-1) in comparison with other models. One can wonder if this estimation can be biased by the driftin surface temperature evolution since the INMCM4 model is one of the two models with the largest driftin surface temperature evolution in the course of the preindustrial control simulation. Indeed, the INMCM4 driftis on the order of 20.03 K century-1 (over a period of =00 yr) against a model ensemble mean absolute value of 0.02 K century-1 and a standard deviation of 0.014 K century-1. However, after removing the temperature trend, the C0 estimate for INM-CM4 still remains largely outside the range of the model ensemble with a value of 271 W yr m-2 K-1. All other parameters of this model and all the parameters of the other models are not significantly impacted by the temperature driftcorrection. Further investigation would be needed to explain the INM-CM4 behavior. By excluding this model, the ensemble mean C0 value is 91 W yr m-2 K-1 with a much smaller standard deviation of 27 W yr m-2 K-1. The heat exchange coefficient g ranges from 0.= to 1.2 W m-2 K-1 (0.9 W m-2 K-1 without the GISS-E2-R model) with an ensemble mean of about 0.7 W m-2 K-1. These values are somewhat larger than the zero-layer EBM heat exchange coefficient k values estimated by Raper et al. (2002) and Gregory and Forster (2008) and of the same order of magnitude as the estimates of Plattner et al. (2008). One could expect that the introduction of the deep-ocean temperature perturbation T0 in the two-layer EBM reduces the contribution of the temperature difference term to the deep-ocean heat uptake H×g(T - T0) formulation (for a given H: T - T0 < T, so that g > k). Fast and slow time responses are also given in Table 4. The fast time constant is on the order of 4 yr and the slow response is on the order of 2=0 yr. These values are consistent with previous estimations of climate system time scales (see, e.g., Olivie et al. 2012). The intermodel standard deviation for the slow relaxation time is about 13= yr. It is reduced to 63 yr by omitting the large value of ts (caused by the large C0) of the INM-CM4 model. The estimates of these climate system parameters could be biased as a consequence of the biases in the radiative parameters estimated using the method of Gregory et al. (2004). The sensitivity of these estimates to a more refined formulation of the two-layer model is explored in Part II. 3) GLOBAL MEAN SURFACE AIR TEMPERATURE RESPONSE The comparison between the analytical model calibrated from abrupt 43CO2, the AOGCM responses to the abrupt 43CO2, and the AOGCM responses to the 1% yr-1 CO2 increase up to 43CO2 is shown in Fig. 2. For CNRM-CM= and GFDL-ESM2M, a 23CO2 stabilization scenario is also available. Note that the analytical EBM results for the 1% yr-1 CO2 and the stabilization cases are computed using the parameters tuned using the abrupt 43CO2 experiment, and are therefore independent of the corresponding AOGCM experiments. All values are temperature change with respect to the mean control values over the whole 1=0-yr period. The simple analytical model is able to reproduce the evolution of surface air temperature in response to both a step-forcing and a gradual-forcing scenario. The fit seems to accurately represent the behavior of the surface temperature in a case of an abrupt forcing, not only at the beginning and at the end of the period (used in the tuning), but also in the intermediate period of transition between the two modes. However, for some models, a slight overestimation is observed for the 1% yr-1 CO2 scenario (CSIRO-Mk3.6.0, MIROC=, MPI-ESM-LR, NorESM1-M) and for the 23CO2 stabilization (GFDLESM2M). It may be due to the imperfect logarithmic dependency between the radiative forcing and the carbon dioxide concentration (e.g., because of tropospheric adjustment) or to limitations inherent to the linear two-layer model such as the use of a single feedback parameter for all radiative forcing amplitudes, the assumption of linearity between the radiative imbalance, and the surface temperature change during a climate transition, or an oversimplified representation of ocean heat uptake. It is possible that using a median scenario to fit the EBM's parameters would give more accurate results. The abrupt 43CO2 case is an extreme case and an intermediate CO2 increase scenario such as a doubling of carbon dioxide concentration may give more adequate results. Overall, it appears that the climate response depicted by the AOGCMs can be captured by a properly tuned two-layer climate model. c. Upper- and deep-ocean heat-uptake contributions to the fast and slow responses In this section, the concepts of upper- and deepocean heat-uptake temperatures are introduced. The heat-uptake temperature TH [Eq. (=)] can be decomposed into the sum of an upper-ocean heat-uptake temperature TU and a deep-ocean heat-uptake temperature TD with ... (23) ... (24) The contribution of these two components to the fast and the slow responses are quantitatively examined for two forcing functions. 1) STEP FORCING In the case of a step forcing, by using Eq. (5), the heatuptake temperature TH is ... (25) The heat-uptake temperature tends exponentially to zero with slow and fast relaxation times ... (26) with fU + fD×1 and sU + sD×1. Each mode (slow and fast) is respectively written as the sum of the contribution of each component heat uptake (the subscript U refers to the first layer and the subscript D to the second layer). Indeed, fU ( fD) and sU (sD) are the partial contributions of the upper (deep) component respectively to the fast and the slow responses in: ... (27) ... (28) where ... (29) The order of magnitude and the sign of the fractional contributions are given in section 3c(3). 2) LINEAR FORCING In the case of a linear forcing, the heat-uptake temperature is ... (30) which can be rewritten as ... (31) where lf and ls are a fractional contribution of the fast and the slow terms (lf + ls×1). Their expression is given in Table 1. Note that fU, fD, sU, and sD are the same as previously, and hU×C(C + C0) and hD×C0(C + C0) correspond to the fractional contribution of upper and lower layers to the asymptotic heat-uptake temperature, which is proportional to the sum of the two heat capacities ... (32) 3) QUANTITATIVE ESTIMATES OF FRACTIONAL CONTRIBUTIONS Figure 3a shows the fractional contributions of the fast and slow modes to the maximum amplitude of the heat-uptake temperature for the step forcing, af and as. For all models except one (CSIRO-Mk3.6.0), the percentage of TH caused by the fast response is larger than that caused by the slow response for a step forcing but with a similar order of magnitude. The multimodel mean value of af is =9%. The contributions of the upper- and lower-layer heat uptake to the fast ( fU and fD) and the slow (sU and sD) terms are depicted in Figs. 3b and 3c. For the fast mode, the role of the two components of the system is opposite but with similar amplitude. For all models, the amplitude of the atmosphere/land/upper-ocean contribution TU is larger than that of the deep ocean. For the slow mode, the contribution of TU is negligible (i.e., sUsD). Then, the temperature slow response is driven exclusively by the deep-ocean heat uptake. The fast and slow modes of the deep-ocean heatuptake temperature TD are of opposite sign with equal initial amplitude. During a step-forcing transient regime, TD decreases from zero toward negative values (the heat uptake H increases from zero) until the fast mode becomes negligible. Then TD increases slowly and tends asymptotically toward zero. This nonmonotonic time evolution results from the fact that the surface and the deep-ocean temperature perturbations T and T0 associated to the fast response have opposite signs (ff< 0). The heat flux between the lower and upper layer is upward: the deep ocean warms the surface in the fast response, as pointed out in section 2b(1). In the case of a linear forcing, the contribution lf of the fast term is negligible (Fig. 3d) with a multimodel mean value of 0.03%, because the fractional amplitudes lf and ls are proportional to their respective relaxation times. The heat-uptake temperature is driven by the deepocean heat-uptake temperature slow term (sU ? sD) and by the asymptotic term TH. The upper-ocean heat-uptake temperature fast termreaches its asymptotic value, which represents on average 8%of the asymptotic heat-uptake temperature TH. Figure 3e shows the multimodel contributions to TH of both upper ocean and deep ocean. However, on a centennial scale, the asymptotic deepocean heat-uptake temperature is not reached. As a result, the contribution of TD relative to TU is smaller during the transient regime (Fig. 3f). The upper-ocean heat-uptake temperature TU is on average 18% of the heat-uptake temperature at the time of 23CO2 (t = 70 yr) and 12% at the time of 43CO2 (t×140 yr). Removing the upper-ocean heat-uptake contribution, the transient climate response (i.e., T at the time of 23CO2) would be on average 11% larger, which corresponds to a temperature difference on average of 0.2 K (and a range of 0.1-0.4 K). 4. Conclusions In this study, we describe the analytical solutions of a two-layer energy-balance model for different idealized forcings and propose a method to tune the parameters of this simple climate model to reproduce the behavior of individual coupled atmosphere-ocean general circulation models. In this simple idealized framework, the global mean surface response change consists of the sum of an instantaneous equilibrium temperature and a disequilibrium temperature, the heat-uptake temperature, which is a sum of two modes. One mode responds very quickly to changes in forcing, whereas the other mode has a longer relaxation time. By analyzing the results of 16 AOGCM's experiments from CMIP=, we show that this decomposition in an equilibrium term and two modes can be derived for any AOGCM by a calibration method using only a step-forcing scenario. We first show that this decomposition can reproduce well the behavior of AOGCM's response to a step 43CO2 forcing scenario over the 1=0-yr period covered by the CMIP= simulations. We also find that the simple model calibrated with a step-forcing experiment is able to represent gradual CO2-increase idealized scenarios because the analytic response exhibits a satisfactory fit for the scenario with a 1% yr-1 CO2 increase and stabilization when available. We found the clear separation of time scales highlighted by Held et al. (2010), since the fast relaxation time multimodel mean is about 4 yr while the slow time scale is about 2=0 yr. An analysis of the contribution of the two layers' heat uptake to the fast and the slow modes shows that the upper-ocean heat uptake contributes only to the fast mode that is shown to be quite small in the case of a linear forcing. In the case of a step forcing, both layers' heat uptakes contribute to the response amplitude and the upper-ocean heat uptake plays a key role in the representation of the first stages of the temperature and radiative flux responses. Thus, this contribution is important to estimate the amplitude of the forcing from a stepforcing experiment. Moreover, an accurate representation of the temperature response near equilibrium is necessary to estimate the equilibrium climate sensitivity. The twolayer EBM is the simplest tool that incorporates both of these features, and is therefore the simplest adequate model to simulate transient climate change under all kinds of idealized scenarios. However, a main limitation of the simple model used in this study is the intrinsic assumption of a linear dependence between the radiation imbalance at the TOA and the mean surface temperature perturbation. In Part II, the two-layer EBM with an efficacy factor of deep-ocean heat uptake proposed in Held et al. (2010) is used to overcome this problemand applied to CMIP= AOGCMs. Acknowledgments. We gratefully thank Jonathan Gregory for his careful and constructive review of the paper and two anonymous reviewers for their comments that helped to improve the manuscript.We thank Laurent Terray and Julien Boe for helpful discussions and valuable comments on the work. Thanks are also due to Isaac Held for sharing interesting ideas in his blog. This work was supported by the European Union FP7 Integrated Project COMBINE. We acknowledge the World Climate Research Programme's Working Group on Coupled Modelling, which is responsible for CMIP, and the U.S. Department of Energy's Program for Climate Model Diagnosis and Intercomparison, which provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals.We thank the climate modeling groups for producing and making available their model output. REFERENCES Andrews, T., J. M. Gregory, M. J. Webb, and K. E. Taylor, 2012: Forcing, feedbacks and climate sensitivity in CMIP= coupled atmosphere-ocean climate models. Geophys. Res. Lett., 39, L09712, doi:10.1029/2012GL0=1607. Budyko, M. I., 1969: The effect of solar radiation variations on the climate of the Earth. Tellus, =, 611-619. Dickinson, R., 1981: Convergence rate and stability of ocean- atmosphere coupling schemes with a zero-dimensional climate model. J. Atmos. Sci., 38, -112-120. Dufresne, J.-L., and S. Bony, 2008: An assessment of the primary sources of spread of global warming estimates from coupled atmosphere-ocean models. J. Atmos. Sci., -1, =13=-=144. Friend, A. D., 2011: Response of Earth's surface temperature to radiative forcing over A.D. 1-2009. J. Geophys. Res., 116, D13112, doi:10.1029/2010JD01=143. Geoffroy, O., D. Saint-Martin, G. Bellon, A. Voldoire, D. J. L. Olivie, and S. Tyteca, 2013: Transient climate response in a two-layer energy-balance model. Part II: Representation of the efficacy of deep-ocean heat uptake and validation for CMIP= AOGCMs. J. Climate, 26, 18=9-1876. Good, P., J. M. Gregory, and J. A. Lowe, 2011: A step-response simple climate model to reconstruct and interpret AOGCM projections. Geophys. Res. Lett., 38, L01703, doi:10.1029/ 2010GL04=208. Gregory, J. M., 2000: Vertical heat transports in the ocean and their effect on time-dependent climate change. Climate Dyn., 16, =0=-=1=. _____, and J. F. B. Mitchell, 1997: The climate response to CO2 of the Hadley Centre coupled AOGCM with and without flux adjustment. Geophys. Res. Lett., 24, 1943-1946. _____and P. M. Forster, 2008: Transient climate response estimated from radiative forcing and observed temperature change. J. Geophys. Res., 113, D2310=, doi:10.1029/2008JD01040=. _____,and M. Webb, 2008: Tropospheric adjustment induces a cloud component in CO2 forcing. J. Climate, -1, =8-71. _____, and Coauthors, 2004: A new method for diagnosing radiative forcing and climate sensitivity. Geophys. Res. Lett., 31, L0320=, doi:10.1029/2003GL018747. Hansen, J., and Coauthors, 200=: Efficacy of climate forcings. J. Geophys. Res., 110, D18104, doi:10.1029/200=JD00=776. Hasselmann, K., R. Sausen, E. Maier-Reimer, and R. Voss, 1993: On the cold start problem in transient simulations with coupled atmosphere-ocean models. Climate Dyn., 9, =3-61. Held, I. M., M. Winton, K. Takahashi, T. Delworth, F. Zeng, and G. K. Vallis, 2010: Probing the fast and slow components of global warming by returning abruptly to preindustrial forcing. J. Climate, 23, 2418-2427. Houghton, J. T., G. J. Jenkins, and J. J. Ephraums, Eds., 1990: Climate Change: The IPCC Scientific Assessment. Cambridge University Press, 410 pp. Meinshausen, M., S. C. B. Raper, and T. M. L. Wigley, 2011: Emulating coupled atmosphere-ocean and carbon cycle models with a simpler model, MAGICC6-Part 1: Model description and calibration. Atmos. Chem. Phys., 11, 1417-14=6. Murphy, J. M., 199=: Transient response of the Hadley Centre coupled ocean-atmosphere model to increasing carbon dioxide. Part III: Analysis of global-mean response using simple models. J. Climate, 8, 496-=14. Olivie, D. J. L., G. P. Peters, and D. Saint-Martin, 2012: Atmosphere response time scales estimated from AOGCM experiments. J. Climate, 2=, 79=6-7972. Plattner, G. K., and Coauthors, 2008: Long-term climate commitments projected with climate-carbon cycle models. J. Climate, -1, 27-1-27=1. Raper, S. C. B., J. M. Gregory, and R. J. Stouffer, 2002: The role of climate sensitivity and ocean heat uptake on AOGCM transient temperature response. J. Climate, 1=, 124-130. Schwartz, S. E., 2008: Reply to comments by G. Foster et al., R. Knutti et al., and N. Scafetta on ''Heat capacity, time constant, and sensitivity of Earth's climate system.''J. Geophys. Res., 113, D1=10=, doi:10.1029/2008JD009872. Sellers, W. D., 1969: A global climatic model based on the energy balance of the earth-atmosphere system. J. Appl. Meteor., 8, 392-400. Soden, B. J., and I. M. Held, 2006: An assessment of climate feedbacks in coupled ocean-atmosphere models. J. Climate, 19, 33=4-3360. Taylor, K. E., R. J. Stouffer, and G. A. Meehl, 2011: An overview of CMIP= and the experiment design. Bull. Amer. Meteor. Soc., 93, 48=-498. van Hateren, J. H., 2013: A fractal climate response function can simulate global average temperature trends of the modern era and the past millennium. Climate Dyn., doi:10.1007/s00382- 012-137=-3, in press. Williams,K.D., W. J. Ingram, and J. M. Gregory, 2008: Time variation of effective climate sensitivity in GCMs. J. Climate, -1, =076-=090. Winton, M., K. Takahashi, and I. M. Held, 2010: Importance of ocean heat uptake efficacy to transient climate change. J. Climate, 23, 2333-2344. O. GEOFFROY AND D. SAINT-MARTIN Centre National de Recherches Meteorologiques, Groupe d'etudes de l'Atmosphere Meteorologique (CNRM-GAME), Toulouse, France D. J. L. OLIVIE Center for International Climate and Environmental Research-Oslo, and University of Oslo, Oslo, Norway A. VOLDOIRE, G. BELLON, AND S. TYTECA Centre National de Recherches Meteorologiques, Groupe d'etudes de l'Atmosphere Meteorologique (CNRM-GAME), Toulouse, France (Manuscript received×April 2012, in final form 28 August 2012) Corresponding author address: Olivier Geoffroy, Centre National de Recherches Meteorologiques (CNRM-GAME), 42 av. G. Coriolis, 310=7 Toulouse, France. E-mail: [email protected] (ProQuest: ... denotes formulae omitted.) (c) 2013 American Meteorological Society |