Abstract

Climate feedbacks vary strongly among climate models and continue to represent a major source of uncertainty in estimates of the response of climate to anthropogenic forcings. One method to evaluate feedbacks in global climate models is the radiative kernel technique, which is well suited for model intercomparison studies because of its computational efficiency. However, the usefulness of this technique is predicated on the assumption of linearity between top-of-atmosphere (TOA) radiative fluxes and feedback variables, limiting its application to simulations of small climate perturbations, where nonlinearities can be neglected. This paper presents an extension of the utility of this linear technique to large forcings, using global climate model simulations forced with CO2 concentrations ranging from 2 to 8 times present-day values. Radiative kernels depend on the model’s radiative transfer algorithm and climate base state. For large warming, kernels based on the present-day climate significantly underestimate longwave TOA flux changes and somewhat overestimate shortwave TOA flux changes. These biases translate to inaccurate feedback estimates. It is shown that a combination of present-day kernels and kernels computed using a large forcing climate base state leads to significant improvement in the approximation of TOA flux changes and increased reliability of feedback estimates. While using present-day kernels results in a climate sensitivity that remains constant, using the new kernels shows that sensitivity increases significantly with each successive doubling of CO2 concentrations.

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 Tas, 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 = QF, where Q is the absorbed SW radiation, then G = −ΔR in equilibrium. Along with changes in F, ΔTas 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:

 
formula

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

 
formula

where stands for global average surface air temperature, Ts 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/∂Xi, where Xi = [Ts, 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 ∂Xi 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:

 
formula

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

 
formula

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).

Fig. 1.

Clear-sky test comparing clear-sky model flux anomalies, normalized by global mean surface air temperature change (solid line), to flux anomalies derived using the 1×CO2 kernel (dashed line). Units are W m−2 K−1. (a)–(c) Zonally averaged SW flux anomalies for (a) 2×CO2 − CNTL, (b) 4×CO2 − 2×CO2, and (c) 8×CO2 − 4×CO2. (d)–(f) As in (a)–(c), but for LW flux anomalies.

Fig. 1.

Clear-sky test comparing clear-sky model flux anomalies, normalized by global mean surface air temperature change (solid line), to flux anomalies derived using the 1×CO2 kernel (dashed line). Units are W m−2 K−1. (a)–(c) Zonally averaged SW flux anomalies for (a) 2×CO2 − CNTL, (b) 4×CO2 − 2×CO2, and (c) 8×CO2 − 4×CO2. (d)–(f) As in (a)–(c), but for LW flux anomalies.

Table 1.

Differences between global average GCM- and kernel-derived flux anomalies, normalized by ΔTas, for the three CO2 doublings—2×CO2 − CNTL, 4×CO2 − 2×CO2 and 8×CO2 − 4×CO2—using the 1×CO2 kernel and a combination of the 1×CO2 and 8×CO2 kernels in units of W m−2 K−1.

Differences between global average GCM- and kernel-derived flux anomalies, normalized by ΔTas, for the three CO2 doublings—2×CO2 − CNTL, 4×CO2 − 2×CO2 and 8×CO2 − 4×CO2—using the 1×CO2 kernel and a combination of the 1×CO2 and 8×CO2 kernels in units of W m−2 K−1.
Differences between global average GCM- and kernel-derived flux anomalies, normalized by ΔTas, for the three CO2 doublings—2×CO2 − CNTL, 4×CO2 − 2×CO2 and 8×CO2 − 4×CO2—using the 1×CO2 kernel and a combination of the 1×CO2 and 8×CO2 kernels in units of W m−2 K−1.

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.

The magnitude of Re—in percent—in Eq. (2) for clear-sky conditions can be estimated using the feedback parameter calculated based on kernels,

 
formula

and a feedback parameter derived from model fluxes. Combining Eqs. (1) and (3), we obtain

 
formula

We compute Gc using the kernel technique. We run the offline radiative transfer model with present-day and doubled CO2 concentrations, and then take the difference in TOA fluxes between the two runs. Equation (7) shows %Re (Re in percent) is the normalized difference between the feedback parameters:

 
formula

We find the residual for 2×CO2 − CNTL to be 9%, comparable to the value of 10% obtained by Shell et al. (2008); Re increases to 41% for 4×CO2 − 2×CO2 and 49% for 8×CO2 − 4×CO2. Thus, the linear kernel based on the present-day climate does not adequately reproduce flux anomalies in response to forcings of 4×CO2 and larger.

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/∂Xi 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.

a. Comparison of 1×CO2 and 8×CO2 climate base states

Compared with the present-day base climate, the new 8×CO2 climate is characterized by substantially higher global average surface and atmospheric temperatures. Global average Ts rises by 7.9 K, with the largest increases at high latitudes. Atmospheric temperatures increase most in the tropical upper troposphere, while the stratosphere cools. Specific humidity increases throughout the atmospheric column—with the strongest increases near the surface, as one would expect from Clausius–Clapeyron—while the relative humidity field exhibits regions of both increases and decreases throughout the troposphere. A strong decrease in sea ice fraction in the high latitudes of both hemispheres leads to a slight decrease in global average surface albedo. Finally, the vertically integrated total cloud fraction increases. This increase is explained by an increase in high cloud fraction, while mid- and low-level cloud fractions decrease on a global average.

b. Comparison of 1×CO2 and 8×CO2 kernels

Figure 2 shows 1×CO2 kernels, while differences between the 1×CO2 and 8×CO2 kernels are depicted in Fig. 3. Figures 2a–f, 3a–f are latitude–pressure plots of the zonal average water vapor and air temperature kernels and kernel differences (8×CO2 − 1×CO2) in units of W m−2 K−1 (100 hPa)−1 for clear-sky (top) and all-sky (bottom) conditions. For the two-dimensional surface temperature and albedo kernels, clear-sky and all-sky kernels and differences in W m−2 per unit standard anomaly (1 K for surface temperature and 1 % for albedo) are plotted together; see Figs. 2, 3g,h.

Fig. 2.

Annual and zonal averages of present-day kernels in units of W m−2 per unit standard anomaly, per 100 mb. (a)–(f) 3D LW and SW water vapor, and air temperature kernels for (top) clear-sky and (bottom) all-sky conditions. (g),(h) 2D surface temperature and albedo kernels. Clear-sky (dashed line) and all-sky (solid line) kernels are combined in one plot.

Fig. 2.

Annual and zonal averages of present-day kernels in units of W m−2 per unit standard anomaly, per 100 mb. (a)–(f) 3D LW and SW water vapor, and air temperature kernels for (top) clear-sky and (bottom) all-sky conditions. (g),(h) 2D surface temperature and albedo kernels. Clear-sky (dashed line) and all-sky (solid line) kernels are combined in one plot.

Fig. 3.

Annual and zonal averages of differences between 1×CO2 and 8×CO2 kernels in units of W m−2 per unit standard anomaly, per 100 mb. (a)–(f) 3D LW and SW water vapor, and air temperature kernels for (top) clear-sky and (bottom) all-sky conditions. (g),(h) 2D surface temperature and albedo kernels. Clear-sky (dashed line) and all-sky (solid line) kernels are combined in one plot.

Fig. 3.

Annual and zonal averages of differences between 1×CO2 and 8×CO2 kernels in units of W m−2 per unit standard anomaly, per 100 mb. (a)–(f) 3D LW and SW water vapor, and air temperature kernels for (top) clear-sky and (bottom) all-sky conditions. (g),(h) 2D surface temperature and albedo kernels. Clear-sky (dashed line) and all-sky (solid line) kernels are combined in one plot.

In an attempt to further narrow down which changes in the base state are responsible for the differences in kernels, we calculate hybrid kernels from a series of mixed base states, where all variables but one come from the 1×CO2 climate. That variable is then substituted from the 8×CO2 climate. These substitutions are made at each time step (every 3 h) and at each grid point. Zonal averages of these hybrid kernels are compared with the 1×CO2 and 8×CO2 kernels in Figs. 4, 5. Figure 4 shows clear-sky LW kernels, while Fig. 5 shows clear-sky SW kernels, as well as the all-sky albedo kernel.

Fig. 4.

(top) Zonal averages of 1×CO2 and 8×CO2 LW kernels in units of W m−2 K−1 compared to hybrid kernels computed from 1×CO2 base states, with one variable at a time substituted from 8xCO2 base state. (bottom) Sum of differences between hybrid and 1×CO2 kernels compared to 8×CO2 kernel. 3D atmospheric temperature and water vapor kernels are summed over the height of the atmosphere.

Fig. 4.

(top) Zonal averages of 1×CO2 and 8×CO2 LW kernels in units of W m−2 K−1 compared to hybrid kernels computed from 1×CO2 base states, with one variable at a time substituted from 8xCO2 base state. (bottom) Sum of differences between hybrid and 1×CO2 kernels compared to 8×CO2 kernel. 3D atmospheric temperature and water vapor kernels are summed over the height of the atmosphere.

Fig. 5.

(top) Zonal averages of 1×CO2 and 8×CO2 SW kernels in units of W m−2 per unit standard anomaly compared to hybrid kernels computed from 1×CO2 base states, with one variable at a time substituted from 8×CO2 base state. (bottom) Sum of differences between hybrid and 1xCO2 kernels compared to 8xCO2 kernel. 3D water vapor kernels are summed over the height of the atmosphere.

Fig. 5.

(top) Zonal averages of 1×CO2 and 8×CO2 SW kernels in units of W m−2 per unit standard anomaly compared to hybrid kernels computed from 1×CO2 base states, with one variable at a time substituted from 8×CO2 base state. (bottom) Sum of differences between hybrid and 1xCO2 kernels compared to 8xCO2 kernel. 3D water vapor kernels are summed over the height of the atmosphere.

For most of the plots, we focus our attention on the clear-sky hybrid kernels, since in the all-sky case the calculations of hybrid kernels are complicated by clouds. Clouds do not depend on a single model variable, and as such their effect can only be examined by substituting several variables simultaneously. The variables we use include cloud fraction, emissivity, in-cloud ice and liquid water paths, and effective liquid droplet and ice particle radii. Other variables, such as T, are likely to also impact clouds in the radiation model. However, using a large number of variables detracts from the purpose of these calculations, which is to isolate the impacts of individual variables on the kernel. Substituting the variables listed above, we find that the cloud hybrid kernels contribute significantly to the difference between 1×CO2 and 8×CO2 kernels only for albedo, shown in Figs. 5e,f.

The solid lines in Figs. 4, 5 represent the 1×CO2 kernel (black) and 8×CO2 kernel (red). For the LW kernels in Fig. 4, we consider substitutions of specific humidity (green dashed line), T (gray dotted line), Ts (purple dotted line), and CO2 (light blue dashed–dotted line). For the SW kernels in Fig. 5, we substitute the albedo (orange dashed–dotted line), specific humidity and temperature (green dashed line), and clouds (gray dotted line). Specific humidity and temperature need to be substituted simultaneously for the SW kernels, since both variables are used by the radiation model to determine relative humidity, which in turn is used in aerosol calculations. Changing specific humidity alone results in a mismatch between water vapor and temperature fields, and in a bias in aerosols, which translates into biased SW fluxes. The albedo is composed of solar SW direct and diffuse and solar LW direct and diffuse albedo terms. In conjunction with the albedo variables, we have also substituted ice fraction and snow height over land, although those variables have no impact on the kernel we calculate. The remaining base state variables, namely, surface pressure and land fraction, also do not impact kernel calculations.

The clear-sky differences between present-day and hybrid kernels combine linearly to give the difference between the present-day and 8×CO2 kernels (bottom panels of Figs. 4, 5). For the all-sky albedo kernel, adding the cloud hybrid kernel also results in good agreement (Fig. 5f). An issue to consider when making these substitutions is the decorrelation of fields, which results from substituting a field from one climate into another at short time intervals. In the PRP method, this issue can be resolved by using a two-sided approach, where the average of substitutions in both directions between the two climates is considered. This is not feasible in the present case, since we are combining a field substitution with a uniform perturbation of a feedback variable. However, because the sum of hybrid kernel differences and the 8×CO2 kernels show good agreement, we assume that the decorrelation of fields is not a significant issue in the calculations presented here.

The LW water vapor kernel (Figs. 2a,b) is positive almost everywhere, meaning that an increase in specific humidity leads to more absorption of upwelling LW radiation in the atmosphere. The largest contributions come from the tropical troposphere, where moist conditions enhance absorption through self-broadening of water vapor absorption lines (Soden et al. 2008). The kernel increases in magnitude in an 8×CO2 climate, meaning that OLR decreases more because of the same standard anomaly (Figs. 3a,b; black and red lines in Figs. 4a,b). The largest changes occur in the upper troposphere and at high latitudes, with another small maximum in the tropical midtroposphere. Thus, changes in water vapor are more efficient at trapping radiation in the troposphere in the 8×CO2 climate, which has higher concentrations of greenhouse gases. Figure 4a shows that the increase in water vapor kernel can be attributed to changes in water vapor and surface temperature. Substituting 8×CO2 specific humidity increases LW absorption and strengthens the water vapor kernel. Changing the surface temperature also increases the kernel magnitude by increasing the amount of upwelling radiation from the surface to be absorbed by water vapor. Substituting 8×CO2 air temperature increases OLR and thus decreases the kernel, particularly in the tropics. To a lesser extent, changing CO2 also decreases the kernel, since the increased absorption by CO2 decreases the amount of upwelling radiation available for absorption. The kernels and patterns of change are fairly similar for clear-sky and all-sky kernels. Under all-sky conditions, cloud masking is responsible for a decreased kernel magnitude (Fig. 2b) and decreased difference between 1×CO2 and 8×CO2 kernels (Fig. 3b), in particular near the surface.

The SW water vapor kernel is associated with the absorption of solar radiation by water vapor, with maximum flux changes over snow- and ice-covered areas under clear-sky conditions (Fig. 2c). The high reflectivity of these surfaces means that more SW radiation is available for absorption by water vapor. Clouds increase the magnitude of the SW water vapor kernel in low latitudes because they too increase the atmospheric radiation pathlengths through reflection. For the 8×CO2 base state, the clear-sky SW water vapor kernel decreases in areas where sea ice and snow melt, reducing reflected SW radiation (Fig. 3c). Increases in high cloud amount, as well as in tropical convective clouds, lead to an increase of the all-sky SW water vapor kernel where these clouds are present (Fig. 3d). Note that the kernel and changes in kernel are an order of magnitude smaller than for the LW water vapor kernel. Figure 5a shows that zonal average differences between 1×CO2 and 8×CO2 kernels are confined to high latitudes and explained by changing albedo in the Northern Hemisphere and changing specific humidity in the Southern Hemisphere.

The 1×CO2 clear-sky surface (Fig. 2g) and air temperature (Fig. 2e) kernels are negative, since rising temperatures lead to more OLR. While more LW radiation is emitted by the surface in the 8×CO2 case, less of it has an effect at the top of the atmosphere because more of it is absorbed by higher concentrations of greenhouse gases now present in the atmospheric column, resulting in a decrease in Ts kernel (the negative kernel becomes less negative, Fig. 3g). Figure 4e shows that in the tropics, this decrease is largely due to higher concentrations of water vapor. In the mid- and high latitudes, the combination of increased water vapor and CO2 concentrations leads to a decrease, which is partially balanced by an increase in kernel due to higher surface temperatures.

The clear-sky air temperature kernel increases in magnitude in the upper troposphere and exhibits areas of decrease close to the surface in the 8×CO2 climate (Fig. 3e). Increases in temperature at all levels throughout the troposphere increase OLR but increases at higher levels have a stronger effect, since the upwelling radiation has to pass through less atmosphere to reach the TOA. The net effect at the top of the atmosphere is an increase in clear-sky T kernel. In the tropics, this increase is largely due to an increased amount of OLR emitters (water vapor) (Fig. 4c), while in the mid- and high latitudes, increasing atmospheric temperatures and specific humidity contribute equally to increased kernel magnitude. All-sky changes are negative in the upper troposphere, with positive values near the surface, where increases in upwelling LW radiation are masked by changes in emission from cloud tops (Fig. 3f).

The surface albedo kernel (Fig. 2h) is negative, that is, an increase in albedo leads to more reflection and thus a decrease in absorbed solar radiation. The effect at the TOA of a uniform albedo change is largest in the tropics, where incident solar radiation has a maximum (Fig. 2h). The positive change in clear-sky kernel from present day to 8×CO2 implies a decrease in the kernel that is well correlated with a decrease in downwelling solar radiation (Fig. 3h). In the tropics and midlatitudes, this decrease is solely due to changes in specific humidity (Fig. 5c), leading to more absorption of incoming solar radiation in the atmosphere and decreasing the amount of solar radiation available for reflection at the surface. In high latitudes, a decrease in albedo due to melting snow and sea ice also contributes to the decrease in kernel. When clouds are included in kernel calculations, the differences between 1×CO2 and 8×CO2 kernels are positive in the tropics and high latitudes with negative areas in the midlatitudes (Fig. 3h). These large zonal variations in kernel differences indicate a strong masking of albedo and water vapor changes by cloud changes. These cloud changes account for roughly half of the kernel change in the tropics and midlatitudes, and for most of the change in the southern high latitudes (Fig. 5e).

Repeating the clear-sky linearity test for the 8×CO2 − 4×CO2 doubling using the 8×CO2 kernels results in substantially better agreement with global average model flux anomalies (Table 1; Figs. 6c,d). For the 4×CO2 − 2×CO2 doubling, we use an average of both kernels, which also improves agreement with model flux anomalies (Table 1; Figs. 6a,b). We expect that calculating a 4×CO2 kernel would further improve this agreement. However, this would detract from the advantage of the radiative kernel technique, which lies in its computational efficiency.

Fig. 6.

Difference between clear-sky-model- and kernel-derived flux anomalies for (a),(b) 4×CO2 − 2×CO2 using the 1×CO2 kernel (solid line) and an average of the 1×CO2 and 8×CO2 kernels (dashed line) and (c),(d) 8×CO2 − 4×CO2 using the 1×CO2 kernel (solid line) and the 8×CO2 kernel (dashed line).

Fig. 6.

Difference between clear-sky-model- and kernel-derived flux anomalies for (a),(b) 4×CO2 − 2×CO2 using the 1×CO2 kernel (solid line) and an average of the 1×CO2 and 8×CO2 kernels (dashed line) and (c),(d) 8×CO2 − 4×CO2 using the 1×CO2 kernel (solid line) and the 8×CO2 kernel (dashed line).

Repeating the calculation of Re with the new kernels, we find that Re remains close to 10% for all three doublings. Hence, at least under clear-sky conditions, additional kernels computed using a different forcing base state can be used to overcome the linearity limitation of the kernel technique and applied to large forcings.

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.

Table 2.

Feedbacks in W m−2 K−1 for 2×CO2 − CNTL and 8×CO2 − 4×CO2 computed from 100-yr average model data using 1×CO2 and 8×CO2 kernels, where λ and λc are the total feedbacks and s is the climate sensitivity in units of K (W m−2)−1.

Feedbacks in W m−2 K−1 for 2×CO2 − CNTL and 8×CO2 − 4×CO2 computed from 100-yr average model data using 1×CO2 and 8×CO2 kernels, where λ and λc are the total feedbacks and s is the climate sensitivity in units of K (W m−2)−1.
Feedbacks in W m−2 K−1 for 2×CO2 − CNTL and 8×CO2 − 4×CO2 computed from 100-yr average model data using 1×CO2 and 8×CO2 kernels, where λ and λc are the total feedbacks and s is the climate sensitivity in units of K (W m−2)−1.

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 Ts 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 Ts 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.

Acknowledgments

The authors thank Andrew Dessler and two anonymous reviewers for their helpful feedback. This work was supported by NASA headquarters under the NASA Earth and Space Science Fellowship Program Grant “10-Earth10R-35,” by the National Science Foundation under Grant ATM-0904092, as well as by the NCAR Advanced Study Program’s Graduate Student Visitor Program. Computing resources were provided by the Climate Simulation Laboratory at NCAR’s Computational and Information Systems Laboratory (CISL), sponsored by the National Science Foundation and other agencies.

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Footnotes

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The National Center for Atmospheric Research is sponsored by the National Science Foundation.