a. System of equations and analytical solution

In this part, we consider the following two-layer EBM with an efficacy factor for deep-ocean heat uptake ɛ proposed by Held et al. (2010):

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where C, C₀, and γ are the first-layer (atmosphere/land/upper ocean) heat capacity, the second-layer (deep ocean) heat capacity (both expressed per unit area), and the heat-exchange coefficient between the two layers, respectively. The term γ(T − T₀) is the heat flux H exchanged between the two layers and is equal to the deep-ocean heat uptake H = γ(T − T₀). Since the change in the heat content of the first layer C(dT/dt) is driven by the sum of the heat flux exchanged with the deep ocean −H and the heat flux exchanged with the external system N, the net radiative flux at TOA evolves as

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In the following, EBM-1 will refer to the standard energy-balance model analyzed in Part I and EBM-ɛ to the model described above. The presence of an additional radiative flux term, −(ɛ − 1)H, in the evolution of N constitutes the difference with the EBM-1.

By introducing and γ′ = ɛγ, the system can be written as follows:

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which is the same mathematical system as that of the EBM-1 except for the primes. As pointed out by Held et al. (2010), the effect of the deep-ocean efficacy factor is equivalent to modifying ocean properties such that the deep-ocean heat capacity and the heat-exchange coefficient between the two layers are scaled by a factor ɛ. Note that EBM-ɛ is physically different from the EBM-1 because it includes one additional process. As a result, all the physical parameters estimated on the basis of this model can be different from their counterparts estimated within the framework of EBM-1. The derivation of the analytical solution of EBM-ɛ is straightforward. All the formulations of the mode parameters given in Part I are still valid by replacing C₀ and γ with and γ′, respectively. These parameters are noted with the primes in the following. The mean surface temperature response for a step forcing is

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and for a linear forcing with an increase rate k is

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where , , , and are the fast and slow mode parameters defined in Part I and expressed as functions of λ, C, , and γ′.

b. EBM-ɛ underlying hypothesis

1) Global budget

In this section, the hypotheses underlying the introduction of an efficacy factor ɛ are presented. Within the framework of a two-layer simple climate model, the change in the heat content of the climate system is the sum of the atmosphere/land/upper-ocean instantaneous heat uptake C(dT/dt) and the deep-ocean instantaneous heat uptake C₀(dT₀/dt). This change is equal to the net radiative imbalance at the top of the climate system N,

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The net radiative imbalance N is the sum of the radiative forcing and of the radiative response R_T of the climate system induced by the surface air temperature change

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Thus, the radiative response R_T can be decomposed as the sum of three radiative responses, an equilibrium radiative response R_eq = − and the radiative responses R_U and R_D equal to the instantaneous rate of heat storage in the upper and the deep oceans, respectively. Similarly to the forcing that constitutes a source of heat (in the case of a positive radiative forcing) for the first layer, the ocean heat uptake constitutes a sink of heat and is associated with a surface temperature change. The equilibrium radiative response can be expressed as a linear function of the equilibrium temperature response. Analogously, we introduce the temperature changes T_U and T_D associated with the upper-ocean and the deep-ocean heat uptake, respectively, such that R_U and R_D are linear functions of T_U with a feedback parameter λ and of T_D with a feedback parameter λ_D, respectively. This leads to the following system of equations:

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The deep-ocean heat-uptake temperature is associated with a different feedback parameter λ_D because the spatial pattern of T_D differs from the equilibrium temperature response pattern. This point is discussed in more detail in the next section.

By assuming additivity of the temperature response patterns (Forster et al. 2000; Boer and Yu 2003a), the surface air temperature response is the sum of the three balance responses,

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The sum T_U + T_D is the disequilibrium temperature difference associated with the deep-ocean heat uptake T_H = T − T_eq (Winton et al. 2010; Part I). The sum of the radiative responses associated with T_U and T_D is equal to the radiative imbalance, R_U + R_D = N.

Summing Eqs. (10)–(12) gives the first layer energy budget,

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By using the decomposition of T [Eq. (13)], Eq. (14) leads to

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Following Hansen et al. (2005), Held et al. (2010), and Winton et al. (2010), an efficacy factor ɛ for deep-ocean heat uptake is introduced,

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By using this definition and T_D = −H/λ_D, Eq. (15) is equivalent to Eq. (1).

In the case of a gradual increase of the external perturbation, C(dT/dt) is small (see section 3c in Part I); in the limit of negligible C(dT/dt), N = H and Eq. (3) leads to the formulation of Winton et al. (2010) [see their Eq. (3)]

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with the equilibrium temperature response defined as T_eq = .

2) Local budget

To understand why the feedback strength may vary with the temperature pattern, it can be useful to examine the evolution of the local energy balance in transient regime. As pointed out by Boer and Yu (2003a), the change in heat content of a climate system column is equal to the local radiative imbalance and the local convergence of the horizontal energy

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where dhⁱ/dt and are the local change in the heat content of the first and the second layer, respectively; Tⁱ, , and λⁱ are the local temperature response, the local forcing, and the local feedback parameter; respectively; and and are the local convergence of the horizontal energy flux of the first and the second layer, respectively. The superscript i denotes local values. The average over the earth’s surface of dhⁱ/dt or is the change in the heat content of the first layer C(dT/dt) or of the second layer C₀(dT₀/dt), respectively. The global mean of the local forcing is . The global average of each local energy convergence and is 0. Note that the local heat flux from the upper ocean to the deep ocean Hⁱ satisfies

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Equation (18) can be viewed as the heat budget in response to the sum of three “forcings”: the external forcing , and two sink terms, −dhⁱ/dt and , considered “internal forcings.” By assuming additivity of the temperature response patterns, the local surface temperature response is expressed as the sum of the three balance responses, , and the local budget (18) can be decomposed in the following system of equations:

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where and are the local temperature perturbations associated with the upper-ocean and the deep-ocean heat uptake. Here, , , and , are the associated convergences of horizontal energy fluxes in the first layer, with . Similarly, , , and are the convergences of horizontal energy fluxes in the second layer. Note that the global average of each energy-flux convergence and is zero. Assuming that is 0 leads to . Also, the decomposition of T and A_t in sums, Eqs. (18) and (20)–(22), leaves 1 degree of freedom in the definition of , , , and .

Introducing the normalized equilibrium temperature amplitude function , the local heat budget at equilibrium is

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One can note that the equilibrium temperature pattern (i.e., ) depends on the local forcing, the local feedback, and the amplitude of the local energy convergence. Thus, the total feedback parameter λ is the average of the local feedback parameter weighted by the equilibrium temperature pattern,

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where ds is the surface area element and S is the world surface area. The parameter λ will be referred as the equilibrium feedback parameter in the following.

By assuming the separability of time and space variables for and , they can be decomposed into the product of a time-varying global average T_x by a spatial pattern . The initial conditions in the case of a step forcing impose T_U(0) = −T_eq(0) and . As a result T_U and T_eq have the same pattern, . Note that the initial conditions of a step-forcing case [T_U(0) = −T_eq(0) and R_U(0) = −R_eq(0)] also directly impose that the feedback parameter associated with T_U in Eq. (11) is the equilibrium feedback parameter. Because the pattern of the deep-ocean heat uptake is different from the pattern of the radiative forcing, is assumed to be associated with a pattern, . Averaging Eq. (22) over the earth’s surface leads to Eq. (12) with the following formulation of λ_D:

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The weight coefficient is different from the one in the equilibrium feedback parameter expression. If the strength of the local feedbacks is large (which corresponds to low values of λⁱ) in regions where the ocean heat uptake induces a small temperature increasing rate (resulting in high values of ), then λ_D is lower than λ. Consequently, for a given amplitude of T_U and T_D, R_D is smaller (i.e., the climate system accumulates less heat). Note that the assumption of separability of and in space and time is not necessary. However, in the case when is time dependent, another assumption is required to obtain a constant λ_D in Eq. (25).

To conclude section 2b, the introduction of an efficacy factor for the deep-ocean heat uptake ɛ = λ/λ_D is the result of a decomposition of the temperature pattern as the sum of the temperature response patterns to the radiative forcing, the upper-ocean and the deep-ocean heat uptakes assuming a linear relationship between these forcings and their associated temperature responses. Because the spatial pattern of the temperature response to the deep-ocean heat uptake differs from the equilibrium pattern, the spatial heterogeneity of the radiative feedbacks strength implies that the magnitude of the global radiative feedback varies in time during a climate transition.

c. Effect of efficacy factor of deep-ocean heat uptake

In case of a step forcing, the analytical solutions for the upper-ocean and deep-ocean heat-uptake temperatures are

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The expression, the order of magnitude, and the sign of the fractional contributions , , , , , and are given in Part I (by replacing C₀ and γ with and γ′ in the expressions, respectively).

The theoretical temporal evolutions of T, T_U, and T_D in the case of a step forcing are represented at the top of Fig. 1 for three values of efficacy factor: ɛ < 1, ɛ = 1, and ɛ > 1, with other parameters unchanged. The upper-ocean heat-uptake temperature T_U increases with the characteristic time scale , and after a few years it tends to zero since the contribution of the slow mode to T_U is negligible, the upper-ocean reservoir is saturated. Concerning the deep-ocean heat-uptake temperature, the contributions of the slow and fast modes ( and ) are comparable but of opposite signs. The fast mode is predominant in the first few years and induces a decrease in T_D; that is, the heat flux exchanged between the two layers H increases because T increases faster than T₀. After this first phase (with a characteristic duration of ), the slow mode becomes dominant and T_D increases slowly back to zero because the deep ocean accumulates less and less heat.

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EBM-ɛ results. (top) Time evolution of global mean surface air temperature response (thick black line), upper-ocean heat-uptake temperature response T_U (thick gray line), and deep-ocean heat-uptake temperature response T_D (dot-dashed gray line) for a step-forcing case; the black dotted line shows the equilibrium temperature response T_eq. (middle) Global mean net radiative flux change (thick black line) and its decomposition into R_U = −λT_U (thick gray line) and R_D = −λ_DT_D (dot-dashed gray line) as functions of the global mean surface air temperature response. (bottom) Global mean net radiative flux change (thick black line; same as above) and its decomposition into the linear term − λT (dotted gray line) and the deviation term −(ɛ − 1)H (dashed gray line). Plots are for the three values of ɛ indicated on the panels and = 3.9 W m⁻², λ = 1.3 W m⁻² K⁻¹, C = 8 W yr m⁻² K⁻¹, C₀ = 100 W yr m⁻² K⁻¹, and γ = 0.7 W m⁻² K⁻¹.

Citation: Journal of Climate 26, 6; 10.1175/JCLI-D-12-00196.1

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The middle panels of Fig. 1 represent the theoretical relationship between the radiative imbalance N and the mean surface temperature perturbation T during the transient regime, for the same values of ɛ. The intercept and the x axis intersection are independent from the value of ɛ. Per definition, the intercept at T = 0 is the amplitude of the forcing (Gregory et al. 2004). Similarly, the x axis intersection is the equilibrium temperature response (the equilibrium climate sensitivity in the case of a 2×CO₂ perturbation per definition). Only the path to join these two points is altered when ɛ is modified.

With ɛ = 1, the net radiative flux varies linearly with the temperature. For ɛ ≠ 1, the plots suggest that there are two distinct stages in the (N, T) response to an abrupt forcing. To understand this behavior, it is convenient to decompose the net flux into the sum of its two components contribution R_U and R_D. In Fig. 1 (middle), the evolutions of (R_U, T) and (R_D, T) are plotted respectively with gray solid lines and gray dash-dotted lines.

During the first period, corresponding to the fast mode response time scale, the two components (upper and deep oceans) contribute with similar amplitude but with opposite trends to the temperature response and N varies roughly linearly with T. Indeed, neglecting the slow response term during this period, the time evolutions of R_U and R_D are proportional to that of T_H (and T); the scale factors are and , with and , respectively. Accordingly, the radiative imbalance N as the sum of these two contributions evolves roughly linearly with T.

During the second period, the contribution of the upper ocean is negligible and the net radiative flux is simply the contribution of the deep-ocean heat-uptake temperature, −λ_DT_D. Then, since T_D ≈ T − T_eq, the radiative flux varies also roughly linearly with T. The sharp change in the trend of the (N, T) line corresponds to a time similar to the fast relaxation time. This analysis suggests that linear fits of the two asymptotes of the (N, T) curve performed separately as in Gregory et al. (2004) give a good approximation of the radiative forcing (as the intercept of the first fit), the equilibrium temperature T_eq (as the x-axis intersection of the second fit), and λ_D = λ/ɛ (as the slope of the second fit).

The net radiative flux at the top of the atmosphere can also be decomposed as the sum of prognostic variables and physical parameters of the EBM-ɛ as shown in Eq. (3). The radiative imbalance N is the sum of a linear term − λT and a fraction 1 − ɛ of the instantaneous rate of heat storage in the deep ocean H. Their evolution in the (N, T) space is illustrated in Fig. 1 (bottom). The linear term takes into account the fact that the surface temperature is not in equilibrium, which induces a radiative imbalance. The second term is a deviation from this linear radiative flux due to the nonlinear evolution of the temperature pattern. The magnitude of H reflects the magnitude of this deviation.

Initially, H = 0, T = 0, and the radiative imbalance is equal to the forcing. In equilibrium, as H is zero, the assumption of linear dependence between the radiative imbalance and the surface temperature remains valid. But during the transient regime, the net radiative flux is affected by the deep-ocean heat uptake. The parameter usually referred to as the effective feedback parameter λ_eff = ( − N)/T varies in time (if ɛ ≠ 1) and needs to be distinguished from the equilibrium feedback parameter λ. It is equal to the transient radiative feedback factor λ_t, such that

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The efficacy factor can be determined from gradual perturbation AOGCMs simulations [by neglecting C(dt/dt)] but requires prior knowledge of the equilibrium climate sensitivity and feedback parameter (Winton et al. 2010). On the other hand, all the EBM-ɛ radiative and thermal inertia parameters can be consistently computed from only a step-forcing AOGCM experiment (and a control simulation), by taking into account the time evolution of the transient radiative feedback factor. In the next section, the method used to calibrate the EBM-ɛ physical parameters to a given AOGCM is briefly described.

d. Method for EBM-ɛ parameter calibration

In comparison with the EBM-1, the EBM-ɛ has an additional radiative parameter ɛ that needs to be tuned consistently with the reference radiative forcing amplitude (e.g., for a 2×CO₂ perturbation) and the equilibrium feedback parameter λ from the N–T evolution. The physical parameters of the EBM-ɛ are computed iteratively using a step-forcing experiment. The parameters are initially set to the EBM-1 values (ɛ = 1, and parameters computed in Part I). For each iteration i, the deep-ocean heat uptake H⁽ⁱ⁻¹⁾ is first evaluated using the analytical solutions and the thermal parameters computed at iteration (i − 1). Then, using Eq. (3), a multilinear regression of N (AOGCM values) against the AOGCM surface temperature response T and H⁽ⁱ⁻¹⁾ provides the values of , λ⁽ⁱ⁾, and ɛ⁽ⁱ⁾,

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Finally the thermal inertia parameters C⁽ⁱ⁾, and γ′⁽ⁱ⁾ are tuned by performing two fits of the surface temperature response following the methodology used for the EBM-1 calibration (see details in section 3 of Part I). Only a few iterations are found to be sufficient to obtain convergence. This method for estimating the equilibrium climate sensitivity, radiative parameters, and thermal inertia parameters from a short-term step-forcing simulation will be referred to in the following as the EBM-ɛ method. In the next section, the EBM-ɛ method is applied to 16 CMIP5 AOGCMs using the abrupt 4×CO₂ experiments and results are compared with the EBM-1 estimates [which, for the radiative properties, correspond to the estimates from Gregory et al. (2004)’s method].

a. Radiative parameters and TOA net flux: Comparison with the EBM-1

For the same 16 AOGCMs of the CMIP5 database analyzed in Part I (see Table 4 in Part I for model expansions), the EBM-ɛ method is applied and radiative parameter values are reported in Table 1. The values of the deep-ocean heat-uptake efficacy factor are mostly greater than 1 (see also Fig. 3a). Excluding the BNU-ESM model that has a value of ɛ very close to 1, only two models (INM-CM4 and CNRM-CM5.1) have values of ɛ smaller than unity. The heat-uptake efficacy factor ranges from 0.83 to 1.82 with a multimodel mean value of 1.28 and an intermodel standard deviation of 0.25. These results are in very good agreement with the estimates of Winton et al. (2010) for some CMIP2 and CMIP3 model’s analysis despite methodological differences. Winton et al. (2010) derived the value of ɛ from 1% yr⁻¹ CO₂ increase experiments using equilibrium climate sensitivity mainly derived from AGCMs coupled with a mixed-layer ocean model and using forcing estimates taken from Solomon et al. (2007). The latter were computed from different sources and they took into account either only the stratospheric adjustment or both stratospheric and tropospheric adjustments [through the method of Gregory et al. (2004)], depending on cases. In this study, the efficacy factor ɛ, the radiative forcing, and the equilibrium climate sensitivity are derived jointly in the single framework of the EBM-ɛ.

Table 1.

The 4×CO₂ radiative forcing , total feedback parameter λ, efficacy factor for deep-ocean heat uptake ɛ, and 4×CO₂ equilibrium temperature estimates in the framework of the EBM-ɛ of the 16 CMIP5 models used in this paper, and their multimodel mean and standard deviation.

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Figure 2 compares for each model the N–T plot for AOGCM results, EBM-ɛ fit, and the linear regression of Gregory et al. (2004). For models with an efficacy factor near 1 (BNU-ESM, CNRM-CM5.1, and IPSL-CM5A-LR), the assumption of linearity between N and T is valid and the results from EBM-ɛ are close to that of the linear model. For models with a large ɛ (CSIRO-Mk3.6.0, GISS-E2-R, HadGEM2-ES, MPI-ESM-LR, and NorESM1-M), the results from EBM-ɛ largely improve the fit of radiative imbalance versus temperature response compared to a linear fit. In particular, the EBM-ɛ is able to reproduce the two-stage behavior of these models in the parameter space (N, T).

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Global annual mean net radiative flux change at TOA N as a function of global mean surface air temperature response T for the abrupt 4×CO₂ experiments (black dots; large dots for the first 15 yr), for the 16 AOGCMs. The thick black line is the EBM-ɛ fit. The dotted and dashed black lines show the linear contribution − λT and the deviation contribution −(ɛ − 1)H, respectively. The gray line is the linear fit of Gregory et al. (2004). The thin black line shows N = 0. Note that the range of T can differ from one panel to another.

Citation: Journal of Climate 26, 6; 10.1175/JCLI-D-12-00196.1

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Figures 3b–d compare the values of the 4×CO₂ radiative forcing , λ, and the 4×CO₂ equilibrium temperature response obtained within the framework of the EBM-ɛ and those derived with the method described in Gregory et al. (2004). The three AOGCMs with ɛ larger than 1.5 are indicated with filled red square, circle, and triangle symbols. For these models, the radiative forcing amplitude and the equilibrium climate sensitivity are larger than in the standard linear model estimate. Indeed, for CSIRO-Mk3.6.0 and HadGEM2-ES, the equilibrium temperature response for a 4×CO₂ perturbation is up to 2 K warmer than the value derived from the linear assumption. The multimodel mean is 0.5 K warmer. The radiative forcing is 1–2 W m⁻² larger for large ɛ models and the multimodel mean is 0.7 W m⁻² larger. Most models have a forcing lower than 8.5 W m⁻² except CCSM4, GISS-E2-R, MPI-ESM-LR, and MIROC5. The three latter have a forcing on the order of 9 W m⁻², which suggests a strong effect of the tropospheric adjustment. The change in the ECS is mainly due to a change in the forcing with the radiative feedback parameters being less impacted (except for the GISS-E2-R model). Moreover, unlike the forcing and the equilibrium temperature, the sign of the difference in λ between the EBM-1 and the EBM-ɛ estimates is independent of the sign of ɛ-1. For example, for HadGEM2-ES and CSIRO-Mk3.6.0, λ is larger and smaller in the EBM-ɛ framework than in the EBM-1 framework, respectively, whereas both have an ɛ value greater than 1. The multimodel radiative forcing and radiative feedback parameter standard deviations are roughly unchanged whereas that of the equilibrium temperature increases from 1.6 to 2.1 K. The improved match of the temperature response and radiative imbalance evolution between the AOGCMs and the simple EBM suggests that the values estimated from the EBM-ɛ method are more accurate. However, a complete assessment of the EBM-ɛ would require to extend AOGCM experiments until equilibrium (i.e., over a period of 1000–1500 years). Indeed, the strength of the feedbacks can saturate, which would impact the equilibrium temperature response (Li et al. 2013).

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Parameter estimates of (a) ɛ values for the 16 AOGCMs and (b) EBM-ɛ estimates as a function of EBM-1 estimates for , (c) λ, (d) , (e) τ_f, (f) τ_s, (g) γ, (h) C₀, and (i) C. Superscripts 1 and ɛ denote estimates from the EBM-1 and the EBM-ɛ methods, respectively. The dotted line shows ɛ = 1 in (a) and the solid lines indicate a perfect match between EBM-ɛ and EBM-1 estimates in (b)–(i). Each symbol corresponds to one AOGCM.

Citation: Journal of Climate 26, 6; 10.1175/JCLI-D-12-00196.1

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b. Thermal inertia parameters and temperature: Comparison with the EBM-1

The thermal inertia physical parameters and the relaxation times are given in Table 2 and represented as a function of their EBM-1 counterparts in Figs. 3e–i. The fast relaxation time scale τ_f is not impacted by the inclusion of the efficacy of deep-ocean heat uptake, whereas the slow relaxation time scale τ_s is. The change in τ_s is mainly due to change in the heat-exchange coefficient γ rather than in the deep-ocean heat capacity C₀. Models with ɛ > 1 have a lower γ than in the EBM-1 framework. The inclusion of the effect represented by the deviation term (1 − ɛ)H in the temperature response amounts to modifying the deep-ocean heat uptake such that the heat-exchange coefficient is ɛγ. The lack of efficacy factor in the EBM-1 is compensated by a large γ for models with ɛ > 1 in the EBM-ɛ framework.

Table 2.

The atmosphere/land/upper-ocean heat capacity C, deep-ocean heat capacity C₀, heat-exchange coefficient γ, and fast and slow relaxation times estimates in the framework of the EBM-ɛ of the 16 CMIP5 models used in this paper, and their multimodel mean and standard deviation.

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The EBM-1 also underestimates the upper-ocean heat capacity C. The estimate of C depends on the forcing estimation since it is evaluated through an estimation of the temperature tendency at t = 0 that is equal to /C. Consequently, an underestimation of leads to an underestimation of C. These results suggest that the lack of radiative effect associated with deep-ocean heat uptake introduces a bias in the EBM-1 estimates of the thermal inertia parameters. The standard deviations of γ, C₀, and C are reduced with the EBM-ɛ from 0.18 to 0.15 W m⁻² K⁻¹, 62 to 52 W yr m⁻² K⁻¹ (but slightly increased from 27 to 29 W yr m⁻² K⁻¹ if INM-CM4 is excluded), and 1.1 to 0.9 W yr m⁻² K⁻¹, respectively. This shows that introducing a new degree of freedom reduces slightly the intermodel spread.

Figure 4 shows the temperature response of the three AOGCMs with the largest ɛ estimates (CSIRO-Mk3.6.0, NorESM1-M, and HadGEM2-ES) for the abrupt 4×CO₂ and the 1% yr⁻¹ CO₂ experiments, as well as the EBM-1 and the EBM-ɛ analytical solutions using the parameters estimated by the corresponding method on the basis of the abrupt 4×CO₂ experiment. The temperature responses are identical for both EBMs in both the abrupt 4×CO₂ and the 1% yr⁻¹ CO₂ simulations over the first 150 years, and they match the AOGCM responses. But, for the step-forcing scenario, the EBM-ɛ response diverges from the EBM-1 response after about 300 years. Only the second phase of the temperature evolution, the one driven by the slow component of the system, is modified by the introduction of an efficacy factor. This is consistent with the fact that only the slow relaxation time scale varies between the EBM-1 and the EBM-ɛ methods. The EBM-1 calibrated with the abrupt simulation is accurate enough to represent the temperature evolution over the centennial scale. However, compared to the EBM-ɛ estimates, the EBM-1 parameters are biased as a result of a bias in radiative parameters estimated following the method of Gregory et al. (2004).

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Temperature response of AOGCMs with the highest ɛ value for the abrupt 4×CO₂ (red lines) and 1% yr⁻¹ CO₂ (blue lines) experiments and corresponding fit for EBM-1 (dashed green) and EBM-ɛ (solid black). Note that the EBM-1 and the EBM-ɛ solutions are superposed for the 1% yr⁻¹ CO₂ and the beginning of the abrupt 4×CO₂ experiments. The dotted lines denote the equilibrium temperature for EBM-1 (orange) and EBM-ɛ (red).

Citation: Journal of Climate 26, 6; 10.1175/JCLI-D-12-00196.1

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c. Parameter dependency

In this section, the question of potential relationships between the EBM-ɛ parameters is investigated. Table 3 shows the multimodel correlations between parameters of the EBM-ɛ, and also between these parameters and the equilibrium temperature response. For the set of 16 models, a correlation coefficient higher than 0.50 is significant at the 95% confidence level (two-tailed test). Note that the statistical test assumes that the AOGCMs are independent. As expected, the anticorrelation between T_eq and λ is high, with a correlation coefficient of −0.84. No correlation is found between and λ, suggesting that the effect of fast tropospheric adjustment is independent of the surface temperature feedback. Consistently, the equilibrium temperature is independent of the adjusted forcing magnitude.

Table 3.

Multimodel correlations between the equilibrium temperature at 4×CO₂ and the physical parameters F, λ, ɛ, γ, C₀, and C of the EBM-ɛ for the 16 CMIP5 AOGCMs. Boldface correlations are significant at the 95% confidence level.

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Raper et al. (2002) suggested a negative correlation between their heat-exchange coefficient κ of the one-layer model (that is similar to the parameter γ) and the radiative feedback parameter λ but the analysis of CMIP3 models by Gregory and Forster (2008) and Plattner et al. (2008) did not find such a correlation. Including an interactive deep ocean changes the formulation of deep-ocean heat uptake and impacts the relationship between the heat-exchange coefficient (κ or γ) and the radiative feedback parameter λ. Indeed, the EBM-ɛ estimates of λ and γ are positively correlated, contrary to the results of Raper et al. (2002), with a correlation coefficient of 0.64. This value is above the significant level. However, excluding the GISS-E2-R model, which is largely outside the range of the model ensemble, the correlation is not significant with a value of 0.43, showing the limited robustness of this correlation. The remaining interparameter correlations are found to not be significant.

d. Decomposition in longwave and shortwave contributions

In this section, the net TOA radiative flux is decomposed into longwave (LW) and shortwave (SW) components N^LW and N^SW, respectively. The net radiative imbalance N is the right-hand side of Eq. (15). It can be expressed as follows [from Eqs. (12) and (16)]:

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We introduce LW and SW radiative feedback parameters associated with the deep-ocean heat-uptake temperature, and , and we assume the decomposition in an upper-ocean and a deep-ocean radiative contribution is valid for each component separately. These assumptions yield the following equations:

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where ^LW, ^SW, λ^LW, and λ^SW are the LW and SW components of the radiative forcing and of the radiative feedback parameter, respectively. Unlike in the case of the total feedback, we do not define an SW or LW efficacy factor ɛ_SW or ɛ_LW. Indeed, although the total feedback is necessarily different from zero, it is possible that λ^SW is zero while (λ^SW) is not. In such a case, a shortwave efficacy factor ɛ_SW would make no sense.

Each LW and SW component is calculated by multilinear regression of the corresponding net radiation flux as a function of temperature (both from the AOGCM abrupt 4×CO₂ experiment) and (from the EBM-ɛ estimation). Values of the tuned LW and SW radiative parameters are reported in Table 4 and resulting fits for each model separately shown in Fig. 5. These figures reflect the large intermodel spread in both forcing and radiative feedback parameters LW and SW components.

Table 4.

The LW and SW components of the radiative forcing, ^LW and ^SW, of the total feedback parameter, λ^LW and λ^SW, and of the deep-ocean heat-uptake feedback parameter, and , estimates in the framework of the EBM-ɛ of the 16 CMIP5 models used in this paper, and their multimodel mean and standard deviation.

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Global annual mean net LW (gray) and SW (black) radiative flux change at TOA as a function of global mean surface air temperature response T for the abrupt 4×CO₂ experiments (black dots; large dots for the first 15 years), for the 16 AOGCMs. The thick gray and black lines are the EBM-ɛ fits of the LW and the SW radiative flux, respectively. The dashed gray lines and the dot-dashed black lines show the LW and SW components of the −(ɛ − 1)H term, respectively. The thin black line shows N = 0. Note that the range of T can differ from one panel to another.

Citation: Journal of Climate 26, 6; 10.1175/JCLI-D-12-00196.1

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AOGCMs that have a large SW forcing contribution can have a large LW contribution (MPI-ESM-LR) or a small LW contribution (IPSL-CM5A-LR). The 4×CO₂ LW forcing ranges from 3.4 to 8.7 W m⁻² with an ensemble mean of 6.5 W m⁻² and a standard deviation of 1.2 W m⁻². Except for three models (CSIRO-Mk3.6.0, FGOALS-s2, INM-CM4), the 4×CO₂ SW forcing is mostly positive with a mean value of 1.2 W m⁻². Its standard deviation (1.1 W m⁻²) is on the same order as that of the LW contribution. By comparison with estimates taking into account the stratospheric adjustment only, the forcing is found to be lower in the LW and larger in the SW. Indeed, Forster and Taylor (2006) found a forcing estimate of 3.45 W m⁻² in the LW for a 2×CO₂ experiment (corresponding to 6.90 W m⁻² for a 4×CO₂ experiment). The instantaneous SW forcing is on the order of −0.06 W m⁻² (Myhre et al. 1998). These estimates confirm Gregory and Webb (2008) and suggest a nonnegligible effect of the fast change in the cloud component (among the other feedbacks) on the radiative forcing adjustment.

The LW contribution λ^LW to the feedback parameter is positive (i.e., negative feedback) for all models because the radiative imbalance is restored by increased LW emission associated with the temperature increase. The SW contribution to the feedback parameter λ^SW is negative (i.e., positive feedback) for all models except GFDL-ESM2M, which has a negligible λ^SW, and GISS-E2-R. For most AOGCMs, λ^SW is above (in absolute value) the 0.2–0.4 W m⁻² K⁻¹ typical range of the albedo feedback, suggesting a positive feedback of clouds in the SW.

The deep-ocean heat-uptake feedback parameter is generally of the same order of magnitude as λ^LW but is smaller than λ^SW. This suggests that the value of ɛ > 1 obtained for the majority of the models is mainly due to the shortwave radiation, with low clouds as a good candidate to explain most of the difference between EBM-ɛ and EBM-1. Further analysis is necessary to understand which components of the climate system are responsible for the differences and to quantify each contribution. But the results of such a simple SW/LW decomposition suggest that the EBM-ɛ framework can be used to decompose the radiative fluxes such as a cloud/clear-sky decomposition or more complex decompositions such as partial radiative fluxes.