Abstract

Climate projections reveal global-mean surface warming increasing nearly linearly with cumulative carbon emissions. The sensitivity of surface warming to carbon emissions is interpreted in terms of a product of three terms: the dependence of surface warming on radiative forcing, the fractional radiative forcing from CO2, and the dependence of radiative forcing from CO2 on carbon emissions. Mechanistically each term varies, respectively, with climate sensitivity and ocean heat uptake, radiative forcing contributions, and ocean and terrestrial carbon uptake. The sensitivity of surface warming to fossil-fuel carbon emissions is examined using an ensemble of Earth system models, forced either by an annual increase in atmospheric CO2 or by RCPs until year 2100. The sensitivity of surface warming to carbon emissions is controlled by a temporal decrease in the dependence of radiative forcing from CO2 on carbon emissions, which is partly offset by a temporal increase in the dependence of surface warming on radiative forcing. The decrease in the dependence of radiative forcing from CO2 is due to a decline in the ratio of the global ocean carbon undersaturation to carbon emissions, while the increase in the dependence of surface warming is due to a decline in the ratio of ocean heat uptake to radiative forcing. At the present time, there are large intermodel differences in the sensitivity in surface warming to carbon emissions, which are mainly due to uncertainties in the climate sensitivity and ocean heat uptake. These uncertainties undermine the ability to predict how much carbon may be emitted before reaching a warming target.

1. Introduction

Climate model projections reveal a simple emergent relationship for our future climate: surface warming increases nearly linearly with the cumulative CO2 emitted since the preindustrial age. This relationship has been illustrated in terms of how surface warming increases with the cumulative CO2 emission using a climate model emulator (Allen et al. 2009), an Earth system model of intermediate complexity (Matthews et al. 2009; Zickfeld et al. 2009), and a suite of Earth system models forced by an annual rise in atmospheric CO2 (Gillett et al. 2013).

Our aim is to understand how the sensitivity of surface warming to cumulative carbon emissions from fossil fuels is controlled: this sensitivity ΔTI is defined by the ratio of the change in global-mean surface air temperature ΔT (K) and the change in fossil-fuel cumulative carbon emission ΔI (in petagrams of carbon denoted by PgC) since the preindustrial age. This sensitivity of surface warming to carbon emissions relates to two climate metrics: for experiments with only atmospheric CO2 forcing, ΔTI is identical to the transient climate response to cumulative carbon emissions (TCRE) on decadal to centennial time scales (Gillett et al. 2013; Collins et al. 2013) and approaches the equilibrium climate response to cumulative carbon emissions (ECRE) on centennial to millennial time scales (Frölicher and Paynter 2015).

The sensitivity of surface warming to carbon emissions is examined here, exploiting our recent theory highlighting the role of the ocean in taking up heat and carbon (Goodwin et al. 2015; Williams et al. 2016). Two sets of Earth system model experiments are diagnosed: 10 Earth system models with an annual 1% rise in atmospheric CO2 and nine Earth system models following representative concentration pathways (RCPs) (Moss et al. 2010) until year 2100, including non-CO2 radiative forcing and carbon emissions from land-use and land-cover changes (Collins et al. 2013). In both sets of experiments, the sensitivity of the surface warming to fossil-fuel cumulative carbon emissions ΔTI remains nearly constant in time for each individual model (Figs. 1a,b), although its value differs for each model.

Fig. 1.

Change in global-mean surface air temperature ΔT(t) (K) vs change in fossil-fuel cumulative carbon emissions ΔI (PgC) since the preindustrial age (a) for 10 different Earth system models assuming an annual 1% increase in atmospheric CO2 and integrated for 140 yr and (b) for nine different Earth system models for the historical period (starting year 1850 for most cases, 1860 for HadGEM2, and 1861 for the GFDL models) and RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The diagnostics are for model projections in (a) for 140 yr and in (b) until year 2100; note that ΔI is greater in (a) than in (b). The dashed gray lines connect a 2°C warming back to a maximum permitted cumulative carbon emission (PgC).

Fig. 1.

Change in global-mean surface air temperature ΔT(t) (K) vs change in fossil-fuel cumulative carbon emissions ΔI (PgC) since the preindustrial age (a) for 10 different Earth system models assuming an annual 1% increase in atmospheric CO2 and integrated for 140 yr and (b) for nine different Earth system models for the historical period (starting year 1850 for most cases, 1860 for HadGEM2, and 1861 for the GFDL models) and RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The diagnostics are for model projections in (a) for 140 yr and in (b) until year 2100; note that ΔI is greater in (a) than in (b). The dashed gray lines connect a 2°C warming back to a maximum permitted cumulative carbon emission (PgC).

This relationship between surface warming and cumulative carbon emissions ΔTI is important in providing guidance to policy makers on how to avoid dangerous climate (Meinshausen et al. 2009; Zickfeld et al. 2009, 2012; Matthews et al. 2012; Friedlingstein et al. 2014). For example, in our subset of models, a 2°C warming target is only avoided if the fossil-fuel cumulative carbon emissions remain below a range of 445–911 PgC for RCP8.5 (Fig. 1b); the Intergovernmental Panel on Climate Change (IPCC) provides a similar model-mean limit of cumulative CO2 emissions of 790 PgC from all anthropogenic sources since year 1870 (Collins et al. 2013).

While the sensitivity of surface warming to fossil-fuel cumulative carbon emissions ΔTI is widely highlighted as being important (IPCC 2013), there is a lack of understanding as to why there is such a range in its value for individual Earth system models (Figs. 1a,b).

In this study, the sensitivity of this climate metric ΔTI is examined using diagnostics of climate model projections. We explore the view that the cause of these intermodel differences in ΔTI may partly arise from intermodel differences in the ocean uptake of heat and carbon. We start by setting out our theory (section 2) and then diagnose the two sets of Earth system models, including an annual increase in atmospheric CO2 or following RCP scenarios plus non-CO2 radiative forcing until year 2100 (section 3). The sensitivity of the climate metric is examined exploiting our theory, focusing on the effects of the heat uptake and carbon undersaturation of the global ocean (section 4). Intermodel differences in the sensitivity of surface warming on carbon emissions are then interpreted (section 5), and, finally, the implications are summarized (section 6).

2. Theory for how surface warming depends on carbon emissions

The sensitivity of surface warming on cumulative carbon emissions ΔTI may be connected to the surface warming and radiative forcing. For a regime where the radiative forcing is only affected by changes in atmospheric CO2, ΔTI may be written in terms of the product of two terms: the ratio of the surface warming and the radiative forcing ΔTR and the ratio of the radiative forcing and the cumulative carbon emissions ΔRI:

 
formula

For a regime where the radiative forcing is also affected by other non-CO2 radiative contributions, the climate metric ΔTI is related to the product of three terms: the ratio of the surface warming and the radiative forcing ΔTR, the ratio of the radiative forcing and the radiative forcing from atmospheric CO2 ΔRRCO2, and the ratio of the radiative forcing from atmospheric CO2 and the cumulative carbon emissions ΔRCO2I (Williams et al. 2016):

 
formula

In the limit of the radiative forcing only being from atmospheric CO2, ΔRRCO2 becomes unity and (1) and (2) become equivalent.

Theory is next applied for the ratios of the surface warming to radiative forcing ΔTR and the radiative forcing from atmospheric CO2 to cumulative emissions ΔRCO2I, so that these terms may be viewed as dependences with functional forms connecting the variables.

a. Dependence of surface warming on radiative forcing

For a long-term equilibrium, an increase in radiative forcing since the preindustrial age ΔR (in W m−2) is expected to drive an increase in global-mean, surface air temperature ΔT, such that

 
formula

where λ is the equilibrium climate parameter (in W m−2 K−1) (Knutti and Hegerl 2008; Gregory et al. 2004; Andrews et al. 2012; Forster et al. 2013) and ΔR is defined as positive into the ocean.

For a transient response, the increase in surface temperature ΔT(t) from the increase in radiative forcing ΔR is reduced in magnitude by the planetary heat uptake N(t) (W m−2), as represented in an extension to (3) (Gregory et al. 2004; Winton et al. 2010) by

 
formula

where ε(t)N(t) is the scaled global-mean heat flux into the climate system and the efficacy ε(t) is a nondimensional scaling taken to vary in time (Paynter and Frölicher 2015). The net global heat uptake N(t) is dominated by the ocean contribution and, henceforth, is taken to represent the global ocean heat uptake.

The dependence of surface warming on radiative forcing ΔTR is then directly connected to the scaled ocean heat uptake divided by the radiative forcing:

 
formula

b. Dependence of radiative forcing from atmospheric CO2 on carbon emissions

For a global air–sea equilibrium, the logarithm of the fractional change in atmospheric CO2 is related to the fossil-fuel cumulative carbon emission ΔI and the increase in the terrestrial carbon inventory ΔIter divided by the buffered carbon inventory (Goodwin et al. 2007):

 
formula

where Δ lnCO2(tequilib) = ln[CO2(tequilib)/CO2(to)], to is the time of the preindustrial age, tequilib is the time for the global equilibrium to be reached, ΔI(t) − ΔIter(t) represents the net carbon emissions to the combined atmosphere and ocean, and the buffered atmosphere and ocean carbon inventory IB (PgC) represents the effective available carbon in the combined atmosphere and ocean in the preindustrial age, taking into account ocean chemistry.

Prior to an air–sea equilibrium being reached, the global ocean is undersaturated with respect to the atmosphere, so that the atmospheric CO2 exceeds the equilibrium value CO2(tequilib). Goodwin et al. (2015) extend (6) by taking into account the global ocean undersaturation of carbon to define the transient atmospheric CO2 response by

 
formula

where IUsat(t) in PgC measures the undersaturation of the global ocean with respect to the instantaneous atmospheric CO2(t).

The radiative forcing from atmospheric CO2 is assumed to depend on the change in the logarithm of atmospheric CO2 (Myhre et al. 1998):

 
formula

where a in W m−2 is the effective radiative forcing coefficient (Forster et al. 2013). The dependence of radiative forcing from atmospheric CO2 on fossil-fuel cumulative carbon emissions ΔRCO2I is then defined by combining (7) and (8):

 
formula

which reveals the connection to global ocean undersaturation of carbon IUsat(t) and the increase in the terrestrial carbon inventory ΔIter(t), both scaled by the cumulative carbon emission ΔI(t).

3. Surface warming as a function of cumulative emissions

Our theory highlights how the ratio of surface warming to cumulative carbon emissions ΔTI connects to the ocean heat uptake and the ocean and terrestrial carbon uptake [using (2), (5), and (9)]. Next, we describe how our diagnostics are applied to the Earth system models and then to two different sets of climate projections.

a. Methodology

The climate response is analyzed for two sets of experiments using a subset of CMIP5 models including an active carbon cycle (Table 1); our selection of Earth system models is constrained by the availability of fossil-fuel carbon emissions (Jones et al. 2013), radiative forcing coefficients (Forster et al. 2013), and some ocean variables needed for our diagnostics.

Table 1.

List of models contributing to CMIP5 whose data have been used for this analysis. Note that the selection of models analyzed in this study is based on the subset of models used in Jones et al. (2013) and Collins et al. (2013) for the analysis of compatible emissions under the four RCPs; Y denotes when the corresponding experiment is used here. (Acronym expansions are available online at http://www.ametsoc.org/PubsAcronymList.)

List of models contributing to CMIP5 whose data have been used for this analysis. Note that the selection of models analyzed in this study is based on the subset of models used in Jones et al. (2013) and Collins et al. (2013) for the analysis of compatible emissions under the four RCPs; Y denotes when the corresponding experiment is used here. (Acronym expansions are available online at http://www.ametsoc.org/PubsAcronymList.)
List of models contributing to CMIP5 whose data have been used for this analysis. Note that the selection of models analyzed in this study is based on the subset of models used in Jones et al. (2013) and Collins et al. (2013) for the analysis of compatible emissions under the four RCPs; Y denotes when the corresponding experiment is used here. (Acronym expansions are available online at http://www.ametsoc.org/PubsAcronymList.)

In the first set of diagnostics, the Earth system models have been forced by an annual 1% rise in atmospheric CO2 for 140 yr starting from a preindustrial control; the 10-different coupled climate–carbon models include BNU-ESM1, CanESM2, CESM1(BGC), GFDL-ESM2M, HadGEM2-ES, IPSL-CM5A-LR, IPSL-CM5A-MR, MIROC-ESM, MPI-ESM-LR, and NorESM1-ME. In the second set of diagnostics, the Earth system models have been integrated from the preindustrial age until year 2100 with atmospheric carbon dioxide following the RCP scenarios RCP2.6, RCP4.5, RCP6.0, and RCP8.5 (Moss et al. 2010); the nine different Earth system models include CanESM2, GFDL-ESM2G, GFDL-ESM2M, HadGEM2-CC, HadGEM2-ES, IPSL-CM5A-LR, MIROC-ESM, MPI-ESM-LR, and NorESM1-ME. For all of the models, the atmospheric CO2, surface air temperature, and ocean variables for temperature, salinity, dissolved inorganic carbon DIC, alkalinity, and phosphate are taken from the CMIP5 model archive.

The Earth system models are forced by a rise in atmospheric CO2, rather than by carbon emissions. The models couple the carbon cycle and climate together by sources and sinks of CO2 being affected by atmospheric CO2 and the change in climate (Ciais et al. 2013).

The fossil-fuel cumulative carbon emissions since the preindustrial age are provided by Jones et al. (2013). For experiments involving only an annual rise in atmospheric CO2, ΔI(t) has been diagnosed from the imposed rate of change in atmospheric CO2 plus the globally integrated air–sea and air–land carbon fluxes. For the RCP scenarios, ΔI(t) has been diagnosed from the difference between the imposed atmospheric CO2 following the RCP scenario and the simulated carbon fluxes from the ocean and the land including the effects of land-use and land-cover changes. The change in the terrestrial carbon inventory is taken from the simulated carbon fluxes from the land (Jones et al. 2013).

The radiative forcing ΔR(t) is provided by Forster et al. (2013). The radiative forcing ΔRCO2(t) is diagnosed from atmospheric CO2 using (8) and the radiative forcing coefficient a provided by Forster et al. (2013) (Table 2). For the RCP scenarios, additional anthropogenic forcings are included, involving other non-CO2 greenhouse gases, anthropogenic aerosols, and land-use changes.

Table 2.

Variations in climate and ocean parameters and the implied ECRE for nine Earth system models for RCP8.5: the effective radiative forcing coefficient for CO2 a; the climate parameter λ; and preindustrial values for the global-mean dissolved inorganic carbon in the ocean DIC, the saturated value DICsat, the buffer factor B, and the buffered atmosphere and ocean carbon inventory IB; and the implied ECRE a/(λIB) (Williams et al. 2012).

Variations in climate and ocean parameters and the implied ECRE for nine Earth system models for RCP8.5: the effective radiative forcing coefficient for CO2 a; the climate parameter λ; and preindustrial values for the global-mean dissolved inorganic carbon in the ocean DIC, the saturated value DICsat, the buffer factor B, and the buffered atmosphere and ocean carbon inventory IB; and the implied ECRE a/(λIB) (Williams et al. 2012).
Variations in climate and ocean parameters and the implied ECRE for nine Earth system models for RCP8.5: the effective radiative forcing coefficient for CO2 a; the climate parameter λ; and preindustrial values for the global-mean dissolved inorganic carbon in the ocean DIC, the saturated value DICsat, the buffer factor B, and the buffered atmosphere and ocean carbon inventory IB; and the implied ECRE a/(λIB) (Williams et al. 2012).

The global ocean heat uptake N(t) is evaluated from the tendency in global ocean heat content. The climate feedback parameter λ is diagnosed from a regression between ΔT(t), ΔR(t), and N(t) using (4) following Gregory and Forster (2008) with an initial estimate of ε = 1.1 (Table 2). The nondimensional weighting of ocean heat uptake ε(t) is then evaluated from ΔT(t), ΔR(t), N(t), and λ using (4).

The atmospheric and ocean carbon inventories are diagnosed from atmospheric CO2 and the dissolved inorganic carbon (DIC). The saturated dissolved inorganic carbon DICsat represents the carbon concentration that a fluid parcel would have if moved adiabatically to the sea surface and reaches an equilibrium with the atmosphere. The quantity DICsat is evaluated for the instantaneous atmospheric CO2 and ocean potential temperature, salinity, alkalinity, and phosphate (Table 2). The global ocean undersaturation in carbon IUsat(t) is diagnosed from the difference in the saturated dissolved inorganic carbon DICsat and the actual DIC anomaly over the global ocean, given by IUsat(t) = V[ΔDICsat(t) − ΔDIC(t)], where V is the ocean volume.

The buffered carbon inventory for the atmosphere and ocean, given by IB = Iatmos(t0) + VDICsat(to)/B, at the preindustrial age t0 is diagnosed from the sum of the atmospheric inventory Iatmos plus the saturated ocean carbon inventory VDICsat divided by the buffer factor, given by B = (ΔCO2/CO2)/(ΔDIC/DIC). The IB and B are evaluated for the preindustrial age at year 1861 with an atmospheric CO2 of 286 ppm and an atmospheric carbon inventory of 607 PgC (Table 2).

b. For an idealized increase in atmospheric CO2

For the set of Earth system models forced by an annual 1% rise in atmospheric CO2, the global-mean surface air temperature increases by typically 1.5–3.5 K over 100 yr (Fig. 2a), which is driven by the increase in radiative forcing from 3.5 to 5.5 W m−2 (Fig. 2b) associated with a cumulative carbon emission typically ranging from 1500 to 2000 PgC (Fig. 2c).

Fig. 2.

Climate projections assuming an annual 1% increase in atmospheric CO2 over 140 yr from 10 different Earth system models (with the model mean in black): (a) change in global-mean surface air temperature ΔT(t) (K); (b) radiative forcing from atmospheric CO2 ΔR (W m−2); (c) the cumulative carbon emissions ΔI (103 PgC); (d) the TCRE ΔTI [K (103 PgC)−1], which using Eq. (1) may be separated into the product of (e) the dependence of surface warming on radiative forcing ΔTR [K (W m−2)−1] and (f) the dependence of radiative forcing from atmospheric CO2 on emissions ΔRCO2I [W m−2 (103 PgC)−1]. The diagnostics in (c)–(f) are not shown for the first 10 yr due to the small magnitude of ΔR and ΔI.

Fig. 2.

Climate projections assuming an annual 1% increase in atmospheric CO2 over 140 yr from 10 different Earth system models (with the model mean in black): (a) change in global-mean surface air temperature ΔT(t) (K); (b) radiative forcing from atmospheric CO2 ΔR (W m−2); (c) the cumulative carbon emissions ΔI (103 PgC); (d) the TCRE ΔTI [K (103 PgC)−1], which using Eq. (1) may be separated into the product of (e) the dependence of surface warming on radiative forcing ΔTR [K (W m−2)−1] and (f) the dependence of radiative forcing from atmospheric CO2 on emissions ΔRCO2I [W m−2 (103 PgC)−1]. The diagnostics in (c)–(f) are not shown for the first 10 yr due to the small magnitude of ΔR and ΔI.

The sensitivity of surface warming to cumulative carbon emissions is generally close to 1.8 K for 1000 PgC (Fig. 2d) with a slight decrease of 10% or less from year 45 to 135 (Table 3), consistent with diagnostics of a larger set of 15 different Earth system models using the same CO2 scenario by Gillett et al. (2013).

Table 3.

The sensitivity of surface warming on cumulative carbon emissions ΔTI and its dependences assuming an annual 1% increase in atmospheric CO2 diagnosed from 10 different Earth system models for two 10-yr periods centered on years 45 and 135 together with the cumulative carbon emission ΔI for a 2°C warming.

The sensitivity of surface warming on cumulative carbon emissions ΔT/ΔI and its dependences assuming an annual 1% increase in atmospheric CO2 diagnosed from 10 different Earth system models for two 10-yr periods centered on years 45 and 135 together with the cumulative carbon emission ΔI for a 2°C warming.
The sensitivity of surface warming on cumulative carbon emissions ΔT/ΔI and its dependences assuming an annual 1% increase in atmospheric CO2 diagnosed from 10 different Earth system models for two 10-yr periods centered on years 45 and 135 together with the cumulative carbon emission ΔI for a 2°C warming.

The near constancy of ΔTI is achieved by two nearly compensating changes in (1), involving the product of the sensitivities of surface warming to radiative forcing ΔTR and radiative forcing to cumulative carbon emissions ΔRI. First, the ratio of the surface warming to radiative forcing ΔTR increases in time for all the models, typically rising from 0.5 to 0.7 K (W m−2)−1 (Fig. 2e; Table 3). Second, the ratio of the radiative forcing on cumulative carbon emissions ΔRI decreases in time for all the models, typically from 3.7 to 2.5 W m−2 (103 PgC)−1 (Fig. 2f; Table 3). This partial compensation leads to a near constancy in ΔTI for both the running mean of all the models (Fig. 2d, thick line), as well as the individual model responses.

c. Following representative concentration pathways

For the set of Earth system models following RCP scenarios, there is an increase in the global-mean surface air temperature typically ranging from 1.5 to 5 K by year 2100 (Fig. 3a), increasing with the fossil-fuel cumulative carbon emission of typically 500–2000 PgC from RCP2.6 to RCP8.5 (Fig. 3b). For this set of Earth system models following RCP8.5, a 2°C warming is reached after a fossil-fuel cumulative carbon emission ranging from 445 to 911 PgC (Table 4; Fig. 1b, gray dashed lines).

Fig. 3.

Climate projections for this century for nine different Earth system models (gray lines) for RCP2.6, RCP4.5, RCP6.0, and RCP8.5 (model means are black, blue, green, and red, respectively): (a) surface warming ΔT(t) (K); (b) fossil-fuel cumulative carbon emissions ΔI(t) (103 PgC); (c) the sensitivity of surface warming on cumulative carbon emissions ΔTI [K (103 PgC)−1]. The ΔTI is equivalent to the product of (d) the dependence of surface warming on radiative forcing ΔTR [K (W m−2)−1], (e) the dependence of radiative forcing from atmospheric CO2 on emissions ΔRCO2I [W m−2 (103 PgC)−1], and (f) the ratio of the radiative forcing and the radiative forcing from atmospheric CO2, ΔRRCO2. Each term is controlled by different mechanisms: (d) varies with ocean heat uptake and climate sensitivity, (e) varies with ocean carbon undersaturation and carbon uptake by the terrestrial system, and (f) varies with radiative forcing from non-CO2 greenhouse gases and aerosols.

Fig. 3.

Climate projections for this century for nine different Earth system models (gray lines) for RCP2.6, RCP4.5, RCP6.0, and RCP8.5 (model means are black, blue, green, and red, respectively): (a) surface warming ΔT(t) (K); (b) fossil-fuel cumulative carbon emissions ΔI(t) (103 PgC); (c) the sensitivity of surface warming on cumulative carbon emissions ΔTI [K (103 PgC)−1]. The ΔTI is equivalent to the product of (d) the dependence of surface warming on radiative forcing ΔTR [K (W m−2)−1], (e) the dependence of radiative forcing from atmospheric CO2 on emissions ΔRCO2I [W m−2 (103 PgC)−1], and (f) the ratio of the radiative forcing and the radiative forcing from atmospheric CO2, ΔRRCO2. Each term is controlled by different mechanisms: (d) varies with ocean heat uptake and climate sensitivity, (e) varies with ocean carbon undersaturation and carbon uptake by the terrestrial system, and (f) varies with radiative forcing from non-CO2 greenhouse gases and aerosols.

Table 4.

The sensitivity of surface warming on cumulative carbon emissions ΔTI and its dependences diagnosed for RCP8.5 from nine climate models for two 10-yr periods centered on years 2010 and 2095, together with the fossil-fuel cumulative emissions ΔI for a 2°C warming.

The sensitivity of surface warming on cumulative carbon emissions ΔT/ΔI and its dependences diagnosed for RCP8.5 from nine climate models for two 10-yr periods centered on years 2010 and 2095, together with the fossil-fuel cumulative emissions ΔI for a 2°C warming.
The sensitivity of surface warming on cumulative carbon emissions ΔT/ΔI and its dependences diagnosed for RCP8.5 from nine climate models for two 10-yr periods centered on years 2010 and 2095, together with the fossil-fuel cumulative emissions ΔI for a 2°C warming.

There is a limited temporal trend in the sensitivity of the surface warming to cumulative carbon emissions ΔTI for the model mean for each RCP (Fig. 3c, colored lines); for example, for RCP8.5, there is an overall decrease in the model mean of ΔTI from 3 to 2.5 K (103 PgC)−1 from years 2010 to 2095 (Table 4). The evolution of ΔTI varies between the models (Fig. 4): it decreases in time for GFDL-ESM2G, GFDL-ESM2M, IPSL-CM5A-LR, and MPI-ESM-LR; it increases in time for HadGEM2-CC, HadGEM2-ES, and both CanESM2 and MIROC-ESM for RCP2.6; and it remains relatively constant for NorESM1-ME and both CanESM2 and MIROC-ESM for RCP4.5, RCP6.0, and RCP8.5.

Fig. 4.

The sensitivity of surface warming on cumulative carbon emissions ΔTI [K (103 PgC)−1] for this century for (a)–(i) nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The ΔTI is equivalent to the product of the terms ΔTR, ΔRCO2I, and ΔRRCO2 (later inFigs. 7, 10, and 11). The magnitude of ΔTI decreases in time in (b), (c), (f), and (h) but increases slightly in (d) and (e) due to the inverse fractional radiative forcing from atmospheric CO2 ΔRRCO2.

Fig. 4.

The sensitivity of surface warming on cumulative carbon emissions ΔTI [K (103 PgC)−1] for this century for (a)–(i) nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The ΔTI is equivalent to the product of the terms ΔTR, ΔRCO2I, and ΔRRCO2 (later inFigs. 7, 10, and 11). The magnitude of ΔTI decreases in time in (b), (c), (f), and (h) but increases slightly in (d) and (e) due to the inverse fractional radiative forcing from atmospheric CO2 ΔRRCO2.

The temporal trends in ΔTI may be understood by considering the product of ΔTR, ΔRCO2I, and ΔRRCO2 from (2). The ratio of the surface warming on radiative forcing ΔTR generally increases in time (Fig. 3d); the model mean for RCP8.5 increasing from 0.53 to 0.66 K (W m−2)−1 over this century (Table 4). This enhanced surface warming is consistent with a general decline in the ratio of the ocean heat uptake divided by the radiative forcing. This surface-intensified warming is consistent with the expected increase in upper-ocean stratification from enhanced radiative forcing.

The ratio of the radiative forcing from atmospheric CO2 on carbon emissions ΔRCO2I decreases in time across all RCP scenarios (Fig. 3e); the model mean for RCP8.5 decreasing from 4.8 to 3.0 W m−2 (103 PgC)−1 over this century (Table 4). This dependence ΔRCO2I decreases as a result of the effect of ocean and terrestrial uptake of carbon. This decrease is primarily due to how the global ocean becomes less undersaturated in carbon relative to the increase in cumulative carbon emissions.

The ratio of the radiative forcing and radiative contribution from atmospheric CO2 ΔRRCO2 is nearly constant for RCP8.5 but slightly declines for the other RCPs (Fig. 3f). This term depends on the effects of non-CO2 greenhouse gases and aerosols. The trends in this term are relatively small compared with its large intermodel scatter.

d. Evolution of ΔT/ΔI toward an equilibrium value

The different intermodel evolution of ΔTI may be partly understood in terms of differences in their final equilibrium value. This equilibrium state assumes that ocean heat uptake and ocean carbon undersaturation have declined in time after emissions cease. For this equilibrium [combining Eqs. (1), (3), (6), and (8) and taking ΔRRCO2 = 1], the surface temperature change asymptotically approaches

 
formula

where ΔI(t) − ΔIter(t) represents the net carbon emissions to the combined atmosphere and ocean.

The climate parameters a/(λIB) determining the equilibrium state (Williams et al. 2012) vary in the model ensemble from 1.2 to 2.8 K (103 PgC)−1 due to variations in a from 4.2 to 6.2 W m−2, IB from 3360 to 3550 PgC, and λ from 0.4 to 1.1 W m−2 K−1 (Table 2 for RCP8.5); our diagnostics of λ for RCP8.5 are slightly lower than the corresponding values for CO2 forcing alone, which range from 0.6 to 1.4 W m−2 K−1 (Forster et al. 2013).

For most of the integrations of the Earth system models, ΔTI evolves toward the equilibrium value a/(λIB) so is closer at year 2095 than at year 2010 (Tables 2 and 4). For example, for GFDL-ESM2G for RCP8.5, there is a decrease in ΔTI from 3.1 to 1.8 K (103 PgC)−1 over this century, which is toward the equilibrium value a/(λIB) of 1.3 K (103 PgC)−1. For HadGEM2-ES for RCP8.5, there is an increase in ΔTI from 2.1 to 2.4 K (103 PgC)−1 over this century, which is the same as the equilibrium value a/(λIB) of 2.4 K (103 PgC)−1.

Hence, the climate parameters a/(λIB) affect the evolution of ΔTI for each Earth system model over this century and provide a measure of the ECRE (Frölicher and Paynter 2015).

4. Mechanisms controlling the sensitivity of surface warming on carbon emissions

Our focus is now to highlight the underlying mechanisms controlling the evolution of ΔTI, focusing on the response of individual Earth system models following RCP scenarios.

a. Ocean heat and carbon uptake

The surface warming response to cumulative carbon emissions ΔTI is controlled by the effects of radiative forcing and the heat and carbon transfers within the climate system. The ocean uptake of anthropogenic heat and carbon is stored in the same manner, preferentially within the upper ocean including the surface mixed layer and stratified thermocline, illustrated here by the tendencies in global heat content and DIC (Figs. 5a,b) for RCP8.5 from IPSL-CM5A-LR as a representative example of an Earth system model.

Fig. 5.

Global ocean heat and carbon diagnostics over depth layers vs year for Earth system model IPSL-CM5A-LR for RCP8.5: (a) tendency of global ocean heat content ∂Q(t)/∂t (TW); (b) tendency of global ocean DIC content V∂DIC(t)/∂t (PgC yr−1); (c) normalized ocean heat uptake by radiative forcing [∂Q(t)/∂t]/ΔR(t); and (d) global ocean carbon undersaturation divided by fossil-fuel cumulative emissions IUsat(t)/ΔI(t). Percentages are provided for the nondimensional ratios in (c) and (d).

Fig. 5.

Global ocean heat and carbon diagnostics over depth layers vs year for Earth system model IPSL-CM5A-LR for RCP8.5: (a) tendency of global ocean heat content ∂Q(t)/∂t (TW); (b) tendency of global ocean DIC content V∂DIC(t)/∂t (PgC yr−1); (c) normalized ocean heat uptake by radiative forcing [∂Q(t)/∂t]/ΔR(t); and (d) global ocean carbon undersaturation divided by fossil-fuel cumulative emissions IUsat(t)/ΔI(t). Percentages are provided for the nondimensional ratios in (c) and (d).

The dependence of the surface warming on radiative forcing ΔTR varies with the ratio of the ocean heat uptake to the radiative forcing N(t)/ΔR(t) from (5). The fraction of radiative heat taken up by the ocean interior at depths greater than 100 m declines after year 2020 (Fig. 5c). Hence, ΔTR increases in time as a slightly larger proportion of the radiative forcing is used to warm the surface, rather than the ocean interior.

The dependence of the radiative forcing from atmospheric CO2 on carbon emissions ΔRCO2I varies with the ratio of the global ocean undersaturation and the carbon emissions IUsatI(t) from (9). The rise in atmospheric CO2 leads to a faster increase in saturated dissolved inorganic carbon in the ocean DICsat(t) than the actual DIC(t) (Table 2), so that the global ocean undersaturation in carbon, given by IUsat(t) = V[ΔDICsat(t) − ΔDIC(t)], increases in time. The rate of increase in the ocean carbon undersaturation IUsat(t) is, though, less than the increase in cumulative carbon emission ΔI(t), so that the normalized ocean carbon undersaturation IUsat(t)/ΔI(t), progressively decreases in time (Fig. 5d) for all depth ranges. Hence, ΔRCO2I decreases in time as the ocean becomes less undersaturated relative to the increase in carbon emissions.

These responses are next worked through for the collection of Earth system models experiencing RCP forcing.

b. Dependence of surface warming on radiative forcing

The surface climate response is driven by the increase in radiative forcing ΔR(t); for RCP8.5, the model mean of ΔR(t) increases from 1.9 to 7.2 W m−2 from years 2010 to 2095 (Fig. 6, full black line; Table 5). The increase in ΔR(t) is primarily due to the atmospheric CO2 contribution ΔRCO2(t), rising from 1.6 to 5.7 W m−2 from years 2010 to 2095 (Fig. 6, gray dashed line; Table 6).

Fig. 6.

Diagnostics of the surface climate response in (4) for this century for nine different Earth system models following RCP8.5: the increase in radiative forcing since the preindustrial age ΔR(t) (W m−2; black line), drives an increase in ocean heat uptake ε(t)N(t) (blue line) and a surface climate response λΔT(t) (red line); λ is taken from a regression (Gregory and Forster 2008), and ε(t) is diagnosed to ensure that (4) holds. The increase in ΔR(t) is dominated by the increase in ΔRCO2(t), rather than ΔRnonCO2(t) (dashed and dotted gray lines, respectively). The ratio of ocean heat uptake and increase in radiative forcing ε(t)N(t)/ΔR(t) decreases in time for all cases.

Fig. 6.

Diagnostics of the surface climate response in (4) for this century for nine different Earth system models following RCP8.5: the increase in radiative forcing since the preindustrial age ΔR(t) (W m−2; black line), drives an increase in ocean heat uptake ε(t)N(t) (blue line) and a surface climate response λΔT(t) (red line); λ is taken from a regression (Gregory and Forster 2008), and ε(t) is diagnosed to ensure that (4) holds. The increase in ΔR(t) is dominated by the increase in ΔRCO2(t), rather than ΔRnonCO2(t) (dashed and dotted gray lines, respectively). The ratio of ocean heat uptake and increase in radiative forcing ε(t)N(t)/ΔR(t) decreases in time for all cases.

Table 5.

Radiative forcing ΔR(t), climate response λΔT(t), scaled ocean heat uptake εN(t), and the dependence of surface warming on radiative forcing ΔTR, diagnosed from nine Earth system models for two 10-yr periods centered on years 2010 and 2095, including the nondimensional ratios of εNR; all diagnostics are for RCP8.5.

Radiative forcing ΔR(t), climate response λΔT(t), scaled ocean heat uptake εN(t), and the dependence of surface warming on radiative forcing ΔT/ΔR, diagnosed from nine Earth system models for two 10-yr periods centered on years 2010 and 2095, including the nondimensional ratios of εN/ΔR; all diagnostics are for RCP8.5.
Radiative forcing ΔR(t), climate response λΔT(t), scaled ocean heat uptake εN(t), and the dependence of surface warming on radiative forcing ΔT/ΔR, diagnosed from nine Earth system models for two 10-yr periods centered on years 2010 and 2095, including the nondimensional ratios of εN/ΔR; all diagnostics are for RCP8.5.
Table 6.

Radiative forcing from atmospheric CO2 ΔRCO2(t) and its dependence on fossil-fuel cumulative carbon emissions ΔI(t), together with the nondimensional ratios of ocean undersaturation of carbon and change in the terrestrial store of carbon divided by cumulative carbon emissions, IUsatI and ΔIterI, respectively, from nine Earth system models for two 10-yr periods centered on years 2010 and 2095; all diagnostics are for RCP8.5.

Radiative forcing from atmospheric CO2 ΔRCO2(t) and its dependence on fossil-fuel cumulative carbon emissions ΔI(t), together with the nondimensional ratios of ocean undersaturation of carbon and change in the terrestrial store of carbon divided by cumulative carbon emissions, IUsat/ΔI and ΔIter /ΔI, respectively, from nine Earth system models for two 10-yr periods centered on years 2010 and 2095; all diagnostics are for RCP8.5.
Radiative forcing from atmospheric CO2 ΔRCO2(t) and its dependence on fossil-fuel cumulative carbon emissions ΔI(t), together with the nondimensional ratios of ocean undersaturation of carbon and change in the terrestrial store of carbon divided by cumulative carbon emissions, IUsat/ΔI and ΔIter /ΔI, respectively, from nine Earth system models for two 10-yr periods centered on years 2010 and 2095; all diagnostics are for RCP8.5.

The increase in radiative forcing ΔR(t) drives both the expected surface warming response λΔT(t) and the scaled ocean heat uptake ε(t)N(t) (Fig. 6, red and blue lines) from (4). For RCP8.5, the model means of λΔT are 0.8 and 3.5 W m−2 at years 2010 and 2095, respectively, while for ε(t)N(t) are 1.1 and 3.7 W m−2 at years 2010 and 2095 (Table 5). The surface climate response λΔT(t) is less than the scaled ocean heat uptake, ε(t)N(t) for HadGEM2-CC, HadGEM2-ES, and NorESM1-ME, or the terms are comparable to each other for the other models (Fig. 6).

The dependence of surface warming on radiative forcing ΔTR typically ranges from 0.3 to nearly 0.7 K (W m−2)−1 by year 2010 and increases to 0.5–0.9 K (W m−2)−1 by year 2095 (Fig. 7; Table 5). In nearly all the Earth system models, the scaled ocean heat uptake ε(t)N(t) takes up a slightly smaller fraction with time of the radiative forcing ΔR(t) so that the model mean of their ratio ε(t)N(t)/ΔR(t) decreases from 0.6 to 0.5 from years 2010 to 2095 for RCP8.5 (Table 5). This reduction is particularly marked for GFDL-ESM2G, HadGEM2-CC, and IPSL-CM5A-LR, which leads to a corresponding increase in ΔTR from (5) and only does not occur for MIROC-ESM (Fig. 7). These changes from ε(t)N(t) are mainly due to changes in ocean heat uptake but also involve temporal changes in ε(t) over the first few decades then with ε(t) approaching a value close to 1 in the latter half of the century.

Fig. 7.

Dependence of surface warming on radiative forcing ΔTR [K (W m−2)−1], for this century for nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The ΔTR has an overall positive trend in nearly all model cases. This positive trend is due to an increasing fraction of the radiative forcing warming the surface and a corresponding decrease in the ratio of scaled ocean heat uptake and radiative forcing ε(t)N(t)/ΔR(t) {since ΔTR is equivalent to λ−1[1 − ε(t)N(t)/ΔR(t)]}.

Fig. 7.

Dependence of surface warming on radiative forcing ΔTR [K (W m−2)−1], for this century for nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The ΔTR has an overall positive trend in nearly all model cases. This positive trend is due to an increasing fraction of the radiative forcing warming the surface and a corresponding decrease in the ratio of scaled ocean heat uptake and radiative forcing ε(t)N(t)/ΔR(t) {since ΔTR is equivalent to λ−1[1 − ε(t)N(t)/ΔR(t)]}.

Hence, the temporal decline in the fraction of radiative forcing raising ocean heat content leads to a temporal increase in the dependence of surface warming on the radiative forcing ΔTR: the radiative forcing initially warms much of the upper ocean, and the fraction of radiative forcing taken up below the upper 100 m of the ocean then weakens and the surface warming strengthens. This response is consistent with the upper ocean becoming more stratified with increased radiative forcing (Capotondi et al. 2012).

c. Dependence of radiative forcing from CO2 on carbon emissions

The cumulative carbon emissions from fossil fuels ΔI(t) drives an increase in the sum of the atmospheric, ocean, and terrestrial inventories of carbon, referred to as ΔIatmos(t), ΔIocean(t), and ΔIter(t), respectively:

 
formula

In all the models, the atmospheric carbon inventory increases as prescribed (Fig. 8, black line) and the increase in the ocean inventory is usually larger than the increase in terrestrial inventory (Fig. 8, blue and green lines), apart from HadGEM2-CC, HadGEM2-ES, and IPSL-CM5A-LR where these changes are comparable. The relatively small magnitude of the terrestrial change ΔIter is partly due to including opposing anthropogenic contributions from land-use changes and increases in terrestrial storage.

Fig. 8.

Diagnostics of the carbon inventory for this century for nine different Earth system models following RCP8.5: the increase in fossil-fuel cumulative carbon emissions since the preindustrial age ΔI(t) (PgC; red line) drives an increase in the carbon inventories in the atmosphere ΔIatmos(t) (black line), the ocean ΔIocean(t) (blue line), and terrestrial system ΔIter(t) (green line); the terrestrial response is of variable magnitude and sometimes positive or weakly negative. The undersaturation of carbon in the ocean IUsat(t) (PgC; dashed line) increases due to the fast pace of emissions.

Fig. 8.

Diagnostics of the carbon inventory for this century for nine different Earth system models following RCP8.5: the increase in fossil-fuel cumulative carbon emissions since the preindustrial age ΔI(t) (PgC; red line) drives an increase in the carbon inventories in the atmosphere ΔIatmos(t) (black line), the ocean ΔIocean(t) (blue line), and terrestrial system ΔIter(t) (green line); the terrestrial response is of variable magnitude and sometimes positive or weakly negative. The undersaturation of carbon in the ocean IUsat(t) (PgC; dashed line) increases due to the fast pace of emissions.

There is a more striking increase in the carbon undersaturation of the ocean IUsat(t) with respect to the instantaneous atmospheric CO2(t) for all the Earth system models for RCP8.5 (Fig. 8, gray dashed line): IUsat(t) increases typically from 700 to over 2000 PgC from years 2010 to 2095 (Table 6), which far exceeds the actual inventory changes of the ocean and terrestrial system. This carbon undersaturation of the ocean is defined relative to instantaneous atmospheric CO2, so is directly affected by the ocean uptake of carbon and indirectly by changes in the land storage of carbon.

The carbon inventory changes are often understood in terms of the airborne, oceanborne, or landborne fractions of carbon (Jones et al. 2013):

 
formula

There is a decrease in the airborne fraction ΔIatmosI in GFDL-ESM2G, GFDL-ESM2M, MIROC-ESM, and MPI-ESM-LR, which is mainly offset by an increase in the landborne fraction ΔIterI (Fig. 9). On the other hand, there are relatively small changes in the airborne fraction ΔIatmosI in CanESM2, HadGEM2-CC, HadGEM2-ES, and IPSL-CM5A-LR, where an increase in the landborne fraction ΔIterI is offset by a decrease in the oceanborne fraction ΔIoceanI (Fig. 9).

Fig. 9.

Diagnostics of the airborne, oceanborne, and landborne fractions of emissions for this century for nine different Earth system models following RCP8.5. There is often a slight decline in the airborne and oceanborne fractions, ΔIatmosI (black line) and ΔIoceanI (blue line), respectively, as well as a slight increase in the landborne fraction ΔIterI (green line). The ratio of the ocean undersaturation of carbon and the emitted carbon IUsat(t)/ΔI(t) (dashed line) declines as the ocean gradually takes up a larger fraction of the emissions.

Fig. 9.

Diagnostics of the airborne, oceanborne, and landborne fractions of emissions for this century for nine different Earth system models following RCP8.5. There is often a slight decline in the airborne and oceanborne fractions, ΔIatmosI (black line) and ΔIoceanI (blue line), respectively, as well as a slight increase in the landborne fraction ΔIterI (green line). The ratio of the ocean undersaturation of carbon and the emitted carbon IUsat(t)/ΔI(t) (dashed line) declines as the ocean gradually takes up a larger fraction of the emissions.

The ratio of the ocean carbon undersaturation and the cumulative carbon emissions IUsat(t)/ΔI(t) decreases for all models over the next century (Fig. 9, gray dashed line); the model mean reducing from 2.1 to 1.1 for years 2010 to 2095 (Table 6). There is a marked decrease in IUsat(t)/ΔI(t) for GFDL-ESM2G, GFDL-ESM2M, MIROC-ESM, and MPI-ESM-LR and smaller decreases for CanESM2, HadGEM2-ES, and IPSL-CM5A-LR (Fig. 9).

The dependence of radiative forcing from atmospheric CO2 on cumulative carbon emissions ΔRCO2I decreases in time from a model mean of 4.8–3.0 W m−2 (103 PgC)−1 at years 2010–95 (Fig. 10; Table 6). The principal reason for this decrease is the progressive decrease in the ratio of the ocean undersaturation and cumulative carbon emissions IUsat(t)/ΔI(t) from (9). There are smaller variations in the landborne fraction ΔIter(t)/ΔI(t), which are more significant for GFDL-ESM2G and GFDL-ESM2M (Figs. 9b,c, green line).

Fig. 10.

Dependence of radiative forcing from atmospheric CO2 on emissions ΔRCO2I [W m−2 (103 PgC)−1] for this century for nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The ΔRCO2I decreases in time for all model cases, which is mainly due to the ocean carbon uptake. In particular, the ratio of the ocean unsaturation of carbon to the emissions IUsat(t)/ΔI(t) decreases in time, since ΔRCO2I is equivalent to (a/IB){1 +[IUsat(t) − ΔIter(t)]/ΔI (t)}.

Fig. 10.

Dependence of radiative forcing from atmospheric CO2 on emissions ΔRCO2I [W m−2 (103 PgC)−1] for this century for nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The ΔRCO2I decreases in time for all model cases, which is mainly due to the ocean carbon uptake. In particular, the ratio of the ocean unsaturation of carbon to the emissions IUsat(t)/ΔI(t) decreases in time, since ΔRCO2I is equivalent to (a/IB){1 +[IUsat(t) − ΔIter(t)]/ΔI (t)}.

d. Interpretation of changes in ΔT/ΔI

The contrasting trends of ΔTI for different Earth system models may be understood via its separate dependences in (2). The larger changes in ΔTI correspond to changes in either the dependence of surface warming on radiative forcing ΔTR or the dependence of radiative forcing from atmospheric CO2 on carbon emissions ΔRCO2I. For example, the significant drop in ΔTI for GFDL-ESM2M is achieved by a reduction in ΔRCO2I involving a substantial decrease in the fractional ocean undersaturation of carbon IUsat(t)/ΔI(t) (Table 6; Figs. 4b, 9b, and 10b). In contrast, the significant increase in ΔTI for HadGEM2-CC is due to an increase in ΔTR involving a reduction in ocean heat uptake divided by radiative forcing εN(t)/ΔR(t) (Table 5; Figs. 4d and 7d). For the other models, the changes in ΔTI are relatively small, involving a partial compensation from the ΔTR and ΔRCO2I contributions.

The additional factor affecting ΔTI is the non-CO2 radiative forcing, involving both effects from non-CO2 greenhouse gases and aerosols (Boucher et al. 2013; Pierrehumbert 2014). The relative importance of this non-CO2 radiative forcing on ΔTI is approximately represented by the ratio of ΔRRCO2, equivalent to 1 + ΔRnonCO2RCO2 (Fig. 11). The non-CO2 radiative forcing contribution is relatively small and positive, ΔRnonCO2(t), typically ranging from less than 1 to 2 W m−2 (Fig. 6, gray dotted line), implying that the positive radiative contribution from non-CO2 greenhouse gases exceeds the negative radiative contribution from aerosols. The model mean for ΔRRCO2 does not significantly alter in time, varying from 1.2 to 1.3 (Table 4 for RCP8.5). However, there is a variable response of individual models: this ratio decreases in time for GFDL-ESM2G, GFDL-ESM2M, and MPI-ESM-LR but instead increases in time for HadGEM2-CC, HadGEM2-ES, and MIROC-ESM (Fig. 11).

Fig. 11.

The inverse fractional radiative forcing from atmospheric CO2 ΔRRCO2 for this century for (a)–(i) nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. There are large variations in ΔRRCO2 with different models, an overall decrease in (b), (c), (f), and (h) and conversely an overall increase in (d), (e), and (g).

Fig. 11.

The inverse fractional radiative forcing from atmospheric CO2 ΔRRCO2 for this century for (a)–(i) nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. There are large variations in ΔRRCO2 with different models, an overall decrease in (b), (c), (f), and (h) and conversely an overall increase in (d), (e), and (g).

5. Intermodel uncertainty in the surface climate response

Our framework is now applied to understand the sources of intermodel variability of ΔTI and its contributions following the uncertainty analysis of Hawkins and Sutton (2009). The uncertainty is taken from the intermodel standard deviation divided by the model mean for ΔTI and its contributions across the ensemble of Earth system models.

a. Uncertainty in the sensitivity of surface warming to emissions ΔTI

The model uncertainty in ΔTI is initially large and then decreases in time for both sets of forcing scenarios (Figs. 12a,b, black line). For the annual 1% increase in atmospheric CO2, the uncertainty in ΔTI decreases from 0.23 to 0.15 from years 45 and 135 (Table 3). For RCP8.5, the uncertainty in ΔTI likewise decreases from 0.31 to 0.22 for years 2010 and 2095 (Table 4). The uncertainty for the RCP8.5 scenario remaining high until year 2020 may possibly be due to decadal variability being relatively important close to the start of the RCP forcing scenario.

Fig. 12.

Intermodel uncertainty in the sensitivities of surface warming to cumulative emissions ΔTI and surface warming to radiative forcing ΔTR for (a),(c) 10 Earth system models for an annual 1% increase in atmospheric CO2 and (b),(d) nine Earth system models for this century for RCP8.5. The uncertainty is represented by the ratio of the standard deviation and mean of the model responses. In (a) and (b), the uncertainty in ΔTI (black line) decreases in time. In (a), the dominant contribution to the uncertainty is from ΔTR (red line) and there is only a comparable contribution from ΔRCO2I (blue line) after 80 yr. In (b), there are dominant contributions from ΔTR (red line) over most of the record, from ΔRRCO2 (green line) over the historical record and from ΔRCO2I (blue line) after year 2020. In (c) and (d), the intermodel uncertainty in ΔTR (dashed red line) is due to variability in the equilibrium climate parameter λ−1 (black line) augmented by the fraction of the radiative forcing taken up by ocean heat flux εNR (blue line).

Fig. 12.

Intermodel uncertainty in the sensitivities of surface warming to cumulative emissions ΔTI and surface warming to radiative forcing ΔTR for (a),(c) 10 Earth system models for an annual 1% increase in atmospheric CO2 and (b),(d) nine Earth system models for this century for RCP8.5. The uncertainty is represented by the ratio of the standard deviation and mean of the model responses. In (a) and (b), the uncertainty in ΔTI (black line) decreases in time. In (a), the dominant contribution to the uncertainty is from ΔTR (red line) and there is only a comparable contribution from ΔRCO2I (blue line) after 80 yr. In (b), there are dominant contributions from ΔTR (red line) over most of the record, from ΔRRCO2 (green line) over the historical record and from ΔRCO2I (blue line) after year 2020. In (c) and (d), the intermodel uncertainty in ΔTR (dashed red line) is due to variability in the equilibrium climate parameter λ−1 (black line) augmented by the fraction of the radiative forcing taken up by ocean heat flux εNR (blue line).

If the uncertainties from each dependence, ΔTR, ΔRRCO2, and ΔRCO2I, are assumed to combine in the same manner as random errors, then the uncertainty squared for ΔTI should be comparable to the sum of the uncertainty squared for each contribution. However, the uncertainty for ΔTI is initially only comparable to the uncertainty for ΔTR but eventually becomes comparable to the contributions of both ΔTR and ΔRCO2I (Figs. 12a,b, red and blue lines). Hence, these contributions to the uncertainty are not random but partially compensate for each other.

In more detail for RCP8.5, the most important contribution to the uncertainty in ΔTI is from the contribution of ΔTR, varying strongly in time over the first 20 yr (Fig. 12b, red line); its uncertainty reduces from 0.28 to 0.20 for a decadal average centered on years 2010 and 2095, respectively (Table 4). This contribution is controlled by variations in ocean heat uptake and the equilibrium climate parameter for each Earth system model. Over most of the century, the next largest source of the uncertainty is from the contribution of ΔRCO2I, the uncertainty reducing from 0.23 to 0.20 for years 2010–95 (Table 4). This uncertainty smoothly reduces in time (Fig. 12b, blue line) owing to the progressive reduction in the carbon undersaturation of the ocean divided by the cumulative carbon emissions. At the start of the century, the uncertainty from ΔRRCO2 is important, reaching 0.25 in year 2010, but then declines to 0.10 by year 2095 (Table 4; Fig. 12b, green line). This contribution is controlled by variations in radiative forcing from non-CO2 greenhouse gases and aerosols between the Earth system models.

b. Uncertainties in ΔTR and ΔRCO2I

The uncertainty in the dependence of surface warming on radiative forcing ΔTR based on (5) is determined by the uncertainty for the equilibrium climate sensitivity λ−1 and the normalized ocean heat uptake ε(t)N(t)/ΔR(t) (Figs. 12c,d).

For the annual 1% increase in atmospheric CO2, the uncertainty in ΔTR is always larger than the contribution for the normalized ocean heat uptake ε(t)N(t)/ΔR(t) (Fig. 12c, red and blue lines) and so is mainly determined by intermodel differences in λ−1.

For the RCP8.5 scenario, the uncertainty for ΔTR is affected by the uncertainty in λ−1, augmented by the normalized ocean heat uptake ε(t)N(t)/ΔR(t) over the first couple of decades (Fig. 12c, red and blue lines). For year 2010, the uncertainty for ΔTR of 0.28 compares with an uncertainty for ε(t)N(t)/ΔR(t) of 0.18 (Table 5), while for year 2095, the uncertainty decreases for ΔTR to 0.20 and for ε(t)N(t)/ΔR(t) to 0.12. Hence, intermodel differences in ΔTR are initially affected by both λ−1 and ε(t)N(t)/ΔR(t) but later dominated by λ−1.

The intermodel differences in the dependence of radiative forcing from atmospheric CO2 on emissions ΔRCO2I based on (9) are affected by contributions from IUsat(t)/ΔI(t) and ΔIterI(t), as well as by intermodel differences in a and IB (Tables 2 and 6). For the fractional ocean undersaturation and change in the terrestrial store of carbon, IUsat(t)/ΔI(t) and ΔIterI(t), the standard deviation of their intermodel differences are 0.49 and 0.21 for year 2010 and reduce to 0.12 and 0.09 for year 2095, respectively (Table 6). Hence, the intermodel differences in ΔRCO2I are initially mainly affected by differences in the fractional ocean undersaturation IUsat(t)/ΔI(t) and later also by differences in the landborne fraction ΔIterI(t). The uncertainties in the landborne fraction ΔIterI(t), though, are larger than the uncertainties in IUsatI(t) (Table 6).

6. Conclusions

A framework is applied to understand the sensitivity of surface warming on cumulative carbon emissions ΔTI involving the product of three terms: 1) the dependence of surface warming on radiative forcing, 2) the fractional radiative forcing contribution from atmospheric CO2, and 3) the dependence of radiative forcing from atmospheric CO2 on cumulative carbon emissions (Williams et al. 2016). The metric ΔTI is equivalent to the transient climate response to cumulative carbon emissions (TCRE) for climate model integrations including radiative forcing only from atmospheric CO2 (Gillett et al. 2013; Collins et al. 2013) and approaches the equilibrium climate response to cumulative carbon emissions (ECRE) on centennial to millennial time scales (Frölicher and Paynter 2015).

The framework for ΔTI provides additional insight to the TCRE by separating the warming response into different dependences, which may be mechanistically connected to heat and carbon uptake (Goodwin et al. 2015; Williams et al. 2016), as well as including the effects of non-CO2 radiative forcing. The framework provides a useful way to interpret and compare the response of Earth system models including a range of radiative forcing, although additional weighting of different radiative contributions from other non-CO2 agents may also be needed (Hansen et al. 1984).

The evolution of ΔTI in our diagnostics of a suite of Earth system model experiments, 10 models for radiative forcing from only atmospheric CO2 and nine models forced by representative concentration pathways, may be understood in terms of an evolution toward an equilibrium response. Initially ΔTI is strongly affected by ocean heat uptake and ocean carbon undersaturation (Goodwin et al. 2015) but eventually evolves toward an equilibrium state where the equilibrium climate response to emissions is determined by a combination of climate parameters a/(λIB) (Williams et al. 2012), depending on effective radiative forcing from CO2, climate sensitivity and the amount of available carbon in the atmosphere and ocean.

The ocean affects the evolution of ΔTI by sequestering both heat and carbon (Solomon et al. 2009; Goodwin et al. 2015). Our diagnostics of one of the Earth system models highlights how the ocean sequestration of additional heat and carbon preferentially occurs over the upper ocean for the next century, involving the transfer of fluid from the mixed layer to the underlying thermocline. This ventilation process is particularly active in the North Atlantic (Marshall et al. 1993) and Southern Oceans (Frölicher et al. 2015), so that the global climate response is likely to be disproportionally affected by these well-ventilated regions. There is also the possibility that the long-term ΔTI is affected by changes in the sequestration of heat and carbon into the deep ocean. There are also surface warming feedbacks from changes in atmospheric CO2, which may itself be altered by other physical and biogeochemical mechanisms.

The intermodel variation in ΔTI is important in affecting the maximum permitted fossil-fuel cumulative carbon emissions to avoid dangerous climate, which for a 2°C warming typically range from 500 to 900 PgC for RCP8.5 (Table 4). Intermodel differences in projections for the maximum permitted carbon to meet warming targets may then be interpreted in terms of our dependences of surface warming on radiative forcing, the fractional radiative forcing contribution from atmospheric CO2, and radiative forcing from atmospheric CO2 on cumulative carbon emissions. The uncertainty in ΔTI based on intermodel differences decreases in time over the century. At the start of the century, intermodel differences in the ΔTI are determined by differences in the climate sensitivity λ−1, ocean heat and carbon uptake, and non-CO2 radiative forcing contributions. Toward the end of the century, the intermodel differences in the ΔTI are more affected by differences in λ−1 and changes in the terrestrial store of carbon. The larger intermodel spread in ΔTI at the start of the century may be reduced by constraining the estimates of ocean heat uptake and the ocean undersaturation of carbon used in the Earth system models.

In summary, our framework may be used to understand the surface warming response to cumulative carbon emissions ΔTI and intermodel differences in its value. The mechanisms by which heat and carbon are sequestered in the climate system help determine ΔTI by altering how surface warming depends on radiative forcing and how radiative forcing from atmospheric CO2 depends on emissions. Intermodel differences in ΔTI arise then from intermodel differences in how heat and carbon are cycled in the climate system, as well as from differences in the equilibrium climate parameters for the Earth system models.

Acknowledgments

The authors acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, responsible for CMIP, together with support and software development by the U.S. Department of Energy Program for Climate Model Diagnosis and Intercomparison, and the Global Organisation for Earth System Science Portals, as well as software infrastructure from the IPSL modeling group. The authors thank Chris Jones (Met Office) for providing compatible emissions and Piers Forster for providing the adjusted radiative forcing. This study benefited from feedback from three referees and detailed comments from the editor, Karen Shell. Our derived data products are available from the British Ocean Data Centre (enquiries@bodc.ac.uk). RGW, VR, and PG acknowledge support from the U.K. Natural Environmental Research Council (NE/N009789/1). PG also acknowledges support from U.K. NERC Grant NE/P01495X/1, and LB acknowledges support from the EU H2020 CRESCENDO project (Grant 641816).

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Footnotes

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