An assessment of methods for computing radiative forcing in climate models

Radiative forcing quantifies the perturbation in radiative fluxes caused by changes in forcing agents. In addition to its instantaneous effect on the flow of radiation, a forcing agent can also cause changes in both the troposphere and stratosphere that further modify the upwelling radiative fluxes at the top-of-atmosphere (TOA). Because these changes occur independent of any change in global-mean surface temperature, they are generally classified as a component of the radiative forcing rather than a radiative feedback. This decoupling from the surface response also implies that radiative 'adjustments' to a forcing agent occur on a much shorter time-scale than radiative feedbacks and thus are also referred to as 'rapid adjustments'.

The cooling of the stratosphere in response to increased CO₂ is the classical example of an adjustment to a forcing agent (Manabe and Wetherald 1967, Hansen et al 1981, 1997). The potential for cloud radiative properties to change in response to aerosol forcing has also been recognized for decades (e.g. Albrecht 1989, Penner et al 1992, Zelinka et al 2014). However more recent studies have identified the potential for clouds (and other tropospheric variables) to undergo rapid changes in response to CO₂ forcing. It has also been noted that rapid adjustments to CO₂ forcing are distinctly different from those to solar forcing (e.g. Andrews et al 2009, Bala et al 2010). Because these changes are believed to occur independent of changes in surface temperature, they have also been regarded as an adjustment to the forcing rather than as a feedback (e.g. Gregory et al 2004, Gregory and Webb 2008, Andrews et al 2009, 2012, Bala et al 2010, Colman and McAvaney 2011, Vial et al 2013, Zelinka et al 2013).

Recent studies have developed methods to infer radiative forcing indirectly using archived output from climate model simulations (Gregory et al 2004, Hansen et al 2005, Chung and Soden 2015). Based on these indirect calculations of forcing, several studies have found that differences in 'adjusted' radiative forcing introduce uncertainties in model projections of the climate response that are comparable to that resulting from the uncertainty in model-simulated climate sensitivity (e.g. Andrews et al 2012, Forster et al 2013, Vial et al 2013). These studies have identified cloud adjustments as the key contributor to the spread in adjusted forcing (e.g. Forster et al 2013, Zelinka et al 2013, Vial et al 2013, Ringer et al 2014). However, Chung and Soden (2015) have further suggested that a significant portion of the intermodel spread in adjusted forcing may actually result from differences in the instantaneous (or direct) forcing from CO₂. We further investigate this issue by comparing estimates of radiative forcing for a quadrupling of CO₂ using four different methods.

The instantaneous radiative forcing has been traditionally diagnosed by performing two sets of offline radiative transfer calculations for a limited set of atmospheric profiles (e.g. Collins et al 2006). Since these sets of 'double call' radiative transfer calculations are mostly performed separately from the actual climate model simulation which uses that forcing, it is difficult to compare explicit calculations of radiative forcing between models, even for the most idealized forcing scenarios.

Alternatively, Gregory et al (2004) developed a method for estimating the adjusted forcing by conducting a linear regression between global-mean surface temperature and TOA radiative fluxes from step-change CO₂ experiments (i.e., the 'Gregory' method). Another alternate method for estimating the adjusted forcing (i.e., the 'Hansen' method) uses model simulations in which climate feedback processes are suppressed by prescribing sea surface temperatures (SSTs) while imposing an external forcing (e.g. Hansen et al 2005, Bala et al 2010, Held et al 2010, Forster et al 2013). However, it is not possible to separate the instantaneous forcing from the radiative adjustments using either the Gregory or Hansen methods.

Radiative kernels were developed to describe the differential response of TOA radiative fluxes to incremental changes in climate variables (Soden and Held 2006) and have become widely used to quantify the importance of different feedback processes (e.g. Shell et al 2008, Soden et al 2008, Soden and Vecchi 2011, Block and Mauritsen 2013, Dessler 2013, Vial et al 2013, Chung et al 2014). Recently, Chung and Soden (2015) used the radiative kernel technique to separate the instantaneous radiative forcing from radiative adjustments to that forcing, and to isolate the contributions of different radiative adjustment processes in both the troposphere and stratosphere. Because radiative kernels can isolate both the instantaneous forcing and adjusted forcing, they provide a useful tool that can be used as a common reference for comparisons against both the double-call instantaneous forcing and the adjusted forcing estimated from the Gregory or Hansen methods.

In this study, we compare and assess these four different methods (double call, Gregory, Hansen and radiative kernel) for computing the radiative forcing from the instantaneous 4 × CO₂ experiments of the Coupled Model Intercomparison Project phase 5 (CMIP5), where the concentration of CO₂, a well-mixed greenhouse gas, is quadrupled instantaneously from the pre-industrial level (Taylor et al 2012).

2.1. Offline radiative transfer calculations

2.2. Gregory method

2.3. Hansen method

2.4. Radiative kernel method

Figure 1(a) compares global-mean adjusted forcing for a quadrupling of CO₂ estimated from the Gregory or Hansen methods with that estimated using radiative kernels. The adjusted forcing for the Gregory and radiative kernel methods is computed from the abrupt4 × CO2 experiment, while the Hansen method is applied to model output from fixed SST experiments (i.e., amip4 × CO2, sstClim4 × CO2 and aqua4 × CO2). The Gregory method estimates are from Forster et al (2013) except for IPSL-CM5A-MR which is computed here following the same method. Filled symbols in figure 1(a) denote models for which double-call results are available.

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The adjusted forcing estimated from both the Gregory and radiative kernel methods ranges from approximately 5–9 W m⁻². The root-mean-square difference (rmsd) between the two methods is 0.42 W m⁻², indicating that the total radiative impact of the step CO₂ increase is reasonably consistent between these two methods. It is important to reiterate that these two methods provide independent estimates of the adjusted forcing. The Gregory method removes the temperature-mediated component of the response (i.e., the feedbacks) by regressing TOA fluxes against the global-mean temperature change, whereas the kernel method regresses the individual feedback variables (i.e., temperature, water vapor, surface albedo, cloud forcing) against the global-mean temperature change and multiplying these changes by the appropriate radiative kernel.

Since the step CO₂ increase was imposed identically among the climate models participating in the CMIP5 (Taylor et al 2012), the intermodel spread in adjusted forcing could be attributable to intermodel differences in 'rapid' adjustments of climate variables to the instantaneous forcing. In particular, recent studies have argued that cloud radiative properties undergo rapid changes in response to step CO₂ increases (e.g. Andrews and Forster 2008, Gregory and Webb 2008, Colman and McAvaney 2011, Andrews et al 2012, Vial et al 2013, Zelinka et al 2013).

In the abrupt4 × CO2 experiment, the response of TOA radiative fluxes represents the sum of both radiative forcing, which is assumed to occur nearly instantaneously, and radiative feedbacks, which are assumed to increase linearly in response to surface warming. In contrast, climate feedback processes are largely suppressed in fixed SST experiments in which SSTs and sea-ice concentration are prescribed. Therefore, in the fixed SST experiments TOA radiative flux differences between perturbed and control states measure, to first order, the total radiative impact of an imposed forcing agent.

Figure 1(a) shows that the strength of the adjusted forcing derived from fixed-SST experiments is different depending on the base climate and land-sea configuration (e.g. systemically higher forcing for aqua4 × CO2). However, the range of estimated adjusted forcing is similar among the three experiments. The rmsd values of adjusted forcing against the radiative kernel method are 0.64, 1.23 and 1.23 W m⁻² for the amip4 × CO2, sstClim4 × CO2 and aqua4 × CO2 experiments, respectively. The rmsd values for amip4 × CO2 and sstClim4 × CO2 are expected to decrease if the land warming effect on TOA radiative flux perturbations is corrected (e.g. Sherwood et al 2015). Part of the difference results from land–sea coupling and inhomogeneity in the initial rate of surface warming in the abrupt4 × CO2 (e.g. Chung and Soden 2015). These different methods of estimating adjusted forcing, however, show a distinct intermodel spread in the adjusted forcing.

Figure 1(b) further decomposes the intermodel spread in the kernel-estimated global-mean adjusted forcing by comparing the sum of instantaneous forcing and stratospheric adjustment with corresponding total adjusted forcing. Note that adjusted forcing (horizontal axis in figure 1(b)) includes tropospheric adjustment in addition to the sum of instantaneous forcing and stratospheric adjustment. The quantitative similarity between the two with an rmsd of ∼0.30 W m⁻² indicates that most of the spread in kernel-estimated adjusted forcing is caused by the spread in the sum of instantaneous forcing and stratospheric adjustment. This means that most of the spread in the total adjusted forcing estimated from the kernels can be explained by considering just the instantaneous forcing and stratospheric adjustment.

The intermodel spread in global-mean adjusted forcing and global-mean instantaneous forcing is presented in figure 2. Symbols denoting the Gregory, Hansen or radiative kernel methods are filled in the case that corresponding double-call results are available for that model. The three alternative methods produce a similar intermodel spread in adjusted forcing (∼4 W m⁻²). Since the double call results available in the CMIP5 archive do not include any adjustment, the strength of double-call-computed instantaneous forcing at the tropopause level is reduced by 15% to approximate the stratosphere-adjusted forcing for comparisons with the adjusted forcing estimated using the alternative methods. For this purpose, we have assumed the tropopause to vary linearly with latitude from 100 hPa at the equator to 300 hPa at the poles, and then applied a 15% reduction to the instantaneous forcing at the tropopause. The forcing estimates are affected by how the tropopause level is selected (Myhre and Stordal 1997). The spread in stratosphere-adjusted forcing obtained from the double-call calculations is roughly 50% smaller (∼2 W m⁻²) than that for the other methods. This suggests that the discrepancy between the double-call method and alternative methods may result from rapid adjustments in the troposphere. However, as shown below, the spread in forcing from the double call calculations is greater at the TOA than at the tropopause level, implying that intermodel differences in instantaneous forcing and rapid adjustments may act to offset each other.

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Figure 2(b) shows distribution of the global-mean instantaneous forcing computed from the double-call results together with that estimated using radiative kernels. The instantaneous forcing estimated from the radiative kernels is systematically larger than that for the double-call method (not shown). However, both methods produce an intermodel spread of ∼2.5 W m⁻². Therefore, both the radiative kernels and offline double-call methods suggest that a substantial portion of the intermodel spread in adjusted forcing is attributed to intermodel differences in the instantaneous component of the radiative forcing. This means that intermodel discrepancies in the direct radiative impact of a forcing agent may significantly contribute to differences in the model-projected climate change.

To examine the double-call calculations in more detail, figure 3 displays the vertical profiles of change in the global-mean net downward radiative flux from the double-call results. The longwave (dashed lines in blue) and shortwave (dashed lines in purple) components are also presented separately. The longwave component has minima at the surface, increases with altitude in the lower-to-middle troposphere, and remains relatively constant between the mid-troposphere and the lower stratosphere. However, quantitative differences exist among the models. For instance, between 500 hPa and the lower stratosphere, the longwave component varies in strength from 7.5 (IPSL models) to 9.5 (MIROC5) W m⁻². Discrepancies among the models are also evident at levels between 10 hPa and TOA.

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The shortwave component shows generally similar patterns of vertical distribution among the models. Minima and maxima are located at the surface and at the TOA, respectively. The difference between maximum and minimum values is less than 1 W m⁻² except for the IPSL models (∼3 W m⁻²). At the TOA, the shortwave forcing ranges from ∼0 to greater than 1 W m⁻².

To further examine the kernel and double-call results, figure 4 compares global-mean instantaneous forcing estimated using radiative kernels with global-mean instantaneous forcing computed from the double-call method. Both of the methods indicate that the direct radiative impact of a quadrupling of CO₂ shows an intermodel spread of ∼2 W m⁻² for the longwave component. However substantial disagreement between the two methods exists for some models, such as HadGEM2-A and MIROC5 for which the kernel-estimated forcing is nearly twice as large as the double-call calculation. In the case of the shortwave component, most of the models have a very small instantaneous forcing and the estimates agree well for both of the methods. However the two IPSL models exhibit distinctly larger shortwave forcings from the double-call method (∼1.2–1.3 W m⁻²) compared to the radiative kernel method (∼0.3 W m⁻²). These two models have exceptionally large double-call computed shortwave forcing, given that their longwave counterpart is ∼3 W m⁻². A comparison of the stratosphere adjusted forcing (i.e., the sum of instantaneous forcing and stratospheric adjustment) from the kernel method to the stratosphere-adjusted forcing from the double-calls (open circles) indicates better agreement between the two estimates, suggesting that intermodel differences in stratospheric adjustment act to compensate for part of differences in instantaneous forcing.

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The cause of the discrepancy between the double-call and kernel estimates of instantaneous forcing is not immediately clear. Because the radiative transfer model and climatological profiles used to develop the radiative kernels differ from that of the individual CMIP5 models, the kernel-method may not properly account for the contributions of the individual radiative feedbacks to the TOA fluxes. Such errors would result in erroneous estimates of the direct forcing term, since it is computed as a residual of the TOA fluxes and kernel-based calculations of radiative feedbacks and radiative adjustments. However, the adjusted forcing is also computed as the residual of the TOA fluxes and kernel estimates of the radiative feedbacks; the good agreement between the Gregory/Hansen estimates of adjusted forcing with those obtained using the radiative kernels suggests that the error in the kernel estimates is small (∼0.5 W m⁻²). Because the magnitude of radiative adjustments is an order of magnitude smaller than the radiative feedbacks, this implies that the uncertainties in computing the radiative adjustments from kernels are also an order of magnitude smaller (e.g. ∼0.05 W m⁻²). This is consistent with previous intercomparisons of radiative kernels which suggest that intermodel differences in kernel calculations are less than 10% (Soden et al 2008), although this comparison was limited to just four different kernels.

In this study, we compared four different methods for computing radiative forcing using climate model experiments performed for CMIP5. It is found that the adjusted forcing estimated from the alternative methods ranges from 5.5–9 W m⁻² for a quadrupling of CO₂ in climate models of CMIP5. Further analysis using radiative kernels and offline double-call radiative transfer calculations clearly indicates that a significant portion of the intermodel spread in adjusted forcing results from intermodel difference in the instantaneous forcing, confirming the argument of Chung and Soden (2015) that intermodel spread in the instantaneous forcing is likely related to biases in computing the radiative transfer and supports previous findings (e.g. Collins et al 2006, Forster et al 2011).

More in-depth analysis is required to attribute the causes of the intermodel spread in forcing. Potential causes that can be envisaged are uncertainty in spectroscopy or in the formulation of the radiative transfer. For instance, Iacono et al (2008) showed that correlated-k methods tend to overestimate CO₂ forcing relative to line-by-line models. Since the error that results from the approximations made in broadband radiative codes in climate models hampers an in-depth analysis, a coordinated intercomparison between double-call results and line-by-line calculations such as one conducted in Collins et al (2006) could help to reduce this uncertainty. The influence of the background state on the potential differences in instantaneous forcing also warrants further attention.

Our assessment of the intermodel spread in the instantaneous forcing from CO₂ is similar to that obtained by Collins et al (2006) for both the shortwave and longwave components. Collins et al (2006) documented that at the top of model the range of instantaneous forcing for a doubling of CO₂ is ∼1.2 W m⁻² for the longwave part of the electromagnetic spectrum and ∼0.5 W m⁻² for the shortwave part. These ranges, respectively, correspond to ∼2.4 W m⁻² and ∼1.0 W m⁻² for a quadrupling of CO₂ if the curve of growth of forcing with CO₂ holds. This agreement further supports the validity of the kernel methodology. The spread is significantly larger than that obtained by Collins et al using line-by-line calculations, indicating that the spread in forcing calculations does not reflect uncertainties in radiative transfer theory, but in the fidelity of its implementation in climate models.

Given the importance of the radiative forcing in climate change projections, we suggest that radiative forcing be a routinely archived diagnostic of climate model simulations. Including offline 'double-call' radiative transfer calculations of the instantaneous forcing is essential to better documenting the cause of the intermodel differences in the projected climate response.

We would like to thank two anonymous reviewers for their constructive and valuable comments which led to an improved version of the manuscript. We acknowledge the World Climate Research Programme's Working Group on Coupled Modeling, which is responsible for CMIP, and we thank the climate modeling groups (listed in table S1 of this study) for producing and making available their model output. For CMIP the US Department of Energy's Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. Model output analyzed in this study is available from the Earth System Grid Federation (http://cmip-pcmdi.llnl.gov/cmip5/). This research was supported by a grant from the NASA ROSES program and the NOAA Climate Program Office.