Climate Feedbacks in CCSM3 under Changing CO2 Forcing. Part I: Adapting the Linear Radiative Kernel Technique to Feedback Calculations for a Broad Range of Forcings

1. Introduction

Greenhouse gas emissions today already exceed those predicted by all but the highest-emission scenarios (Raupach et al. 2007). Walker and Kasting (1992) show that, without reductions in CO₂ emissions, atmospheric CO₂ could reach concentrations as high as 2200 ppmv, or 8 times preindustrial values, by the twenty-fourth century. With CO₂ being the strongest positive radiative forcing of climate change (Forster et al. 2007), the response of climate to such a large forcing deserves thorough investigation. The response of climate, most commonly defined as the global average surface air temperature change, to an external forcing (such as CO₂ doubling) is referred to as climate sensitivity (Randall et al. 2007; Roe and Baker 2007) and depends on radiative feedbacks in the climate system.

Several previous studies have investigated the behavior of climate sensitivity in global climate models for a wide range of forcing magnitudes. Colman and McAvaney (2009) use the partial radiative perturbation (PRP) method (Wetherald and Manabe 1988) to obtain individual radiative feedbacks from the Centre for Australian Weather and Climate Research (CAWCR) general circulation model simulations forced with CO₂ concentrations varying between one-sixteenth and 32 times present-day values. They find a decreasing climate sensitivity with increased CO₂, mostly due to a weakening albedo feedback. The PRP method is powerful and accurate. However, it requires repeated runs of an offline radiative transfer model, making it computationally expensive.

Colman et al. (1997) propose a modified PRP approach to investigate the nonlinear behavior of climate feedbacks resulting from globally uniform SST perturbations ranging between ±2 K in the CAWCR model. The consideration of higher-order terms allows them to evaluate nonlinearities for individual feedbacks and estimate errors that would result from using linear theory. They find that the largest nonlinearities are associated with changes in lapse rate, water vapor, and high clouds. Boer et al. (2005) investigate the climate response to large variations in solar constant in the National Center for Atmospheric Research (NCAR) Climate System Model (CSM) using the cloud radiative forcing (CRF) approach (Cess and Potter 1987). They are able to distinguish between the clear and cloudy skies as well as longwave (LW) and shortwave (SW) contributions to the feedback parameter. This approach does not, however, allow for the quantification of individual feedbacks. In contrast to Colman and McAvaney (2009), they detect an increase in climate sensitivity with increasing forcing and a runaway warming for increases in solar constant of 25% and above, which they attribute to changes in the shortwave cloud feedback.

All studies discussed above focus their analysis on only one global climate model. While they contribute to our understanding of the behavior of those particular models, due to the well-known spread in climate sensitivities and feedbacks among global climate models, this does not necessarily translate into increased understanding of the sensitivity of the actual climate system under strong climate forcing. Model intercomparison studies, which could help advance such understanding, are complicated by the fact that the current methods for the evaluation of individual nonlinear feedbacks are generally computationally very expensive.

Here, we explore the utility of the computationally more efficient radiative kernel technique (Soden et al. 2008; Shell et al. 2008) for this problem. This technique does make use of the assumption that feedbacks behave linearly with respect to the climate state. The linearity assumption has been shown to be valid for perturbations on the order of magnitude of 2×CO₂ by Shell et al. (2008), who calculated a kernel using the Community Atmosphere Model, version 3 (CAM3). Further applications of the kernel technique include the work of Dessler et al. (2008), who computed the water vapor feedback from observations of present-day climate fluctuations, and Sanderson et al. (2010), who analyzed a large ensemble of transient model simulations with perturbed physical parameters relating to atmosphere, ocean, and the sulphur cycle. The CO₂ forcing used by all these studies is limited to observed and Special Report on Emissions Scenarios (SRES) A1B scenario–based CO₂ concentrations, not exceeding CO₂ doubling.

The kernel technique has not been tested on larger forcings thus far. It is, however, generally accepted that nonlinearities become relevant with increasing forcing (Colman et al. 1997). In this study we attempt to extend the technique to be used for a wider range of forcings, using a set of long Community Climate System Model, version 3 (CCSM3) simulations forced with instantaneous doubling, quadrupling, and octupling of CO₂. We first compare top-of-atmosphere (TOA) flux changes between experiment and control simulations to those derived using radiative kernels. We find increasing disagreement between flux changes with rising CO₂ concentration, suggesting increasing nonlinearity in fluxes, which the standard linear kernel technique omits. To address this issue, we compute a new set of kernels based on an 8×CO₂ climate state. Flux changes computed using a combination of this and the original kernel show much better agreement with model flux changes. We discuss the differences between these two sets of kernels as well as implications for feedbacks obtained using the different kernels.

The focus of this paper is the applicability of the radiative kernel technique to global climate model simulations with large climate forcing. A more in-depth analysis of the behavior of individual feedbacks with increasing CO₂ forcing in CCSM3 is presented in Jonko et al. (2012, unpublished manuscript). Note that other types of feedbacks, for example due to changes in the carbon cycle, also affect the final climate response (Cox et al. 2000; Friedlingstein et al. 2006). However, the feedbacks discussed in the following are associated with physical changes in response to specified CO₂ only, since the version of the global climate model we use does not include an interactive carbon cycle.

2. Model data

We use the low-resolution version of the NCAR CAM3, which is truncated at T31 (3.75° × 3.75°) with 26 vertical levels in the atmosphere (Collins et al. 2006; Yeager et al. 2006). This version of CAM is known to have a low climate sensitivity in comparison with other global climate models (Kiehl et al. 2006). The climate base states for kernel calculations were obtained from CAM3 simulations coupled to a Slab Ocean Model, while the kernels themselves were calculated with the offline radiative transfer model component of CAM3 (Collins et al. 2006). The model fluxes and feedback variables used in the clear-sky test in section 4 and for feedback calculations in section 6 come from simulations with CCSM3, where CAM3 is coupled to a full-depth ocean with a nominal horizontal resolution of 3° and 25 vertical levels (Danabasoglu and Gent 2009). The control run is forced with the observed 1990 CO₂ concentration of 355 ppmv. The instantaneous forcing increases to 710, 1420, and 2840 ppmv in the 2×CO₂, 4×CO₂, and 8×CO₂ simulations, respectively. Since data for all four simulations are available up to year 1450, we analyze the 100-yr period from year 1351 to year 1450. By model year 1450, the global mean surface air temperature has increased by 2.3 K in 2×CO₂, 4.8 K in 4×CO2₂, and 8.0 K in 8×CO₂ compared with the control run.

3. Radiative kernel technique

An increase in surface air temperature T_as, resulting from a positive forcing G, leads to increased outgoing LW radiation (F) in the atmosphere. If the net TOA radiative flux, excluding the forcing itself, is R = Q − F, where Q is the absorbed SW radiation, then G = −ΔR in equilibrium. Along with changes in F, ΔT_as also induces changes in other climate variables that then act to either amplify or dampen the initial temperature change. Thus, the equilibrium sensitivity of climate to a forcing depends not only on the forcing itself but also on radiative feedbacks due to changes in temperature, water vapor, albedo, and clouds (Shell et al. 2008), as shown:

Individual feedbacks can be obtained from a linear decomposition of the feedback parameter λ according to Zhang et al. (1994), as follows:

where

stands for global average surface air temperature, T_s is surface temperature, T is atmospheric temperature, ln(q) is the natural logarithm of specific humidity, representing the water vapor feedback, α is surface albedo, C is clouds, and Re is a residual, which is expected to be small for small climate perturbations (Zhang et al. 1994). The sign convention we use is such that a positive flux change corresponds to a warming. This linear decomposition is essentially a Taylor series expansion, where all higher-order terms have been neglected. Other decompositions are equally valid. It is common, for example, to split the atmospheric temperature feedback into a lapse-rate feedback and a Planck response (Colman and McAvaney 2009; Soden et al. 2008).

Using the radiative kernel technique, the two terms constituting each feedback in Eq. (2) are computed separately. One term, the radiative kernel—∂R/∂X_i, where X_i = [T_s, T, ln(q), α]—is the response of TOA radiative fluxes to incremental changes in feedback variables, referred to as standard anomalies. The other term, , is the climate response; that is, the response of feedback variables to changes in global average surface air temperature. The radiative kernel is calculated by perturbing an offline radiative transfer model by a standard anomaly in feedback variable ∂X_i and computing the change in TOA radiation due to that perturbation.

Here, we use the offline radiative transfer model from CAM3 (Collins et al. 2006) to compute kernels. The climate base state for kernel computation is obtained from a present-day control simulation of CAM3 coupled to a slab ocean model. The fact that a slab ocean model rather than a fully coupled model is used to derive the base state may result in a slightly different kernel than would be obtained using a fully coupled model, since the impact of ocean dynamics is excluded. However, using the fully coupled model to compute the base state would lessen the advantage of the radiative kernel technique being a computationally efficient method. Further, previous studies have shown that the climate sensitivities of CAM and CCSM are comparable (Danabasoglu and Gent 2009). For comparison with the fully coupled model, the climate sensitivity is 2.3 K (Kiehl et al. 2006) with respect to CO₂ doubling and 7.9 K with respect to CO₂ octupling.

We calculate changes in TOA fluxes by running the radiation model twice—with input data from the base state and then perturbing the feedback variable under consideration at each grid point, pressure level, and time step (3 h)—and taking the difference between the fluxes obtained from each simulation. We use standard anomalies of 0.01 for the surface albedo and 1 K for both surface and air temperature. Rather than using a uniform specific humidity perturbation for the water vapor kernel, we compute the change in the natural logarithm of specific humidity corresponding to a 1-K temperature perturbation at constant relative humidity. We use the natural logarithm ln(q) based on the near proportionality of the absorption of radiation by water vapor to ln(q) (Raval and Ramanathan 1989). The radiative kernel is defined as the TOA flux difference for a standard anomaly.

The climate response is the difference in climate variables between the experiment and control global climate model simulations, normalized by the global average surface air temperature change. We obtain TOA flux anomalies for the 3D variables temperature and water vapor by first combining the kernel with the climate response and then summing over the vertical levels of the atmosphere to obtain the total effect. To calculate feedbacks, we sum only over the layers of the troposphere. The tropopause as a function of latitude φ is approximated by 100 hPa + 200 hPa (|φ|/90°), varying between 100 hPa at the equator and 300 hPa at the poles. Because of nonlinearities introduced primarily by cloud overlap, we cannot evaluate the cloud feedback directly using a cloud kernel. The kernel technique can be used to obtain an improved estimate of cloud feedback from the change in cloud radiative forcing (ΔCRF; Shell et al. 2008). The details of this calculation are not the focus of the present study, but we discuss cloud feedbacks in large forcing experiments in Jonko et al. (2012, unpublished manuscript).

4. Clear-sky linearity test

The kernel technique presumes a linear relationship between changes in feedback variables and changes in TOA radiation. Although the applicability of the radiative kernel technique to large forcings has not previously been tested, there is no physical basis for assuming linearity at large forcings. To examine at what forcing magnitude the linear relationship breaks down, we perform a clear-sky linearity test (Shell et al. 2008) comparing TOA flux anomalies derived from model simulations with those derived from kernels. A disagreement between the flux anomalies calculated using the two methods indicates that the kernel does not adequately approximate clear-sky TOA flux changes between the experiment and control model simulations. Agreement between the flux anomalies provides a necessary but not sufficient condition, since we cannot exclude compensating errors.

We consider the SW and LW components of the clear-sky flux anomalies separately. To obtain these from model simulations, we difference the TOA SW and LW clear-sky fluxes from the experiment (EXP) and control (CNTL) simulations,

and

. These flux changes, denoted by a prime, differ from flux changes obtained using the kernel technique, in that they already include the forcing G, as shown:

Equivalent flux anomaly values can be obtained by summing the individual clear-sky anomalies produced by changes in each feedback variable as follows:

The SW flux anomaly consists of contributions from surface albedo and SW water vapor changes, while the LW flux anomaly includes contributions from LW water vapor, surface temperature, and air temperature changes. Both SW and LW flux anomalies also include a CO₂ forcing term that is strictly speaking not a feedback but is included in the test because it contributes to the total energy budget of the climate system (Shell et al. 2008). The SW forcing term is negligible compared to the LW forcing term but is included here for completeness.

Results of the clear-sky test for the three doublings 2xCO₂ − CNTL, 4xCO₂ − 2xCO₂ and 8xCO₂ − 4xCO₂ are shown in Fig. 1. Flux changes resulting from the different forcing cases have been normalized by for plotting. The top part of Figs. 1a–c shows zonally averaged SW flux changes. These are positive everywhere and dominated by contributions from high latitudes, where albedo changes due to melting sea ice and snow result in increased absorption of solar radiation. SW clear-sky flux changes per K increase with increasing forcing in the northern high latitudes, while they initially remain constant and then decrease in the southern high latitudes. Since we use the same kernel for all three doublings, these changes are associated with changes in feedback variables, primarily albedo. While the albedo change increases with each CO₂ doubling in the Arctic, it decreases in southern high latitudes, where the largest decrease in albedo, correlated with the strongest melting of sea ice, occurs for the first doubling, 2×CO₂ − CNTL. There is good agreement between the GCM- and kernel-derived flux anomalies except in the northern high latitudes, where the kernel increasingly overestimates the TOA flux change. Globally averaged, this overestimate amounts to 0.03 W m⁻² K⁻¹ for 4×CO₂ − 2×CO₂ and 0.05 W m⁻² K⁻¹ for 8×CO₂ − 4×CO₂ (Table 1). Shell et al. (2008) found an underestimation of the clear-sky SW flux anomaly in their analysis of climate feedbacks in CAM. This discrepancy is explained by the fact that the surface albedo feedback, which contributes most to the SW flux change, is somewhat nonlinear and depends on the magnitude of the standard anomaly. Shell et al. (2008) use a standard anomaly of 0.001, one order of magnitude smaller than the one used here, which results in a smaller surface albedo feedback. Since the albedo feedback is confined to the polar regions, the bias introduced by this nonlinearity is rather small on the global average. However, its existence does call into question the robustness of albedo feedback calculations, especially at high latitudes (Kay et al. 2012).

The LW clear-sky flux changes (−ΔF) are positive in the tropics and negative in mid- and high latitudes (Figs. 1d–f). A positive flux change corresponds to a warming effect, or a decrease in outgoing longwave radiation (OLR). Thus, LW feedbacks and forcings lead to warming in the tropics and cooling in the extratropics. While for 2×CO₂ − CNTL the kernel-derived zonal flux anomalies are in relatively good agreement with the model anomalies, the kernel-derived anomalies are clearly more negative than GCM-derived values for 8×CO₂ − 4×CO₂. The global average difference between the normalized flux anomalies is −0.05 W m⁻² K⁻¹ for 2×CO₂ − CNTL, −0.32 W m⁻² K⁻¹ for 4×CO₂ − 2×CO₂, and −0.60 W m⁻² K⁻¹ for 8×CO₂ − 4×CO₂ (see Table 1). The kernel-derived flux changes are shifted toward more negative values throughout all latitudes. To ensure the robustness of these calculations, we have repeated them using 50-yr (years 1401–50), 100-yr (years 1351–1450), and 500-yr (years 951–1450) averages and obtained consistent results.

5. 8×CO₂ kernel

The increasing discrepancies between the model- and kernel-derived flux changes in the clear-sky test show that, especially for LW fluxes, ∂R/∂X_i is not a constant. The kernel depends on the radiation model used to calculate it as well as the standard anomalies and the climate base state. Soden et al. (2008) have shown that kernels are relatively insensitive to the choice of radiative transfer model. Hence, we focus on their sensitivity to the climate base state used to calculate them. We use the same model and standard anomalies as were used to compute the present-day kernels to calculate a new set of kernels using an 8×CO₂ climate base state. This new base state is derived from a CAM3 slab ocean model simulation forced with instantaneous octupling of CO₂ and run for 65 yr. All other parameters being equal, differences in kernels are inferred to be solely due to differences in climate base state.

6. Impacts of kernel differences on feedbacks

To evaluate the sensitivity of feedback estimates to the differences in kernels described in the previous section, we compare 8×CO₂ − 4×CO₂ feedbacks computed using the original 1×CO₂ kernel and those calculated with the 8×CO₂ kernel. The sensitivity of feedbacks to forcing magnitude is examined in detail in Jonko et al. (2012, unpublished manuscript). Feedbacks are calculated by combining the kernels with differences in feedback variables between experiment and control model simulations, normalized by the global average change in surface air temperature. We use CCSM3 simulations described in section 2, averaging the last 100 years (1351–1450) to calculate long-term average feedbacks, shown in Table 2. As for the clear-sky test, we have calculated 50-, 100-, and 500-yr averages to verify that these feedback magnitudes are robust.

Increasing surface and atmospheric temperatures lead to an increase in OLR, which stabilizes the climate system, resulting in negative surface and atmospheric temperature feedbacks. Note that we have defined the feedbacks to include the Planck response of the climate system to increased CO₂ concentrations (Bony et al. 2006). When the 1×CO₂ kernel is used with the 8×CO₂ − 4×CO₂ climate response, the all-sky T_s feedback remains approximately constant, increasing only slightly in magnitude from −0.65 W m⁻² K⁻¹ for 2×CO₂ − CNTL to −0.67 W m⁻² K⁻¹ for 8×CO₂ − 4×CO₂. The new 8×CO₂ kernel includes increased greenhouse gas concentrations in the atmospheric column, which diminish the effect of upwelling LW radiation from the surface at the TOA. Thus, the T_s response decreases to −0.49 W m⁻² K⁻¹ for 8×CO₂ − 4×CO₂. The all-sky temperature feedback increases in magnitude with increasing forcing. This increase is larger using the 8×CO₂ kernel (from −2.77 W m⁻² K⁻¹ for 2×CO₂ − CNTL to −3.00 W m⁻² K⁻¹ for 8×CO₂ − 4×CO₂) than using the 1×CO₂ kernel (from −2.77 to −2.88 W m⁻² K⁻¹), since the 1×CO₂ kernel underestimates the change in TOA flux due to a temperature anomaly.

As atmospheric temperatures rise, so does the water vapor content of the atmosphere, strengthening the greenhouse effect and resulting in a positive water vapor feedback, which increases with forcing. The all-sky water vapor feedback increases from 1.56 W m⁻² K⁻¹ for 2×CO₂ − CNTL to 1.96 W m⁻² K⁻¹ for 8×CO₂ − 4×CO₂ using the new kernel. The 1×CO₂ kernel underestimates the LW flux difference for 8×CO₂ and thus underestimates the 8×CO₂ − 4×CO₂ water vapor feedback as 1.62 W m⁻² K⁻¹.

The surface albedo feedback results primarily from perturbations in albedo in areas covered with sea ice and snow. As sea ice and snow melt, they expose underlying areas that typically have much lower albedos, leading to more SW absorption and further temperature increase. The albedo feedback decreases with increasing CO₂ forcing. This decrease is stronger when the 8×CO₂ kernel is used (from 0.30 to 0.16 W m⁻² K⁻¹) than for the 1×CO₂ kernel (from 0.30 to 0.26 W m⁻² K⁻¹). This difference between the two sets of kernels is mainly due to less SW radiation reaching the surface because of increased absorption of SW radiation in the atmospheric column in the 8×CO₂ case.

The cloud feedback is calculated by correcting ΔCRF (Cess and Potter 1987) for noncloud contributions. For this purpose, differences between all-sky and clear-sky feedbacks are subtracted from ΔCRF (Jonko et al. 2012, unpublished manuscript). The estimate of the cloud feedback obtained in this manner is small and positive compared to a negative ΔCRF. It increases from 0.10 W m⁻² K⁻¹ for 2×CO₂ − CNTL to 0.30 W m⁻² K⁻¹ for 8×CO₂ − 4×CO₂ using the 8×CO₂ kernel.

Summing surface and air temperature, water vapor, surface albedo, and cloud feedbacks gives λ from Eq. (2), where λ is negative; that is, it is dominated by stabilizing contributions from surface and air temperature changes. An estimate of climate sensitivity in units of K (W m⁻²)⁻¹ can be obtained directly from λ: s = −1/λ. All-sky climate sensitivity remains constant from the first to the third CO₂ doubling when using the 1×CO₂ kernel but increases by 38%, from 0.68 to 0.93 K (W m⁻²)⁻¹, with the 8×CO₂ kernel. The largest contribution to this increase comes from the change in water vapor feedback followed by effects of clouds, while the increasing atmospheric temperature feedback and the decreased albedo feedback decrease sensitivity.

7. Discussion

We have used CCSM3 simulations with forcings ranging from CO₂ doubling to octupling to investigate the linearity assumption underlying the radiative kernel technique. Deviations from linearity were identified using the clear-sky linearity test. We confirm that for forcing magnitudes on the order of 2×CO₂, there is good agreement between modeled TOA fluxes and TOA fluxes derived using radiative kernels. However, increasing discrepancies between model and kernel fluxes appear at larger forcings, indicating that the relationship between TOA fluxes and feedback variables is not linear. The nonlinearities found in the kernel at large forcings limit the range of simulations to which a single kernel can be applied. The available present-day kernels remain a useful tool for analysis of feedbacks in present day to 2×CO₂ climates. However, based on the results presented here, we recommend the computation of additional kernels in studies investigating a larger range of forcings.

We calculate a new set of kernels using an 8×CO₂ climate base state. Combining these new kernels with the present-day kernels, we obtain better agreement with model TOA fluxes. We use both sets of kernels to compute feedbacks for 8×CO₂ − 4×CO₂ and find that the discrepancies in flux anomalies translate into significant differences in feedback values and estimates of climate sensitivity. All-sky climate sensitivity increases with increasing forcing when the new 8×CO₂ kernel is used, whereas using the 1×CO₂ kernel incorrectly indicates that the sensitivity does not change. This result is in agreement with Boer et al. (2005), who examine the effects of large solar forcing on sensitivity in the NCAR model. However, they find that the majority of this increase in sensitivity is due to changes in planetary albedo and cloud feedbacks, while our analysis suggests that the largest contributions come from the water vapor feedback, followed by the cloud feedback. An increase in climate sensitivity is also suggested by Hansen et al. (2005), who use the Goddard Institute for Space Studies (GISS) model E for their analysis, while Colman and McAvaney (2009) obtain a decrease in climate sensitivity in the CAWCR model. These conflicting results highlight the shortcomings of using any one model to make inferences about the sensitivity of the climate system and the need for a thorough comparison among models. Such comparisons are hindered by limitations in methods currently available to compute feedbacks and climate sensitivity from model simulations, in particular for large perturbations. Accurate methods are computationally very expensive, while fast methods, such as the radiative kernel technique or the Gregory method (Gregory et al. 2004), often have the caveat of linearity associated with them. We have presented one possible way in which such linear methods can potentially be extended to be useful for applications with larger forcing magnitudes, facilitating comparisons of climate feedback and sensitivity estimates across models and for varying forcing magnitudes.