Within the context of the prediction, detection and attribution of climate change, a number of studies have explicitly or implicitly assumed that individual climate responses to individual forcing agents can be linearly added to obtain the total climate response to the sum of the forcing agents. This assumption of the ‘linear additivity of forcing-response relationships’ has been tested by previous studies, but it remains controversial whether linear additivity holds with all combinations of forcing agents, such as ‘greenhouse gases plus indirect effects of anthropogenic aerosols’ or ‘greenhouse gases plus solar irradiance’. This study explored whether linear additivity holds in 5-year mean temperature/precipitation responses to various combinations of forcing agents in the 20th century and in a future-emissions scenario at global and continental scales. We used Model for Interdisciplinary Research on Climate version 3, which includes the first and second indirect effects of aerosols. The forcing factors considered were well-mixed greenhouse gases, the direct and indirect effects of sulphate and carbon aerosols, ozone, land-use changes, solar irradiance and volcanic aerosols (the latter three factors were specified only in the 20th-century runs). By performing and analysing an enormous matrix of forcing runs, we concluded that linear additivity holds in temperature responses for all of the combinations of forcing agents at the global and continental scales, but it breaks down for precipitation responses in certain cases of future projections.
The sum of climate response patterns for different forcing agents is often assumed to be equal to the total climate response to those forcing agents. This concept, the linear additivity of the forcing-response relationship (Wigley, 1994), is one of the most basic assumptions underlying detection–attribution studies (Hegerl et al., 2007; Stott et al., 2010), observationally constrained projections of future climate change (Allen et al., 2000; Stott and Kettleborough, 2002; Stott et al., 2006; Kettleborough et al., 2007), and pattern-scaling climate change projections (Schlesinger et al., 1997; Hulme et al., 2000; Schlesinger et al., 2000; Wigley et al., 2000; Wigley, 2003). Previous studies have addressed the issue of linear additivity by performing equilibrium or transient climate simulations of different coupled atmosphere-ocean general circulation models (AOGCMs), atmospheric general circulation models (AGCMs) coupled to mixed-layer ocean models, or earth system models of intermediate complexity (EMIC). Studies have shown that linear additivity holds in the responses of temperature and/or precipitation to the combined effects of well-mixed greenhouse gases and direct aerosol in both the global mean regional scales [Ramaswamy and Chen, 1997 (using the GFDL R15 AGCM), Haywood et al., 1997 (using the GFDL R15 AOGCM), Gillett et al., 2004 (using the HadCM2 AOGCM)]. Several studies have shown that the global and/or regional linear additivity of temperature and/or precipitation responses break down when applied to combinations of the indirect effects of anthropogenic aerosols and greenhouse gases [Sexton et al., 2003 (using the HadAM3 AGCM), Feichter et al., 2004 (using the ECHAM4 AGCM), Ming and Ramaswamy, 2009 (using the GFDL AM2.1 AGCM)]. In contrast, Jones et al. (2007) have indicated that linear additivity holds when applied to the combined indirect effects of anthropogenic aerosols with greenhouse gases using the HadGEM1 AGCM.
In addition to the combinations of anthropogenic aerosols and greenhouse gases, using the Parallel Climate Model AOGCM, Meehl et al. (2004) examined various combinations of forcing agents in the 20th century, including well-mixed greenhouse gases, the direct effect of sulphate aerosols, ozone, volcanic aerosols and changes in solar irradiance. These researchers showed that linear additivity holds in the global mean temperature responses in various combinations of forcing agents. Using the UVic ESCM EMIC, Matthews et al. (2004) also failed to find any evidence of nonlinearity in the global mean temperature responses to combinations of greenhouse gases, the direct sulphate effect, land-use change, volcanic aerosol, solar variability and orbital changes. However, their AOGCM and EMIC did not include the indirect effects of aerosols. Combined indirect aerosol effects and other forcing agents may not conform to linear additivity. Meehl et al. (2003, 2009) suggested the possibility of nonlinearity for regional precipitation in response to combined solar irradiance and greenhouse gases. Thus, it is not clear that regional additivity holds under all combinations of forcing agents.
Detection–attribution analyses have been narrowed from the global scale to continental and sub-continental scales (Hegerl et al., 2007; Stott et al., 2010). Regional-scale climate projections based on the pattern-scaling method (Schlesinger et al., 1997; Hulme et al., 2000; Schlesinger et al., 2000; Wigley et al., 2000; Wigley, 2003) are important for climate change impact assessments and adaptation strategies. In this study, we tested the linear additivity of forcing-response relationships at the global and continental scales by analysing an enormous matrix of forcing runs in a state-of-the-art AOGCM called Model for Interdisciplinary Research on Climate version 3 (MIROC3) (K-1 model developers, 2004). This AOGCM explicitly considers the indirect effects of aerosols. Because nonlinearity may appear for large forcing agents in the future, we examined runs for 21st-century scenarios in addition to the 20th-century forcing runs. Land-use changes may have influenced past regional climate change (Matthews et al., 2004). Therefore, we also investigated linear additivity with regard to combinations of land-use change and other agents.
We used the MIROC3 AOGCM (K-1 model developers, 2004) for our simulations. The spatial resolution of the atmospheric component is T42 with 20 vertical levels (the height of the model is approximately 30 km), whereas the horizontal resolution of the ocean component is 1.4° in longitude by a variable of 0.56–1.4° in latitude with 44 vertical levels (21 levels in the upper 500 m). Flux adjustment is not required. This AOGCM includes an aerosol transport-radiation model called the Spectral Radiation-Transport Model for Aerosol Species (SPRINTARS) (Takemura et al., 2000; Takemura et al., 2002; Takemura et al., 2005), and it calculates the three-dimensional distribution of five types of aerosols: sulphate, black carbon, organic carbon, sea salt and soil dust. This model represents the direct, semi-direct, first indirect (cloud albedo) and second indirect (cloud lifetime) effects of aerosols.
In the 20th-century runs, the integration period was 1850–2000. We considered both anthropogenic and natural external forcing factors (Table 1) (Nozawa et al., 2005). The specified anthropogenic forcing agents are as follows:
Well-mixed greenhouse gases (CO2, CH4, N2O and 16 species of (H)(C)FCs) (Johns et al., 2003)
Stratospheric volcanic aerosols (Sato et al., 1993)
In the 21st-century runs (the integration period was 2001–2100; the initial conditions were taken from all the forcing runs of the 20th century), we specified the same anthropogenic forcing agents, except land-use change, in accordance with the A2 scenario of the IPCC Special Report on Emissions Scenarios (SRES) [Nakicenovic et al., 2000; Shiogama et al., 2010a (hereafter S10), 2010b]:
Well-mixed greenhouse gases
Black and organic carbon aerosols and
Tropospheric and stratospheric ozone
We computed a large matrix of ensembles forced by external forcing agents (Table 2). For the 20th-century runs, we performed runs with the following forcing agents: all forcing agents (GSCOLSolVol); anthropogenic forcing agents (GSCOL); natural forcing agents (SolVol); well-mixed greenhouse gases (G); sulphate aerosol (S); black and organic carbon aerosol (C); tropospheric and stratospheric ozone (O); land-use change (L); solar irradiance (Sol); volcanic (Vol); anthropogenic aerosols (SC); all forcing agents without anthropogenic aerosols (GOLSolVol); all forcing agents without carbon aerosols (GSOLSolVol) and all forcing agents without land-use change (GSCOSolVol). For the 21st-century runs, we performed runs with the following forcing agents: all forcing agents (GSCO); well-mixed greenhouse gases (G); sulphate aerosol (S); black and organic carbon aerosol (C); tropospheric and stratospheric ozone (O); all forcing agents without well-mixed greenhouse gases (SCO) and the year 2000 commitment runs (Com), in which all the external forcing agents were held constant at the conditions present in the year 2000. The ensemble sizes of these runs were 3, 4, 8 or 10 (Table 2), and we analysed their ensemble means. We also used the 3600-year, stable, pre-industrial control run (CTL) for statistical tests.
In the 20th-century forcing runs, we tested the linear additivity of the forcing-response relationships in the following combinations of forcing agents (Table 3):
Anthropogenic and natural forcing agents (GSCOLSolVol = GSCOL + SolVol)
Individual anthropogenic forcing agents (GSCOL =G + S + C + O + L)
Aerosols and other forcing agents (GSCOLSolVol =GOLSolVol + SC)
Carbon aerosols and other forcing agents (GSCOLSolVol = GSOLSolVol + C)
Sulphate aerosols and all forcing agents other than sulphate/carbon aerosols (GSOLSolVol =GOLSolVol + S)
Sulphate and carbon aerosols (SC = S + C)
Land-use changes and other forcing agents (GSCOLSolVol = GSCOSolVol + L), and
Solar and volcanic forcing agents (SolVol = Sol + Vol)
Table 3. The combinations of external forcing runs examined
20th-century forcing runs
GSCOLSolVol = GSCOL + SolVol
GSCOL = G + S + C + O + L
GSCOLSolVol = GOLSolVol + SC
GSCOLSolVol = GSOLSolVol + C
GSOLSolVol = GOLSolVol + S
SC = S + C
GSCOLSolVol = GSCOSolVol + L
SolVol = Sol + Vol
21st-century forcing runs (IPCC SRES A2 scenario)
GSCO = G + S + C + O − 3Com
GSCO = G + SCO − Com
SCO = S + C + O − 2Com
All of the 21st-century forcing-agent ensembles include the influence of climate change commitment in the 21st century in response to anthropogenic and natural forcing agents by the year 2000 (Wetherald et al., 2001; Wigley, 2005; Meehl et al., 2005; S10). Therefore, to test the additivities, we omitted the climate responses in Com from all of the runs. We examined the additivities of the forcing-response relationships in the following combinations of the 21st-century forcing agents (Table 3):
Individual anthropogenic forcing agents (GSCO = G +S + C + O − 3Com)
Well-mixed greenhouse gases and other anthropogenic forcing agents (GSCO = G + SCO − Com)
Sulphate aerosols, carbon aerosols and ozone (SCO =S + C + O − 2Com)
We analysed the 5-year mean temperature and precipitation changes in eight continental-scale land regions: North America, South America, Africa, Europe, Asia, Australia, Arctic Land and Antarctic Land. These continental-scale regions were based on those defined by Giorgi (2002), but the definitions of Arctic Land (north of 67.5°N) and Antarctic Land (south of 67.5°S) were based on Gillett et al. (2008). We also examined responses averaged over the global land and ocean areas. The analysed periods were 1900–1999 for the 20th century and 2001–2100 for the 21st century. We examined the anomalies from the 1900–1999 mean for the 20th century and from the 1980–1999 mean of the GSCOLSolVol runs for the 21st century. For each 100-year non-overlapped segment of the CTL, its average was subtracted to remove the possible influence of slight climate drifts in the long CTL run from the estimates of natural variability.
Figure 1 shows the 5-year mean 20th-century surface air temperature time series of all forcing runs (GSCOLSolVol) and the sum of the anthropogenic and natural forcing runs (GSCOL + SolVol). When we considered the 20th-century ensembles AB, A and B (with ensemble sizes of NAB, NA and NB, respectively), the time series of AB was compared with that of A + B. We estimated the 5–95% confidence range for differences at each pentad between AB and A + B. Instead of the standard deviation of the residual A + B − AB, that of the CTL was used for the t-test (Gillett et al., 2004). The standard deviation of the CTL was scaled by to allow for averaging over the ensemble members. The assumed degree of freedom was NAB + NA + NB − 3. In other words, the initial conditions of the 20th-century simulations, starting from the 100-year intervals in the controls, and the resulting 21st-century simulations were effectively independent. Most of the 5-year-mean/region values of GSCOL + SolVol fall within the 5–95% confidence range. The linearity of combining the anthropogenic and natural forcing agents does not break down in the 5-year mean temperature changes at either the global or continental scales. The same trend holds for the 21st-century temperature responses for the anthropogenic forcing runs (GSCO) and the sum of individual anthropogenic forcing runs (G + S + C + O − 3Com) (Figure 2). Of the 21st-century ensembles (AB, A and B), AB was compared with A + B − (2 − 1)Com. The standard deviation of CTL was scaled by , where NCom is the ensemble size of Com. The assumed degree of freedom was .
For the 20th-century precipitation responses of all the forcing runs (GSCOLSolVol) and the sum of the anthropogenic and natural forcing runs (GSCOL + SolVol), the additivity holds (Figure 3). In contrast, the precipitation responses in some regions in the G + S + C + O − 3Com run of the 21st century systematically depart from that of GSCO, although they still fall within the 5–95% confidence level range (e.g. Africa) (Figure 4).
For further objective assessments, we also performed an F-test on the ratio of the variance in the residual A + B − AB to that in the CTL. Here, we used the first half-period of the 3600-year CTL run, and its variance was scaled by the factor (1/NAB + 1/NA + 1/NB). The effective sample size in the CTL was estimated by considering the persistence in a first-order autoregression (Wilks, 1995), and we estimated the residual's value by scaling the CTL's value. Figure 5 shows the ratios of variance in the temperature responses. Linear additivity holds in most of the forcing combinations/regions. The proportions of cases lying outside the 5–95% significance levels for the F-test were 7/88 = 8% and 1/33 = 3% for the 20th and 21st-century ensembles, respectively. These anomalous residual variances may not represent clear evidence of nonlinearity because 10% of the regional values are expected to be outside the 5–95% significant levels by chance if the regions are independent of one another. The proportions of cases in which linear additivity appeared to fail in the precipitation responses are greater than those in the temperature responses, i.e. 17/88 = 19% (20th century) and 9/33 = 27% (21st century) (Figure 6).
The above assumption that the regions are independent of one another is false. The global-mean values are the area-weighted averages of the land-mean and ocean-mean values. Land-mean represents the area-weighted averages of all continental regions. Furthermore, in the climate system, considerable spatial correlations of the natural variability can lead to more or less than 10% of areas with significant residual variances. To evaluate the possibility that these proportions of significant cases result from the natural variability alone, we applied a field significance test (Livezey and Chen, 1983; Karoly and Wu, 2005). When we considered the residual A + B − AB, three 100-year segments (ABc, Ac, Bc) were randomly selected from the 18 100-year segments of the latter half of the CTL run. ABc, Ac and Bc were scaled by , and , respectively. The F-test was applied to the residual Ac + Bc − ABc. We performed the same analysis for all the forcing combinations, and we counted the number of cases with significant residual variance according to the F-test. We repeated these procedures 1000 times and counted the probability that these random samples produced proportions of significant cases at least as large as those found from the scenario studies. The probabilities are 33, 85, 10 and 6% that at least 7, 1, 17 and 9 cases would be significant according to the F-test as a result of the natural variability alone for 20th-century temperature, 21st-century temperature, 20th-century precipitation and 21st-century precipitation, respectively. Therefore, there is moderate evidence of nonlinearity only for 21st-century precipitation.
The significantly large residual variances in 21st-century precipitation are mainly observed when the well-mixed greenhouse gases are combined with the other forcing factors (Figure 6(i) and (j)) rather than the combinations of anthropogenic aerosols and ozone (Figure 6(k)). Figure 7 shows the residuals of precipitation and vertically integrated horizontal water vapour flux for G + S + C + O − 3Com minus GSCO (Figure 7(a)) and G + SCO − Com minus GSCO (Figure 7(b)) in the 21st-century ensembles. These patterns are similar to each other. If there are great decreases of precipitation about −100%/Century in some regions, the effects of bounded field (precipitation cannot fall below zero) can cause the nonlinear responses of precipitation. However, there is not such large decrease in GSCO, G, SCO and Com (not shown). The water vapour flux is transported from the Indian Ocean, Africa, the tropical Atlantic Ocean and the northeast of South America to the tropical Pacific region in the individual responses versus the combined response, leading to drying in the former regions and wetting in the tropical Pacific region. The mechanism for these changes in the horizontal distributions of water vapour and precipitation remains unknown and will be further investigated in a future study. It is suggested that nonlinearity is not negligible, at least in studies of the regional precipitation responses to well-mixed greenhouse gases and the other forcing factors in the 21st-century ensembles.
4. Summary and discussion
To test the linear additivity of surface air temperature/precipitation responses at the global and continental scales, we performed and statistically analysed a number of climate simulations of the MIROC3 AOGCM under various combinations of forcing agents from the 20th century and the IPCC SRES A2 scenario. We found little evidence of nonlinearity for the temperature and precipitation responses in the 20th-century simulations and the temperature responses in the 21st-century simulations. In contrast, linear additivity did not hold in the precipitation responses to the combinations of well-mixed greenhouse gases, anthropogenic sulphate/carbon aerosol emissions and changes in ozone in the 21st-century ensembles. This nonlinearity of precipitation changes under the greater forcing expected in the 21st century than that existing in the 20th century could cause significant errors in future climate projections derived from pattern-scaling methods based on the separated climate-response patterns associated with each forcing agent.
In the future projections, we did not include any changes in the solar irradiance or in volcanic aerosols. One proposed approach to geoengineering involves adding reflective aerosols to the stratosphere, mimicking the effect of volcanic eruptions (Ricke et al., 2010; Kravitz et al., 2011). Given the results found in this article, there could be substantial interactions between the precipitation response to other anthropogenic forcings and the response to such geoengineering approaches when performed on a large scale. Additionally, any geoengineering approach would involve an evolving modus operandi that depends, in part, on monitoring the progress of the impacts on anthropogenic climate change. The interpretation of this progress would have to account for nonlinear interactions between the responses of the time-evolving balance of anthropogenic greenhouse gas forcing, other ad hoc anthropogenic forcings, and the influence of geoengineering.
This work was funded by the Global Environment Research Fund (S-10) of the Ministry of the Environment of Japan, the Innovative Program of Climate Change Projection for the 21st century and a Grant-in-Aid 23310014 for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan. The Earth Simulator and an NEC SX-8R at NIES were utilized to perform the MIROC3 simulations. The authors declare no conflict of interest.