1. Introduction
The global carbon cycle is a crucial component of future climate change, closely linking anthropogenic CO2 emissions with future changes in atmospheric CO2 concentration and hence climate (Denman et al. 2007; Ciais et al. 2013). Inclusion of the carbon cycle as an interactive component in comprehensive Earth system models (ESMs) has grown since early coupled studies (Cox et al. 2000) and intercomparisons such as the Coupled Carbon Cycle–Climate Model Intercomparison Project (C4MIP; Friedlingstein et al. 2006) and is now a mainstream component of coordinated climate simulations like phase 5 of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012).
Such coupled climate–carbon cycle ESMs simulate the natural exchange of carbon by the land and ocean with the atmosphere and thus provide a predictive link between emissions and atmospheric concentrations of CO2. They can be used to compute the emissions required to follow a prescribed concentration pathway (Jones et al. 2006; Plattner et al. 2008). This method has become widespread and was recommended by Hibbard et al. (2007) as the experimental design for CMIP5 and has subsequently been used to present compatible emissions from the CMIP5 multimodel ensemble (Jones et al. 2013).
The natural uptake of carbon by land and ocean biospheres is sensitive to both changes in climate and the concentration of atmospheric CO2 and the balance between these large but offsetting effects on decade–century time scales is not well constrained by observations. Comparison between the C4MIP ESMs showed large quantitative uncertainty in the future projections of carbon uptake (Friedlingstein et al. 2006), similarly seen in perturbed parameter simulations (Booth et al. 2012). This spread of results across global climate models (GCMs) has not reduced substantially in CMIP5 (Arora et al. 2013; Jones et al. 2013).
To improve the understanding of future land–atmosphere and ocean–atmosphere exchange of carbon, it is imperative to attribute the variation in these fluxes to their component sources. “GCM variability” originates from an incomplete understanding of physical processes including both climate and ecosystem processes, involved in air–surface carbon exchange and from the limitation of GCMs to represent known behavior. “Scenario variability” arises from uncertainty in future human activity; socioeconomic storylines of population and technology growth are produced by integrated assessment models (IAMs; see van Vuuren et al. 2011b) to provide plausible scenarios of future anthropogenic activity such as energy use (and hence fossil fuel emissions) and land-use change. The “internal variability” of a given GCM represents the natural variability of the climate at daily to multidecadal time scales (Karoly and Wu 2005), owing to the chaotic and nonlinear nature of the carbon flux processes; this variability has been long observed even in a stationary climate (Madden 1976). There also exists the possibility of “GCM–scenario interaction” if the differences in simulated climate between GCMs vary between scenarios.
The aim of this study is to quantify the relative importance with time of GCM and scenario variability and to estimate the future time beyond which scenario variability dominates GCM variability. The importance of the GCM–scenario interaction term will be quantified as a tool to understanding the response of GCMs to different scenarios.
Analysis of variance (ANOVA; Gelman 2005) has been used in several climate studies for quantifying sources of variability (von Storch and Zwiers 2001; Tingley 2012) and more recently has been effectively used to diagnose variability in multimodel ensembles (Yip et al. 2011; Sansom et al. 2013; Hingray and Saïd 2014). The opportunistic nature of the analysis of the CMIP5 carbon fluxes has resulted in a varying number of runs for each GCM–scenario pair (some of which have zero runs)—thus an unbalanced factorial design. An ANOVA method appropriate for this data (Northrop and Chandler 2014) has been used to partition the different sources of variability.
a. RCP scenarios
A set of four representative concentration pathways (RCPs) were developed to provide a common set of future climate scenarios to the scientific community, which would allow for better comparisons between ESM/GCM studies and for ease of communication of GCM results (van Vuuren et al. 2011b). The four RCP scenarios—RCP8.5 (Riahi et al. 2011), RCP6.0 (Masui et al. 2011), RCP4.5 (Thomson et al. 2011), and RCP2.6 (van Vuuren et al. 2011a)—lead to an approximate increase in global radiative forcing by the year 2100 of 8.5, 6.0, 4.5, and 2.6 W m−2, respectively. The scenarios are sufficiently separated in terms of the radiative forcing pathways to provide distinguishable climate results at the global scale (Moss et al. 2010). The RCP scenarios have a harmonized historical period, assumptions for carbon emissions and concentrations, and land-use change.
The dominant driver of future radiative forcing for each RCP is the CO2 concentration pathway (Fig. 1a) along with the fossil fuel emissions associated with that pathway (Fig. 1b) from each IAM that generated the scenario. According to the concentration-driven experiment design in CMIP5 (Taylor et al. 2012), each GCM performs simulations from a preindustrial state (typically representative of 1850) up to 2100, using these CO2 concentrations as a boundary condition to force the GCM.
Shown for each RCP scenario are future (a) atmospheric CO2 concentration pathway, (b) fossil fuel CO2 emissions, (c) crop and pasture land fraction (i.e., the anthropogenic land-use change), and (d) anthropogenic land-use change CO2 emissions [(a)–(c) courtesy of Jones et al. (2013)].
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
Shown for each RCP scenario are future (a) atmospheric CO2 concentration pathway, (b) fossil fuel CO2 emissions, (c) crop and pasture land fraction (i.e., the anthropogenic land-use change), and (d) anthropogenic land-use change CO2 emissions [(a)–(c) courtesy of Jones et al. (2013)].
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
Shown for each RCP scenario are future (a) atmospheric CO2 concentration pathway, (b) fossil fuel CO2 emissions, (c) crop and pasture land fraction (i.e., the anthropogenic land-use change), and (d) anthropogenic land-use change CO2 emissions [(a)–(c) courtesy of Jones et al. (2013)].
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
Such results are of relevance to policy decisions that aim to achieve a given climate target, but they are subject to large uncertainty (Jones et al. 2013). We want to understand the causes of this uncertainty in compatible emissions. The scenarios are designed to be different—they were selected from hundreds of possible scenarios and approximately span the 10th to 90th percentiles of future radiative forcing across published scenarios. They represent very different societal choices around climate targets and how to achieve them. Hence, it is desirable that the consequences of these choices can be distinguished in terms of their impacts on global and regional climate and ecosystems. We might therefore (by the year 2100) expect the differences between scenarios to be bigger than the differences between GCMs whose aim is to represent the same processes (Cox and Stephenson 2007). For example, the variability in global average temperature by 2100 was greater across scenarios than between GCMs for the CMIP3 ESMs running the SRES family of scenarios (Hawkins and Sutton 2009). However, at regional scales this is not always true (e.g., over the British Isles by 2100; Hawkins and Sutton 2009). Our study has explored the variability in carbon cycle behavior between different GCMs and scenarios, along with their interaction, and how this varies by process, by region and through time.
b. Land-use change in the RCP scenarios
While both ocean and land carbon fluxes respond to change in climate and CO2, terrestrial carbon storage is additionally influenced by direct anthropogenic activity to modify vegetation cover. The RCPs include changes in anthropogenic land use during the twenty-first century, and the ESMs attempt to simulate this although with varying degrees of complexity and completeness of process representation (Hurtt et al. 2011; Jones et al. 2013). Differences in land-use change contribute to the spread of results across scenarios, and differences in the representation of land use contribute to the spread between ESMs.
The degree of land-use change in the scenarios is not closely related to global radiative forcing, and so the relative importance of land use in the global carbon balance changes in time differently between scenarios. Figure 1c shows the global-scale evolution of land use in the scenarios in terms of fraction of land given over to agriculture (crop and pasture). The scenarios differentiate between agricultural land for crops and for pasture, whereas GCMs differ in how they treat these two classes of land. A common feature is that conversion of forest land to agricultural land involves the removal of large amounts of carbon (as tree biomass). Subsequent changes of soil carbon and the response of carbon storage to management practices differ between ESMs. This might be expected to lead to a strong GCM–scenario interaction and will be explored explicitly in the results section.
Global time series up to 2100 hide the time evolution of land use at regional scales, which can differ markedly between regions. Unlike global climate and CO2, land use in the RCPs changes most markedly in the early decades of the twenty-first century. We expect the carbon cycle impact from CO2 and climate change to continue to increase under most scenarios throughout the twenty-first century and perhaps to be more similar across scenarios during the early decades. For prescribed land-use change, however, it is the case that the scenarios diverge rapidly and differ most in the early decades with reduced land-use forcing by the end of the century (Fig. 1d).
2. Data
The monthly fields of carbon mass flux out of atmosphere due to net biospheric production on land (output variable name nbp) and surface downward CO2 flux (output variable name fgco2) from seven GCMs participating in CMIP5 have been extracted for this work (Taylor 2013; CCCma 2015; Dunne et al. 2014; Sanderson et al. 2014; Denvil et al. 2016; JAMSTEC et al. 2015; Giorgetta et al. 2012; Bentsen et al. 2012). These GCMs were selected as they had good coverage of the four future scenarios and/or multiple runs for each GCM–scenario pair. Where a particular GCM has been designed with and without coupled atmospheric chemistry (e.g., MIROC-ESM and MIROC-ESM-CHEM), only one version of the GCM was included; it was felt that including multiple versions of a particular GCM could artificially alter the GCM variance. The seven GCMs included in the analysis are listed in Table 1.
Number of runs of each GCM–scenario pair, made available for download. (Acronym expansions are available online at http://www.ametsoc.org/PubsAcronymList.)
In the analysis we have aggregated the monthly fields into decadal atmosphere–ocean carbon fluxes (GtC decade−1) and decadal atmosphere–land carbon fluxes (GtC decade−1); the rational for the choice of time step is given in the discussion (section 5). The first (e.g., decadal atmosphere–land carbon flux) time step was obtained by summing all the monthly (e.g., fgco2) fields over the area of interest for the period 2006–15. The decadal atmosphere–land carbon fluxes can also be thought of as the decadal change in stored carbon. In the analysis we have used raw projection carbon flux data as opposed to change projection data; this is justified in section 5.
The data will be analyzed for the period 2006–95 where all of the simulations listed in Table 1 contain data (the historical period in the GCMs covers the years 1861–2005, and the future period continues to at least 2100 and beyond for some of the GCMs).
3. Methodology
The unbalanced design of the CMIP5 experiment, having different numbers of runs for each combination of GCM i and scenario j, complicates the application of classical ANOVA techniques (e.g., Yip et al. 2011). Unbalanced designs can be addressed using multiple regression (Searle 1987; Sansom et al. 2013). However, the variance attributed to each component depends on the order that the components are entered in to the regression (Davison 2003, chapter 8.5). Therefore, we follow Northrop and Chandler (2014) and use a random effects ANOVA to accommodate the unbalanced design.
The GCM differences αi are modeled as independent, identically distributed normal random variables with mean zero and variance 
The interpretation of this framework is that the GCM effects αi, scenario effects βj, interaction effects γij, and internal variability εijk are samples from some larger superpopulations (Stephenson et al. 2012). Therefore, the variances 
It is also possible to compute variances for the specific sample of GCMs and scenarios being analyzed—for example, 
We are primarily interested in the superpopulation variances since there are many other climate models and scenarios we could consider if data were available. However, the finite-population variances provide a useful alternative view from the current sample.
We follow Northrop and Chandler (2014) and take a Bayesian approach to estimating the population variances. In a Bayesian analysis, we are required to specify our prior beliefs about the quantities of interest. We then update those beliefs after observing the data (i.e., the CMIP5 runs). A vague normal prior (normal with large variance) was specified for the overall mean μ of each flux.
Vague inverse-gamma priors are a common choice for variance parameters in normal models. However, with only seven GCMs and four scenarios in the ensemble, there is limited information to quantify the population variances 
The half-Cauchy distribution has the advantage of concentrating less of the prior mass close to zero than the inverse gamma distribution (Gelman 2006). By controlling the scale parameter A, we can spread the prior mass of the population standard deviations over a plausible range, limiting the possibility of either unreasonably small or large estimates of the population standard deviations (Gelman 2006).
The prior scale parameter A was approximated from the range in decadal fluxes in the year 2100 in Figs. 2a, 3a, 3e, 4a, 4e, 5a, 5e, and 5i. Following Northrop and Chandler (2014), the scale parameter was set to one-quarter of the approximated range (since in the Gaussian approximation 95% of the data should be within two standard deviations of the mean). Values for figures are given in Table 2.
(a) The decadal CO2 fossil fuel emissions for all GCMs and scenarios; (b) the corresponding standard deviation of 
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
(a) The decadal CO2 fossil fuel emissions for all GCMs and scenarios; (b) the corresponding standard deviation of
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
(a) The decadal CO2 fossil fuel emissions for all GCMs and scenarios; (b) the corresponding standard deviation of
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal (a)–(d) ocean and (e)–(h) land carbon fluxes for the period 2006–95. A subset of GCMs is highlighted for the global atmosphere–land carbon fluxes in (e).
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal (a)–(d) ocean and (e)–(h) land carbon fluxes for the period 2006–95. A subset of GCMs is highlighted for the global atmosphere–land carbon fluxes in (e).
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal (a)–(d) ocean and (e)–(h) land carbon fluxes for the period 2006–95. A subset of GCMs is highlighted for the global atmosphere–land carbon fluxes in (e).
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal (a)–(d) Southern Ocean and (e)–(h) North Atlantic Ocean carbon fluxes for the period 2006–95.
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal (a)–(d) Southern Ocean and (e)–(h) North Atlantic Ocean carbon fluxes for the period 2006–95.
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal (a)–(d) Southern Ocean and (e)–(h) North Atlantic Ocean carbon fluxes for the period 2006–95.
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal carbon fluxes of (a)–(d) northern high-latitude land, (e)–(h) northern temperate land, and (i)–(l) tropical land for the period 2006–95. A subset of GCMs is highlighted for the decadal carbon fluxes in (a),(e),(i).
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal carbon fluxes of (a)–(d) northern high-latitude land, (e)–(h) northern temperate land, and (i)–(l) tropical land for the period 2006–95. A subset of GCMs is highlighted for the decadal carbon fluxes in (a),(e),(i).
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
As in Fig. 2, but for decadal carbon fluxes of (a)–(d) northern high-latitude land, (e)–(h) northern temperate land, and (i)–(l) tropical land for the period 2006–95. A subset of GCMs is highlighted for the decadal carbon fluxes in (a),(e),(i).
Citation: Journal of Climate 29, 20; 10.1175/JCLI-D-16-0161.1
Values of prior scale parameter A for figures.
The same scale parameter A was used for all three standard deviations 
Estimating the prior scale parameter from the data is a double use of the data. However, no independent source was available, and the prior is designed only to provide a mild constraint on plausible values of the population standard deviations, so the compromise is acceptable.
4. Results
We follow the convention for graphically comparing variance components using stacked fractional variance plots (Hawkins and Sutton 2009). The fractional variances are based on the posterior medians of the superpopulation variances 
a. Variability in CO2 emissions
From the terms in Eq. (3), the compatible emissions are strongly influenced by the change in atmospheric CO2, which is prescribed as a common forcing to all GCMs and is larger, by a factor of approximately 2–3, than the change in land and ocean carbon storage. Hence, the dominant term in the compatible emissions does not vary between GCMs. This leads to the striking result that the variability in emissions between GCMs is much smaller than the variability between scenarios. However, there is still substantial GCM spread in the compatible emissions for each scenario, and it is desirable to investigate the components of this.
b. Variability in global CO2 fluxes
Subjectively, it is clear that the multimodel ensemble of global land and ocean carbon fluxes behave differently (Fig. 3). For decadal atmosphere–ocean carbon fluxes, scenario variance 
It is interesting to see that the posterior median of the scenario standard deviation 
c. Variability in regional ocean CO2 fluxes
Two key ocean regions of CO2 uptake were considered: the North Atlantic Ocean and the Southern Ocean. The regional extents of these ocean regions are given in Table 3. The main uptake of atmospheric CO2 in the Southern Ocean takes place in the latitudes 60°–40°S (Le Quéré et al. 2000, 2007; Takahashi et al. 2002), but there is much greater variability in the simulated fluxes below 60°S (Lenton et al. 2013). The noticeable difference in the fractional variance plots between Southern Ocean (Fig. 4d) and global ocean (Fig. 3d) carbon fluxes is that 
Extent of regions where carbon fluxes were analyzed.
The North Atlantic Ocean has been identified as a key sink of atmospheric CO2 (Schuster and Watson 2007; Takahashi et al. 2002). The North Atlantic Ocean has unusually large differences between simulated carbon fluxes, as can be seen in the bunching by GCM in these fluxes (Fig. 4e) compared with the bunching by scenario in the global ocean fluxes (Fig. 3a). This explains why 
d. Variability in regional land CO2 fluxes
We performed ANOVA for the three land regions defined in Table 3. The behavior of the northern temperate and northern high latitudes is similar. In the early decades, the carbon fluxes bunch together closely by GCM (represented by the line styles in Figs. 5a,e). Later in the century, the contribution of the GCM–scenario interaction 
In the tropical land region, different scenarios are already displaying different CO2 flux behaviors in the first decade (Fig. 5i) with 
5. Discussion
We made a decision to aggregate the monthly fields into decadal atmosphere–ocean carbon fluxes and decadal atmosphere–land carbon fluxes (see section 2). We are not interested in trying to predict carbon fluxes for individual years; instead, we want to know how long-term cumulative changes in natural stores of carbon affect anthropogenic emissions and/or climate. For this we want to smooth out annual- and subannual-scale variability, but we still care about how much the decadal-scale fluxes are dependent on the initial conditions. The ocean uptake looks much the same on annual (not shown) and decadal time steps. However, the internal variance of annual land carbon fluxes (not shown) is very large, in agreement with expectations, owing to the tropical land response to El Niño–Southern Oscillation (ENSO)-induced climate fluctuations (Jones et al. 2001; Cox et al. 2013). Overall, more robust estimates of the variance components of land carbon fluxes were obtained from the decadal time-step.
We have chosen to use raw projection carbon flux data in the analysis since model differences at 2006 represent genuine uncertainty. For ocean uptake, observational estimates in the 1990s suggest a mean ocean CO2 sink of 2.2 ± 0.4 GtC yr−1, which has a very similar uncertainty to the spread of GCM global ocean carbon fluxes of 2.0 ± 0.4 GtC yr−1 given by Le Quéré et al. (2015). There are no observational constraints on land uptake of carbon in the present; these were instead estimated from the residual of CO2 emissions, which are neither absorbed by the ocean nor remain in the atmosphere (Le Quéré et al. 2015).
Our simple ANOVA model treats each time point separately, so the results may not be robust in the presence of large internal variability; that is, the fractional variance plots may be very noisy rather than varying smoothly. In that case, it might be more appropriate to use a time series ANOVA approach such as the heuristic polynomial method of Hawkins and Sutton (2009) or the quasi-ergodic approach of Hingray and Saïd (2014). In the present work, such a time series approach would likely have produced robust estimates of the variance terms if using variables with annual rather than decadal time steps.
In Eq. (1) the GCMs are assumed to be independent. However, there are likely to be similarities in the way certain processes are parameterized. In that case, 
In contrast to the GCMs, the RCP scenarios are designed to span the 10th and 90th percentiles of likely future radiative forcing. They do not constitute a random sample from the space of possible future scenarios (i.e., they are unlikely to be independent). As a result, the superpopulation variance 
6. Conclusions
a. Interpretation of results
An ANOVA method was used to quantitatively partition the compatible fossil fuel emissions and their component carbon fluxes in terms of their variability across scenarios and between GCMs. For the compatible emissions, the spread of results is dominated by spread across scenarios (scenario variability overtakes GCM variability in the late 2020s), at least partly owing to the large role of prescribed CO2 concentration in the compatible emissions. The CO2 pathways in the RCPs are sufficiently distinct that the scenarios are more different from each other than the differences between GCMs. However, there is still variability between GCMs and therefore uncertainty in how to achieve a given pathway owing to natural land and ocean carbon uptake.
Scenario variability overtakes GCM variability in global ocean fluxes around 2025, while GCM variability remains dominant over scenario variability beyond 2100 for global land fluxes. There are large regional variations in fractional variance (e.g., GCM variability dominating beyond 2100 in the North Atlantic Ocean).
Although the focus here is on simulated carbon fluxes, this explicitly includes the GCM representation of climate processes that will differ regionally and by quantity (temperature, precipitation, etc.); that is, the simulated carbon fluxes may differ owing to different simulated climate as well as owing to differences in carbon cycle representation. Studies that have run vegetation models offline forced by prescribed climate inputs have found substantial differences from both multiple land surface models driven by the emulated climate of a single common GCM (Sitch et al. 2008) and emulated climate changes of multiple GCMs (Huntingford et al. 2013). Similarly, changes to the land surface within a common climate model framework have been shown to drive a broad spread in carbon responses (Booth et al. 2012).
At the moment land carbon flux output from different GCMs are not sufficient to reliably distinguish the consequences of different CO2 pathways, with the disagreement between GCMs growing increasingly large from RCP2.6 to RCP8.5. The reasonable agreement in land carbon fluxes between GCMs in the present day is likely the result of constraining carbon fluxes to observations. On the other hand, the global intramodel agreement suggests that we can be more confident in linking different future socioeconomic pathways to consequences of ocean carbon uptake (such as ocean acidification) than we can for consequences of land carbon uptake.
To aid decision making, GCM output is of use if it allows a distinction to be seen between different courses of action (such as land-use choices or emissions reductions). If GCM variance is greater than scenario variance it reduces confidence in statements around the difference between scenarios and reduces the utility of the GCMs and the simulations. Where the scenarios are more different than the GCM variance this implies we have some confidence in the differences in climate outcomes owing to socioeconomic choices. The smaller we can make the GCM variance, the finer the distinction we can make between policy options.
b. Ideas for further work
In the RCP scenarios there is a progression of radiative forcing from low to high, but the degree of land-use change does not follow this with different rates and even different signs of changes between scenarios and in particular regions. Land-use change storylines also diverge more quickly than CO2 concentration and climate across the RCPs and are likely therefore to be responsible for the early twenty-first-century changes. It has not been possible to sample the range of land-use change scenarios independently of future CO2 scenarios in this analysis; it would be of great use if a designed set of GCM runs could sample this range.
Acknowledgments
MOHC authors were supported by the Joint DECC/Defra Met Office Hadley Centre Climate Programme (GA01101). SY was supported by the Hong Kong Polytechnic University grant “Bayesian Modelling for Quantifying Uncertainty in Climate Predictions” (1-ZV9Z). We are grateful for the feedback from the editor and reviewers, which has greatly improved on the initial work. We acknowledge use of the R software package (R Core Team 2013). We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups for providing their GCM output (listed in Table 1). Support of this dataset was provided by the Office of Science, U.S. Department of Energy.
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