a. Radiative kernel regression

The radiative kernel technique (Soden and Held 2006; Soden et al. 2008) was developed to serve as a computationally efficient method for analyzing radiative changes across multiple models. Radiative kernels K_x are the direct radiative response to a small perturbation of a radiatively relevant state variable x (e.g., temperature, water vapor, surface albedo). They are computed with an offline version of a model’s radiative transfer code, using model data from a simulation as input. To produce K_x, radiative fluxes are calculated at a high frequency [e.g., 3 hourly by Soden et al. (2008)] with a small perturbation in x (at one vertical level for 3-day variables), while all other surface and atmospheric variables required to produce the fluxes remain unperturbed. The calculations are carried out again with no perturbations as a control. The radiative kernel is defined as the difference between the perturbed and control radiative fluxes at a level, the TOA or surface, herein referred to as TOA radiative kernels (TOA K_x) or surface radiative kernels (surface K_x), respectively. Surface radiative kernels will be a primary tool used in this study.

Because some cloud-induced radiative changes are nonlinear, cloud radiative kernels cannot be quantified in the same manner as temperature, moisture, and surface albedo kernels [Zelinka et al. (2012) offer an alternative cloud radiative kernel method, however]. Instead, A_C and λ_C are estimated from changes in cloud radiative effect (CRE) in model output, and adjusted for cloud masking (Soden et al. 2008). We define λ and A as the sum of all individual radiative responses λ_x and radiative adjustments A_x, respectively.

b. Model simulations and radiative kernels

Surface radiative forcing and responses are evaluated in 18 CMIP5 GCMs, using monthly mean data from simulations where CO₂ is abruptly quadrupled from preindustrial concentrations and then held constant (abrupt4xCO₂). Climatic responses in these simulations are estimated relative to a control simulation where a preindustrial forcing scenario is imposed (piControl). This allows for a clear separation of the fast (i.e., subannual) radiative forcing associated with the CO₂ perturbation and the slow (i.e., years to decades) radiative responses to subsequent surface temperature increase. All models used in this study are listed in Table 1. To maintain equal weighting of each model, only the first realization is used, even for models where multiple ensemble members were run.

Table 1.

CMIP5 models used in this study. Archived data from abrupt4xCO2 and piControl experiments was used.

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Following the methodology described in section 2a, ISRF, A_x, and λ_x at the surface are evaluated using surface radiative kernels generated from version 1.1 of the NCAR Community Earth Systems Model (CESM1.1), configured with version 5.0 of the Community Atmosphere Model (CAM5) (Pendergrass et al. 2018). These surface radiative kernels were developed following the methodology outlined by Soden et al. (2008). Specifically, they are the differential radiative response to a 1-K warming at the surface and at each atmospheric level (temperature radiative kernel K_T), an increase in atmospheric specific humidity anticipated from 1-K warming with constant relative humidity (water vapor radiative kernel K_q), and a 1% increase in surface albedo (albedo radiative kernel K_a). Table 2 outlines key notation used throughout the study to describe the radiative kernels and associated radiative calculations.

Table 2.

A description of key notation used to describe radiative kernels and associated calculations throughout the text. When variables outlined below are used in the context of clear-sky conditions they are denoted with a superscript “clr” in the text.

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Figure 1a shows the atmospheric component of the total-sky zonal-mean, annual-mean surface K_T. For clarity, only the response to perturbations in the lowest one-tenth of the atmosphere is shown, since, above this point, the zonal-mean magnitude is negligible. Importantly, surface K_T has a large vertical gradient in these lowest layers of the atmosphere. The atmospheric values are positive, indicating that an increase in atmospheric temperature increases net radiation into the surface. In contrast, the surface component of surface K_T (which we denote as K_Ts) is negative (Fig. 1b), indicating that uniform warming of the surface decreases net radiation into the surface (increases net outgoing from the surface). Because of the large contribution from the surface component, the vertically integrated, global, annual-mean surface K_T is negative.

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Zonal, annual-mean surface radiative kernels, including: (a) total-sky temperature surface radiative kernel [surface K_T; W m² K⁻¹ (100 hPa)⁻¹], (b) the surface-temperature component of surface K_T (surface K_Ts; W m² K⁻¹) for total-sky (solid line) and clear-sky (dashed line) conditions, and total-sky (c) longwave (surface K_q,LW) and (d) shortwave (surface K_q,SW) water vapor radiative kernels [W m² K⁻¹ (100 hPa)⁻¹]. For clearer visualization of low-level features, the atmospheric column is only shown below σ = 0.9 for surface K_T and only below σ = 0.5 for surface K_q,LW and K_q,SW.

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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In principle, the surface K_Ts response should stem entirely from a change in upwelling LW radiation, approximately equal to the first derivative of the Stephan–Boltzmann equation. Assuming realistic climatological surface temperatures and surface emissivity, global-mean surface K_Ts should be about −5.5 W m⁻² K⁻¹. Since atmospheric temperature is not perturbed in this calculation, there should be no change in surface downwelling LW flux. Contrary to these expectations, the global-mean surface K_Ts from this set of kernels is only −3.78 W m⁻² K⁻¹, including an increase in downwelling LW flux of 1.7 W m⁻² K⁻¹ at the surface (which accounts for the reduced magnitude of surface K_Ts relative to Stephan–Boltzmann scaling). The downwelling LW flux is slightly smaller for clear-sky conditions, so total-sky surface K_Ts is 0.03 W m⁻² K⁻¹ smaller than the clear-sky counterpart. The clear-sky and total-sky surface temperature kernels would be identical were it not for this unexpected change in LW downwelling. This feature is an artifact of the methodology used to calculate the radiative fluxes in some radiative transfer models (RTM). When computing the radiative fluxes, the contributions from each model-layer midpoint are determined using average temperatures between adjacent level interfaces. In the course of this conversion to the RTM vertical grid, a prescribed temperature increase at the surface also affects the lowest-layer air temperature, inducing a change in the downwelling radiation at the surface. When the lowest atmospheric layer is perturbed 1 K to calculate surface K_T, a corresponding and compensating additional upwelling LW flux from the surface is induced. This downwelling LW feature also occurs in surface radiative kernels generated from some other RTMs (Previdi 2010; Previdi and Liepert 2012; Soden et al. 2008), but has not been previously documented.

Figures 1c and 1d show total-sky longwave (LW) and shortwave (SW) components of the water vapor surface radiative kernel (surface K_q,LW and K_q,SW, respectively), similar to Fig. 1a, but shown for the lowest half of the atmosphere. Surface K_q,LW is positive almost everywhere and reaches a maximum in the tropical boundary layer. The contribution from high latitudes is small because of lack of moisture. The SW water vapor kernel K_q,SW is negative, indicating that increased SW absorption in the atmosphere leads to decreased SW absorption at the surface.

Since high water vapor concentrations in the boundary layer absorb much of the downwelling radiation emitted from the middle and upper troposphere, the surface radiative kernels are dominated by the very lowest layers of the troposphere, where large vertical gradients in moisture are present. Consequently, estimates of surface radiative responses are quite sensitive to the magnitude of the radiative kernels at the lowest levels, and thus, how vertical interpolation is applied to the radiative kernels. Radiative kernels are typically interpolated to standard pressure levels to match model output, but this results in undefined values at grid points with a surface pressure less than the lowest standard pressure level (1000 hPa), missing part of the near-surface kernel response, which makes a substantial contribution to the radiative flux at the surface. To preclude this problem, in this study, all radiative kernels with a vertical dimension are used on the original CESM hybrid-sigma coordinates, and the accompanying CMIP5 model output used to compute forcings and radiative responses are interpolated to the same vertical coordinate. The sensitivity of surface radiative kernels to vertical interpolation is investigated further in the appendix.

c. Gregory regression

The total temperature–mediated surface radiative response λ is calculated as the sum of individual radiative responses derived from the radiative kernel-regression technique [Eqs. (2) and (4)]. It can also be obtained more directly from model output using linear regression following Gregory et al. (2004), where λ is defined as the slope of the linear regression of net surface radiative imbalance against global-mean ∆T_s.

In Fig. 2, λ is shown for each CMIP5 model, estimated using the radiative kernel-regression method λ_K and Gregory regression λ_G. The linear relationship and strong agreement (r = 0.86, RMSE = 0.17 W m⁻² K⁻¹) suggests that the radiative kernel-regression method produces a reasonable estimate of λ at the surface. Additionally, the difference between the clear-sky radiative responses (${\lambda }_K^{\text{clr}}$ and ${\lambda }_G^{\text{clr}}$) is <10% for 14 of the 18 models (not shown). In the sense that the Gregory regression method is the true measure of λ^clr, this percent error benchmark, termed the clear-sky linearity test, has been used in previous studies to assess the effectiveness of radiative kernels to quantify individual radiative responses in a given model (Vial et al. 2013; Jonko et al. 2012). We note, however, that this test evaluates not only the linearity assumption, but also the consistency between radiative transfer models. The fact that there are three noticeable outliers seems more likely to result from bias between the transmittance calculations of the RTMs, rather than nonlinearities in radiative perturbations that are unique to only those three models.

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Global-mean sum of temperature-mediated radiative responses estimated from the radiative kernel-regression method vs that estimated from the Gregory regression method. Numbered markers correspond to the CMIP5 models listed in Table 1.

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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a. Surface radiative forcing

If a perturbation of a forcing agent is prescribed identically across models, one may expect the resulting radiative forcing to also be identical. However, recent work has indicated that model differences in TOA ERF are an important source of uncertainty in climate projections (Forster et al. 2013; Vial et al. 2013), and that TOA IRF accounts for a substantial portion of that intermodel spread (Chung and Soden 2015a,b). Intermodel differences in ESRF and ISRF have not been previously quantified.

Offline double-call calculations of the instantaneous surface radiative forcing (ISRF_D) are not typically reported and are only available for a small subset of models. Instead, we use radiative kernels to estimate ISRF^clr as a residual calculation [Eq. (4)] and then apply a proportionality constant to retrieve total-sky ISRF in all models, following the methodology used by Chung and Soden (2015a,b) for the TOA. Error in the radiative kernel calculation of λ^clr, which is typically an order of magnitude larger than ISRF^clr (see discussion below), can contribute to error in this residual calculation. However, as noted above, λ^clr can alternatively be estimated using the Gregory regression method (${\lambda }_G^{\text{clr}}$) instead of the radiative kernel-regression method (${\lambda }_K^{\text{clr}}$), reducing the potential for radiative kernel error in the calculation of λ to be aliased into ISRF. Hereafter, we refer to ISRF estimated using ${\lambda }_K^{\text{clr}}$ or ${\lambda }_G^{\text{clr}}$ as ISRF_K or ISRF_G, respectively. Gregory regression cannot similarly be used to diagnose radiative adjustments in isolation, therefore the calculation of ISRF_K and ISRF_G both rely on kernel-derived radiative adjustments.

In Fig. 3, ISRF_D, ISRF_K, and ISRF_G are compared for the 6 CMIP5 models where double-call calculations are available. The difference between ISRF_K and ISRF_D is large (RMSE = 0.94 W m⁻²). For four of the models shown, the errors in ${\lambda }_K^{\text{clr}}$ relative to ${\lambda }_G^{\text{clr}}$ are among the five largest errors across all 18 models used in this study (Fig. 2). The agreement is better between ISRF_G and ISRF_D (RMSE = 0.35 W m⁻²), suggesting that errors in the residual calculation of ISRF mostly stem from the radiative kernel calculation of ${\lambda }_K^{\text{clr}}$. Furthermore, this indicates that error can be reduced by using ${\lambda }_G^{\text{clr}}$ (and thus ISRF_G), instead of ${\lambda }_K^{\text{clr}}$. The disagreement between ISRF_K and ISRF_G also implies that there are systematic differences in the transmittance calculations between the RTMs of the model and the kernel. Differences in the treatment of water vapor are one possible explanation, which has been noted previously for the shortwave radiative fluxes (Collins et al. 2006; DeAngelis et al. 2015). For CanESM2, global-mean ISRF_D (1.74 W m⁻²) differs substantially from both ISRF_K (1.1 W m⁻²) and ISRF_G (0.99 W m⁻²), while the latter two are much more similar. This is the sole outlying case, and suggests that ISRF_D is inconsistent with the simulated fluxes from the CanESM2 abrupt4xCO₂ simulation, potentially because of the details of this particular double-call calculation.

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Global-mean ISRF computed from double-call calculations (ISRF_D) vs that estimated using radiative kernels. Each numbered marker corresponds to a CMIP5 model listed in Table 1. For black markers, radiative kernels were used to estimate both the temperature-mediated radiative response and radiative adjustment (ISRF_K). For red markers, the Gregory regression method was used to estimate the temperature-mediated radiative response, while radiative kernels were used to estimate radiative adjustment (ISRF_G). Results are shown only for those models where data for all three methods are available.

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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Even though radiative kernels are also used to estimate A^clr, error in the diagnosis of ISRF_K relative to ISRF_D is mostly associated with λ^clr. This is because absolute λ^clr, expressed in watts per square meter as in Eqs. (3) and (4), is considerably larger than A^clr. Our explanation is as follows: radiative adjustments occur before any surface warming response, so associated temperature and water vapor changes in the boundary layer are minimal. Since the surface radiative imbalance is insensitive to perturbations above the boundary layer (Fig. 1), A^clr is relatively small. Figure 4 shows the global, ensemble mean of the clear-sky radiative responses (${\lambda }_x^{\text{clr}}$), radiative adjustments ($A_x^{\text{clr}}$), and instantaneous surface radiative forcing (${\text{ISRF}}_K^{\text{clr}}$). The radiative responses are substantially larger than the corresponding adjustments. Since ${\lambda }_x^{\text{clr}}$ is large, there can be a large absolute difference between surface ${\lambda }_K^{\text{clr}}$ and ${\lambda }_G^{\text{clr}}$.

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Global ensemble mean of temperature-mediated (left) surface radiative responses, (center) surface radiative adjustments, and (right) instantaneous surface radiative forcing for clear-sky conditions.

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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ISRF_G is also more accurate than ISRF_K spatially, evaluated against ISRF_D. While ensemble-mean ISRF_G is near zero or positive everywhere, in agreement with ISRF_D, ISRF_K is large and negative in the Arctic (Fig. 5), with an area-weighted, ensemble mean of −4.75 (σ = 2.40) W m⁻² across all models and −3.63 (σ = 2.33) W m⁻² among the 6 models where estimates of ISRF_D are possible. All but one model exhibits a negative Arctic mean IRF_K. In this region, ensemble-mean ${\lambda }_K^{\text{clr}}$ is greater than both A^clr and the sum of all surface radiative changes (not shown), mostly because of the clear-sky surface albedo radiative response ${\lambda }_a^{\text{clr}}$, which is the largest radiative response north of 65°N by a factor of ~3. Similarly, Chung and Soden (2015a) estimated TOA IRF_K to be negative in the Arctic. This discrepancy is due to the inability of radiative kernels to capture nonlinearity in ${\lambda }_a^{\text{clr}}$. As the amount of sea ice decreases, ${\lambda }_a^{\text{clr}}$ weakens (Colman and McAvaney 2009). Previous work shows that TOA K_a derived from a base climate at 1xCO₂ is larger than one derived from a climate with elevated CO₂ concentrations, especially in the Arctic (Jonko et al. 2012; Block and Mauritsen 2013). Although these analyses examine the TOA, this finding should also apply to the surface. Since clear-sky SW absorption in the atmospheric column is small, K_a is similar for the TOA and surface. Consistent with the behavior of ${\lambda }_a^{\text{clr}}$, the magnitude of surface K_a is smaller when less sea ice is present in the baseline climate, like under elevated CO₂ compared to 1xCO₂ (Jonko et al. 2012). Therefore, the 1xCO₂ kernel used here overestimates ${\lambda }_a^{\text{clr}}$ in the abrupt4xCO₂ simulations. ISRF_G is not subject to this kernel-related bias in ${\lambda }_a^{\text{clr}}$, and therefore is positive in the Arctic. Hereafter, for analyses of ISRF across all 18 models, we focus our attention on ISRF_G.

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Zonal ensemble-mean total-sky ISRF in which the temperature-mediated radiative response is quantified using the radiative kernel-regression method (ISRF_K; blue), the Gregory regression method (ISRF_G; red), and offline double-call calculations (ISRF_D; black).

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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Global-mean ISRF_G is shown in Fig. 6 for the full suite of CMIP5 models used in this study. The results are displayed as a vertically oriented dot plot (Fig. 6a), to highlight the intermodel spread, and as a bar chart (Fig. 6b), to highlight the results for each individual model. The intermodel spread in ISRF_G is 2.30 W m⁻², similar to the spread in ISRF_D (2.16 W m⁻²). We note, however, that the distribution of this spread differs. For example, for two of the six models (variants of the same GCM) ISRF_D is notably smaller than the other available double-call calculations. In contrast, ISRF_G is more evenly distributed, since this larger ensemble includes more models with small forcing. Among the subset of models where both estimates are available (red numbers in Fig. 6a), the intermodel spread in ISRF_G (1.83 W m⁻²) is slightly smaller than ISRF_D. To the extent that this finding is representative of the entire ensemble, it suggests that the kernel residual calculation underestimates the spread across models compared to diagnoses of ISRF from double-call calculations. In addition to the spread across models, the ensemble mean of ISRF_G, ISRF_K (not shown), and ISRF_D are very similar. This result may differ between the surface and TOA, since Chung and Soden (2015b) found disagreement between the magnitudes of TOA IRF_K and TOA IRF_D. One potential factor in the difference is that Chung and Soden (2015b) used a different set of radiative kernels developed from the GFDL model (Soden et al. 2008).

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Global-mean intermodel comparison of (a) total-sky ISRF estimated using radiative kernels (ISRF_G in the text) and double-call calculations in a subset of CMIP5 models and (b) the results displayed as a bar chart with models ordered by magnitude of kernel-derived ISRF. Numbered markers in (a) correspond to the CMIP5 models listed in Table 1. For models where both methods are available, markers are red. Model numbers on the abscissa in (b) also correspond to Table 1.

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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In a different, larger ensemble of ${\text{ISRF}}_D^{\text{clr}}$ calculations forced by instantaneous doubling of CO_2, Collins et al. (2006) found a range of ~1.3 W m⁻² in the LW and ~3.5 W m⁻² in the SW. They only evaluated clear-sky fluxes, not total-sky and only for a series of single-column profiles, not globally. Assuming ${\text{ISRF}}_D^{\text{clr}}$ is twice as large for a quadrupling as for a doubling of CO₂, the spread in the Collins et al. ensemble would translate to a range of ~2.6 W m⁻² in the LW and ~7.0 W m⁻² in the SW under 4xCO₂. This is in close agreement with our estimates of spread in LW ${\text{ISRF}}_G^{\text{clr}}$ and ${\text{ISRF}}_D^{\text{clr}}$, but more than twice as large for the SW components. The reduced SW spread indicates that modeling groups have addressed sources of uncertainty in SW parameterization schemes in response to the findings by Collins et al. Additionally, in the Collins et al. ensemble from the Radiative Transfer Model Intercomparison Project, each double-call calculation includes a different RTM, but an identical base climate used to initialize the simulations. The ensemble of CMIP5 ${\text{ISRF}}_D^{\text{clr}}$ used here contains differences in both components across models. Therefore, another potential explanation for the smaller SW range in the CMIP5 models is that spread associated with RTM and base climate differences compensates to some degree, which may have been introduced in the model tuning process (Mauritsen et al. 2012; Hourdin et al. 2017). This warrants further investigation, which is beyond the scope of this study.

Intermodel spread in ESRF is 3.5 W m⁻². This spread is further decomposed in Fig. 7, which compares global-mean ESRF to ISRF_G for each model. There is a strong, linear relationship under total-sky conditions (r = 0.90, RMSE = 0.82 W m⁻²), indicating that much of the intermodel spread in ESRF can be attributed to intermodel differences in ISRF and thus, differences in radiative transfer algorithms across CMIP5 models, or model differences in base climate state. We note the agreement is stronger under clear-sky conditions (r = 0.95, RMSE = 0.39 W m⁻²). It is possible that the radiative kernel technique underestimates spread in ISRF, which we estimate is smaller than spread in ISRF^clr, because of the assumption that cloud masking is a constant across models. It seems unlikely that such an error entirely accounts for the difference in ISRF spread between total-sky and clear-sky conditions, however, or differences in its contribution to ESRF spread. The assumption of constant cloud masking holds well for the subset of models where it is verifiable against double-call calculations (Fig. 6). A more likely explanation is that, in addition to ISRF, cloud adjustments contribute significantly to the spread of ESRF, while noncloud adjustments do not. We will explore this further in the next section by quantifying individual surface radiative adjustments. It is evident that cloud adjustments contribute to the magnitude of ESRF, which is systematically larger than ISRF (Fig. 7a, positive bias of 0.68 W m⁻²), unlike under clear-sky conditions.

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Global-mean ESRF vs ISRF for (a) total-sky and (b) clear-sky conditions. Numbered markers correspond to the CMIP5 models listed in Table 1. Results shown are derived using radiative kernels (ISRF_G in the text).

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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b. Radiative adjustments and radiative responses

For each model, Fig. 8a displays the global-mean surface radiative adjustments. The cloud radiative adjustments have the dominant (1.83 W m⁻²) intermodel spread among the total radiative adjustments at the surface, and also have the largest ensemble mean of any adjustment (0.77 W m⁻²). Some of the surface radiative adjustments shown lack an obvious physical explanation. For example, radiative adjustments as defined here should be independent of surface temperature change. However, most models exhibit a nonzero, slightly negative Planck adjustment, even though the Planck effect is entirely surface-temperature-dependent by definition. Also, surface albedo adjustment is negative (an increase in upwelling SW) in all but one model, which could be due to the spectral dependence of sea ice albedo (e.g., Brandt et al. 2005), but this effect is expected to be small as represented in climate models. Alternatively, both findings may reflect shortcomings associated with the assumptions inherent to using linear regression to separate λ_x from A_x. For example, similarities in the horizontal spatial structure between ensemble-mean A_x and λ_x, and between A_x and initial surface warming patterns (see the online supplemental material), suggest that some of the temperature-driven response is aliased into A_x when linear regression is used. This supports conclusions drawn by Chung and Soden (2015a), who found similar artifacts in the magnitude and spatial structure of A_x at the TOA.

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Intermodel comparison of global-mean (a) surface radiative adjustments and (b) temperature-mediated surface radiative responses estimated using surface radiative kernels. Each blue marker represents a single model. Ensemble means are marked with a black dash. Results are averaged over the 140-yr time period of the integration.

Citation: Journal of Climate 32, 13; 10.1175/JCLI-D-18-0137.1

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The radiative kernel-regression technique also affords the opportunity to estimate surface radiative responses in isolation. Intermodel differences in these responses are displayed in Fig. 8b. The water vapor radiative response λ_q has an ensemble mean of 1.31 W m⁻² K⁻¹, the largest radiative response at the surface. The lapse rate radiative response λ_LR is small in both magnitude and intermodel spread, unlike the analogous λ_LR at the TOA. The surface response is small because, globally, the largest temperature changes occur in the free troposphere. Because of the high opacity of the lower troposphere, the surface does not respond radiatively to these changes. The cloud radiative response λ_C exhibits the largest intermodel spread, but the ensemble mean is small (0.06 W m⁻² K⁻¹) and there is an even distribution of positive and negative λ_C across models. The response in the equatorial Pacific contributes the most to global-mean spread, similar to λ_C defined at the TOA (Chung and Soden 2015a). Split into its components, SW surface λ_C exhibits larger intermodel spread (range of 1.23 W m⁻² K⁻¹) than the LW response (0.55 W m⁻² K⁻¹).

Previously, Previdi and Liepert (2012) used ECHAM5-based radiative kernels to evaluate λ_x in simulations where CO₂ was instantaneously doubled. They did not make a distinction between λ_x and A_x, since finite differencing was used to compute ∆x. They similarly found that λ_q is the largest radiative response at the surface. However, they found λ_T (sum of λ_LR and Planck effect λ_Pl) and λ_q to be smaller in magnitude than our estimates, diagnosing the ensemble mean of each quantity as −0.60 and 0.89 W m⁻² K⁻¹, respectively. To better match their methodology, we estimate the same radiative responses using finite differencing for ∆x, and still find estimates that are substantially larger (−1.06 and 1.35 W m⁻² K⁻¹) than those of Previdi and Liepert (2012). This indicates λ_x may be particularly sensitive to the choice of radiative kernels used. It is also possible that differences in the treatment of the vertical interpolation of the radiative kernels between studies, or the treatment of associated near-surface numerical issues (appendix), contributed to the differing results. The magnitudes of λ_T and λ_q in our study agree more closely with findings by Colman (2015), who combined regression with the PRP approach to evaluate λ_x in a single model.

At the TOA, λ_LR and λ_q are anticorrelated (Soden and Held 2006), and multiple studies have shown that the sum of the TOA λ_LR and λ_q exhibits less intermodel spread than the individual components. Previous literature is inconsistent on whether this compensation also applies to atmospheric (TOA minus surface) radiative responses that constrain the hydrological cycle. O’Gorman et al. (2012), using radiative kernels developed from ECHAM5 (Previdi 2010), found there is compensation between intermodel spread in atmospheric λ_LR and λ_q, while Flaschner et al. (2016), using radiative kernels developed from ECHAM6, found that summing the two components does not markedly reduce intermodel spread. The latter was in agreement with findings by Pendergrass and Hartmann (2014), which diagnosed radiative responses but did not use radiative kernels. Flaschner et al. (2016) show that almost all of the intermodel spread in the sum of atmospheric λ_LR and λ_q stems from model disagreement in the lower troposphere, where the magnitude of the ECHAM5 and ECHAM6 radiative kernels are noticeably different. We find no compensation in surface λ_LR and λ_q. Since these responses are almost solely related to changes in the lower troposphere (Figs. 1 and 2), this implies compensation does not occur for atmospheric radiative responses either, consistent with Flaschner et al. (2016).