Model scope and calibration

FeliX models the complex interconnections among global human and natural systems to identify the probable economic and environmental impacts of trends, policies and technologies in the Anthropocene Era22,23. Fundamental linkages and feedbacks among demographic, economic, land, energy, carbon and climate systems are drawn from published models, articles and sector reports, and codified as differential equations. These equations describe the status and flow of resources, subject to geoclimactic and economic parameters to characterize the present state and co-dependent development of natural and economic systems.

The model is calibrated to available historical data between 1900 and 2015 (refs 24, 25). All FeliX model historical data, parameters, and results are calculated and reported as global averages.

Non-CO 2 greenhouse gases

Although the Paris Agreement encompasses all greenhouse gases, non-CO 2 emissions pathways are not modelled endogenously by FeliX. As a result, this analysis is limited to pathways for CO 2 mitigation. Emissions pathways and associated radiative forcing for non-CO 2 greenhouse gases (that is, CH 4 , N 2 O, HFC and ‘others’) are assumed to follow RCP 4.5, one of the IPCC’s four benchmark pathways for atmospheric greenhouse gas concentrations, through 2100 (refs 6, 7). The warming effects of alternative pathways for non-CO 2 emissions are plotted in Supplementary Fig. 4b.

Population and GDP

The model begins with the medium population projection from the United Nations Department of Economic and Social Affairs26 and historical data on global gross domestic product (GDP) as an indicator of global economic growth27.

Energy sector

Nominal energy demand per capita is linked sigmoidally with GDP per capita28.

The RE-Low and RE-High scenarios are loosely calibrated to plausible energy futures under climate action scenarios, as identified by the Global Energy Assessment (MESSAGE model: geala_450_atr_nonuc, geaha_450_atr_full; IMAGE model: GEA_low_450 and GEA_high_450)8. In this way, we establish soft links between FeliX, and MESSAGE and IMAGE, two specialized models of global energy systems8 (cf. Table 2).

Table 2 Primary energy consumption in year 2100 of FeliX and similar models. Full size table

Energy consumption in these pathways is compared with FeliX in Supplementary Table 3. In addition, we note that bioenergy production rates in the BAU and RE-Low scenarios are compatible with meta-analyses, including ‘medium agreement’ projections from the IPCC WG3, AR5 for year 2050, whereas bioenergy production in the RE-High scenario falls just beyond this range29,30.

Although there are significant uncertainties associated with projecting energy consumption several decades into the future, primary energy profiles are not FeliX model results for the purposes of this analysis. Rather, they are definitional of their respective scenarios, and are intended to illustrate correspondences between R AF and energy sector transformations of various magnitudes.

We model low and high shifts in total primary energy demand as variations on the Fossil Fuels, BAU, RE-Low and RE-High scenarios. Primary energy consumption in 2100 in these scenarios is indicated by the columns to the right of PE profiles in Fig. 2. Carbon dioxide emissions from oil, gas, coal, biomass, solar, and wind energy are calculated as the product of total consumption and carbon intensities which reflect the full life-cycle of each technology8. R AF and warming projections are calculated for each of these variations and presented in Fig. 1 as shaded ranges around each of the central values.

In scenarios with CCS, the technology is implemented not as a step function, but as a sigmoidal expansion over the course of the twenty-first century to the maximum value of 40% ( CCS) or 80% (CCS) of gross energy sector emissions. The expansion of CCS technology through 2100 is shown in Supplementary Fig. 1a.

Kaya factors

To facilitate comparison to other analyses, all FeliX scenarios are decomposed in Supplementary Fig. 6 into the four Kaya factors. In the population plot at top left, dotted lines indicate UNDESA high and low population variants26. In the GDP per capita (top right) and energy intensity of GDP (bottom left) plots, dotted lines project the continuation of trends from the recent past31.

LULUCF sector representation

Land in the FeliX model is distributed among four mutually exclusive and collectively exhaustive categories: agricultural, forest, urban/industrial and ‘other’. Each category is calibrated to FAOSTAT data on a global level (available for 1961–2010 for agricultural and 1990–2012 for forest and other land)24. Although not on a geographically explicit basis, land can be repurposed—most notably, due to changes in demand for agricultural land.

Land categorized as ‘agricultural’ is subdivided into arable land, permanent crops and permanent meadows and pastures. Arable land and permanent crops can be used to produce food, feed or energy crops, while permanent meadows and pastures are used only for feed production. The BAU scenario is calibrated to historical data available on FAO24. Crop and livestock yields are modelled endogenously as a function of input-neutral technological advancement, land management practices (that is, the expansion of high-input agriculture), water availability, pollution (including atmospheric carbon fertilization) and climate change.

Supplementary Fig. 1b plots FeliX model projections for crop yields in the BAU scenario. The model predicts an end to the steady expansion of agricultural land seen in the second half of the last century: through 2050, growth in demand for vegetal and animal products is likely to be satisfied by agricultural intensification. After midcentury, however, the cumulative effects of fertilizer saturation, water scarcity and ozone pollution may cause agricultural yields to stagnate or decline. As demand for food (in particular animal products) continues to grow, agricultural land may begin to expand indefinitely after 2050 at the expense of natural habitats.

Between 2010 and 2100, both supply- and demand-side trends lead to a 17% expansion in agricultural land. As land is a finite resource, transitions are zero sum. In the BAU scenario, agricultural land expands at the cost of natural habitats included in forests and ‘other’ land (that is, grassland). The general category of ‘forest’ includes both natural and managed stands, although emissions from conversion from natural to managed forest (and vice versa) are explicitly accounted for ref. 32.

LULUCF sector emissions

CO 2 emissions from the LULUCF sector include deforestation and forest conversion to managed forests and plantations, net of afforestation. All processes are calculated explicitly in the model as the net effect of global population growth, per capita energy and food demand, and agricultural yields2,32. Relative to energy sector emissions, LULUCF emissions are subject to very large uncertainties (cf. Fig. 4), as shown by divergences among annual and cumulative LULUCF emissions estimates from the IPCC, CMIP5, CDIAC and RCP database.

In the FeliX model, LULUCF emissions per unit area are calibrated using deforestation rates from the FAO24 and historical CO 2 emissions from the CDIAC33. This method produces a historical emissions pathway well within the error range as estimated by IPCC and CMIP5 analyses3,5. Future LULUCF emissions projections fall within the full range of RCP projections.

In the Fossil Fuels scenario, LULUCF emissions drop to zero at the end of the century as easily-deforested areas are depleted and the persistent predominance of fossil fuels limits land-use change for bioenergy production. Greater bioenergy demand in the BAU scenario leads to greater land-use change and associated emissions.

Much of the land-use change in the Fossil Fuels and BAU scenarios can be attributed to ‘unnecessary’ deforestation, which is forest loss from failure to optimize land use in ways that are technically possible. Unnecessary deforestation, an example of squandered resources, is land-use change driven by social and political constraints, including conflict, poor governance, perverse incentives, poverty and shortage of labour or capital9. Consistent with all four RCPs, the RE-Low and RE-High scenarios eliminate unnecessary deforestation based on the assumption that any significant future transition to REs will be coupled with enhanced protection of terrestrial landscapes and their carbon stocks3.

The nominal rate of input-neutral growth is shifted up and down to evaluate the impacts of alternative yields. The magnitude of these shifts is indicated by the shaded region in Supplementary Fig. 1b, but the effect of this shift is smaller than that of primary energy demand, and is therefore suppressed throughout figures in the main paper.

Carbon reservoir flux

The model calculates projected CO 2 emissions based on representations of carbon emissions from the energy and land-use change sectors, as discussed above. These emissions accumulate in the atmosphere until they are absorbed into the biosphere, pedosphere or oceans based on C-ROADS, a Simple Climate Model, which has been used extensively for climate policy impact analysis and decision making by parties to the UNFCCC34,35. Pathways and simplified equations for gross carbon flux among the reservoirs are illustrated in Fig. 7 and discussed below28.

Figure 7: Schematic illustration of gross flows through the global carbon cycle. At left: FeliX model formulas for calculating gross carbon flux from reservoir X to reservoir Y, Σ(X→Y) (PgC per year). FeliX model parameters are based on the C-ROADS model34 and are defined and discussed in further detail in the model’s technical documentation28. Full size image

Supplementary Figs 2 and 3 illustrates the ocean and land sink responses to increasing atmospheric carbon concentration. In Supplementary Fig. 2, cumulative uptake is plotted for both natural sinks relative to atmospheric carbon. In Supplementary Fig. 3, we plot the absolute (a) and fractional (b) responses of net sink flux to a range of constant emissions rates. For example, Supplementary Fig. 3b indicates that a 100% increase in annual emissions generates a 150% increase in net atmospheric carbon flux, an 80% increase in net carbon uptake by the land sink, and only a 50% increase in oceanic carbon uptake. Conversely, a 50% decrease in anthropogenic emissions generates a 66% decrease in net atmospheric carbon flux and 50 and 35% decreases in net land sink and ocean fluxes, respectively.

For the nominal ECS value (3.0 °C/2 × CO 2 ), FeliX projects a transient climate sensitivity of 2.4 °C/2 × CO 2 . For ECS=2.5 °C/2 × CO 2 , the transient climate sensitivity is 2.1 °C/2 × CO 2 .

Land sink carbon flux and feedbacks

Carbon flow from the atmosphere into the land sink begins from preindustrial net primary productivity (initial NPP=85.2 PgC per year). This initial carbon flux is coupled with atmospheric carbon concentration and global surface temperature via bio-stimulation (Λ C ) and climate (Λ T ) coefficients, respectively.

The bio-stimulation feedback mechanism introduces logarithmic NPP-carbon coupling as described in equation (1) (λ C =0.35). This mechanism increases land sink carbon uptake by 1.25 GtC p.p.m.−1 (cf. Table 3) such that a doubling of the carbon content of the atmosphere (C A ) relative to preindustrial generates a 24% increase in NPP before taking into account other feedbacks20.

Table 3 Parameters describing chemical and climate feedback to land and ocean sinks. Full size table

Within the land sink, the carbon stocks of the biosphere (plants) and pedosphere (soil) are modelled separately to account for distinct characteristic residence times. Gross carbon flow out of biomass is equal to biomass carbon stock (C B ) divided by a constant residence time (τ B =10.6 years). Treating this parameter as a constant neglects the impact of water and other nutrient availability in unmanaged ecosystems, which are not represented in the FeliX model20. This flow is distributed between the pedosphere and the atmosphere according to the constant biomass humification fraction (f H =0.428; cf. Fig. 7 (2 and 3)). Gross carbon flow out of the pedosphere is equal to the carbon content of the reservoir (C P ) divided by its residence time (τ P ). This residence time is coupled with climate, as discussed below.

Carbon flux through the land sink is linked with global temperature change at two points in the model, providing feedback to both NPP and pedosphere residence time. NPP–climate coupling is included in the nominal land sink carbon flux through the climate coefficient (Λ T in Fig. 7 (4) and equation (2) below), where the coefficient (λ T =0.012 K−1) is calibrated to match the average value in literature of the land sink-climate feedback mechanism4,5.

Second, the average residence time of carbon in soil ( =27.8 per year for preindustrial climate; cf. Fig. 7 (1)) is linearly coupled with climate, as shown in equation (3) below, where λ P =0.5 year K−1.

The net effect of the land sink-climate coupling reduces land sink carbon uptake by 66 PgC K−1 (cf. Table 3)4. In year 2050 of the BAU scenario, this effect reduces global soil carbon reserves by 26 PgC (measured relative to BAU without the coupling. This figure is in line with the latest global estimates (30±30 PgC)36.

Net land sink carbon flux is plotted for all scenarios in Fig. 5c. The cycling and availability of nutrients including N, P and water represent another important feedback to land sink carbon flow in general and NPP in particular20. These considerations tend to limit NPP response to atmospheric carbon concentrations and should be included in subsequent iterations of this analysis.

In deep decarbonization scenarios, net carbon flux into the ocean and lands reservoirs switches directions, turning these sinks into carbon sources. The simplest physical explanation for this effect is that rapid decarbonization reverses the sign, or direction, of the chemical coupling between the atmosphere, ocean and land sink. This coupling is responsible for increasing the net carbon flux into natural sinks as emissions rise; thus, it should also be expected to cause a net carbon flow out of these sinks if net-negative anthropogenic emissions are achieved and atmospheric carbon concentration begins to drop.

From another perspective, we could invoke Le Châtelier’s principle to predict that the earth system will act to ‘resist’ change. In deep decarbonization scenarios, the ‘change’ is the reduction in atmospheric carbon concentration resulting from net-negative anthropogenic emissions and the natural ‘response’ is the net flow of carbon out of the land and ocean sinks.

Ocean carbon flux and feedbacks

In addition to cycling through terrestrial reservoirs, carbon is removed from the atmosphere through dissolution into the mixed ocean layer (depth 0–100 m) and subsequently propagates through four independently modelled deeper layers (100–400, 400–700, 700–2,000 and 2,000–2,800 m).

The equilibrium dissolved inorganic carbon content of the mixed ocean layer (C ML eq.) is given by equation (4), where is the preindustrial carbon content of the oceans, C A is the present carbon content of the atmosphere and is the preindustrial carbon content of the atmosphere. The carbon content of the mixed layer (C ML ) is assumed to reach equilibrium with the atmosphere with a constant characteristic mixing time of 1 year (τ ML =1 year).

The ocean-climate coupling (Λ O ; λ O =0.0045 K−1; cf. equation (5)) reduces carbon uptake by 46 GtC K−1 (cf. Table 3).

Finally, the ocean-carbon coupling is expressed by the dimensionless Revelle factor (ζ), which expresses the marginal capacity of the oceans to absorb carbon (δ R =0.0045). The Revelle factor increases logarithmically with the carbon content of the oceans, rising from its initial value (ζ I =9.7) to 10.9 in year 2010 of the simulation (cf. equation (6))4. The ocean-carbon coupling increases carbon uptake by 1.23 GtC p.p.m.−1 (cf. Table 3).

Climate gain

Table 3 presents FeliX model parameters including climate sensitivity (α), land sink sensitivity to carbon (β L ) and climate (γ L ), and ocean sensitivity to carbon (β O ) and climate (γ O ), calculated for 2100 as in the Coupled Climate Carbon Cycle Model Intercomparison Project (C4MIP)4,5. Overall gain (g) of the climate system, which quantifies the ratio of temperature change due to these feedback loops to total temperature change, is shown in the column at the far right4,37. For comparison to similar models, we also show the average value for each parameter from the ensemble of 11 models included in the same study. All FeliX parameters show satisfactory agreement with the carbon flux drivers and feedbacks as modelled in this ensemble and in the subsequent iteration, CMIP5 (ref. 5).

Temperature anomalies

HadCRUT4 data on global surface temperature anomaly are used for results validation, and represent observed temperature increases relative to the period (1850–1900) from the Met Office Hadley Center38.

Global surface temperature change is affected by radiative forcings, feedback cooling due to outbound longwave radiation, and heat transfer from the atmosphere and mixed ocean layer to the four deep ocean layers. Net radiative forcing is calculated from the concentration of carbon in atmosphere, a product of CO 2 emissions from the energy and LULUCF sectors and other greenhouse gases, including CH 4 , N 2 O, halocarbons, and other gases and aerosols. Endogenous projections of atmospheric carbon concentration are used to model associated radiative forcing anomaly for carbon dioxide. Forcing anomalies associated with other greenhouse gases are modeled exogenously using RCP 4.5 (ref. 7). Total greenhouse gas forcing is translated into temperature anomalies as in the C-ROADS model34,35 (cf. Fig. 6).

A negative feedback loop incorporates heat transfer from the atmosphere and the upper ocean into space via outbound longwave radiation. The magnitude of this feedback, or cooling, is determined by the ECS, a metric used to characterize the response of the global climate system to a given forcing. ECS is broadly defined as the equilibrium global mean surface temperature change following a doubling of atmospheric CO 2 concentration. In the FeliX model, ECS is nominally set equal to 3.0 °C/2 × CO 2 . In Supplementary Fig. 4a, temperature anomaly projections are shown for the BAU scenario over the full range of probable values for ECS (1.5–4.5) as identified by the IPCC11. In Supplementary Fig. 5, we plot R AF and ΔT projections based on ECS′=2.5 °C/2 × CO 2 . This range of values is consistent with Coupled Climate Carbon Cycle Model Intercomparison Project results4 and the latest estimates from IPCC11.

R AF definition and error propagation

We begin by identifying carbon sources X={FF, LUC, RE} and sinks Y={O, LS} as shown in Supplementary Table 1. Net annual carbon flux from each emissions source (X) is labeled ɛ X , while net annual flux into each reservoir (Y) is labeled Ω Y . For all sources (X), positive values of ɛ X indicate net positive emissions. For all reservoirs (Y), positive values of Ω Y indicate net uptake of carbon. Net annual increases in atmospheric carbon are defined as the sum of emissions and sink fluxes as shown in equation (7):

In accordance with the COP21 text, we define R AF in equation (8):

This equation is used to calculate R AF for all scenarios, as plotted in Fig. 1. We calculate the error on R AF in the standard manner:

Taking the partial derivatives:

We arrive at:

We use equation (12) to calculate errors on R AF using the relative errors measured by the IPCC (cf. Supplementary Table 1). This is a very conservative projection, given the probable advancement of global carbon monitoring technologies and techniques. For scenarios with CCS, errors are calculated on gross emissions from REs.

Data availability

The most recently published version of the FeliX model is freely available for download and use at the model website32. The version of the model used for this analysis will be made available on the same site upon publication of the manuscript. The authors agree to make all scenarios used in this analysis available upon request.