Sensitivity of Global Warming to Carbon Emissions: Effects of Heat and Carbon Uptake in a Suite of Earth System Models

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 dependenceof surface warming on radiative forcing, the fractional radiative forcing from CO 2 , and the dependence of radiative forcing from CO 2 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 anensembleofEarthsystemmodels,forcedeitherbyanannualincreaseinatmosphericCO 2 orbyRCPsuntil year 2100. The sensitivity of surface warming to carbon emissions is controlled by a temporal decrease in the dependence of radiative forcing from CO 2 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 CO 2 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.


Introduction
Climate model projections reveal a simple emergent relationship for our future climate: surface warming increases nearly linearly with the cumulative CO 2 emitted since the preindustrial age. This relationship has been illustrated in terms of how surface warming increases with the cumulative CO 2 emission using a climate model emulator (Allen et al. 2009), an Earth system model of intermediate complexity Zickfeld et al. 2009), and a suite of Earth system models forced by an annual rise in atmospheric CO 2 (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 DT/DI is defined by the ratio of the change in global-mean surface air temperature DT (K) and the change in fossil-fuel cumulative carbon emission DI (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 CO 2 forcing, DT/DI 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 .
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 CO 2 and nine Earth system models following representative concentration pathways (RCPs) (Moss et al. 2010) until year 2100, including non-CO 2 radiative forcing and carbon emissions from land-use and landcover changes (Collins et al. 2013). In both sets of experiments, the sensitivity of the surface warming to fossil-fuel cumulative carbon emissions DT/DI remains nearly constant in time for each individual model (Figs. 1a,b), although its value differs for each model. This relationship between surface warming and cumulative carbon emissions DT/DI is important in providing guidance to policy makers on how to avoid dangerous climate Zickfeld et al. 2009Zickfeld et al. , 2012Matthews et al. 2012;Friedlingstein et al. 2014). For example, in our subset of models, a 28C 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 CO 2 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 DT/DI 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 DT/DI is examined using diagnostics of climate model projections. We explore the view that the cause of these intermodel differences in DT/DI 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 CO 2 or following RCP scenarios plus non-CO 2 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).

Theory for how surface warming depends on carbon emissions
The sensitivity of surface warming on cumulative carbon emissions DT/DI may be connected to the surface warming and radiative forcing. For a regime where the radiative forcing is only affected by changes in atmospheric CO 2 , DT/DI may be written in terms of the product of two terms: the ratio of the surface warming and the radiative forcing DT/DR and the ratio of the radiative forcing and the cumulative carbon emissions DR/DI: For a regime where the radiative forcing is also affected by other non-CO 2 radiative contributions, the climate metric DT/DI is related to the product of three terms: the ratio of the surface warming and the radiative forcing DT/DR, the ratio of the radiative forcing and the radiative forcing from atmospheric CO 2 DR/DR CO2 , and the ratio of the radiative forcing from atmospheric CO 2 and the cumulative carbon emissions DR CO2 /DI (Williams et al. 2016): In the limit of the radiative forcing only being from atmospheric CO 2 , DR/DR CO2 becomes unity and (1) and (2) become equivalent. Theory is next applied for the ratios of the surface warming to radiative forcing DT/DR and the radiative forcing from atmospheric CO 2 to cumulative emissions DR CO2 /DI, 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 DR (in W m 22 ) is expected to drive an increase in global-mean, surface air temperature DT, such that FIG. 1. Change in global-mean surface air temperature DT(t) (K) vs change in fossilfuel cumulative carbon emissions DI (PgC) since the preindustrial age (a) for 10 different Earth system models assuming an annual 1% increase in atmospheric CO 2 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 DI is greater in (a) than in (b). The dashed gray lines connect a 28C warming back to a maximum permitted cumulative carbon emission (PgC).  Andrews et al. 2012;Forster et al. 2013) and DR is defined as positive into the ocean. For a transient response, the increase in surface temperature DT(t) from the increase in radiative forcing DR is reduced in magnitude by the planetary heat uptake N(t) (W m 22 ), as represented in an extension to (3) (Gregory et al. 2004;Winton et al. 2010) by where «(t)N(t) is the scaled global-mean heat flux into the climate system and the efficacy «(t) is a nondimensional scaling taken to vary in time . 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 DT/DR is then directly connected to the scaled ocean heat uptake divided by the radiative forcing: b. Dependence of radiative forcing from atmospheric CO 2 on carbon emissions For a global air-sea equilibrium, the logarithm of the fractional change in atmospheric CO 2 is related to the fossil-fuel cumulative carbon emission DI and the increase in the terrestrial carbon inventory DI ter divided by the buffered carbon inventory (Goodwin et al. 2007): where D lnCO 2 (t equilib ) 5 ln[CO 2 (t equilib )/CO 2 (t o )], t o is the time of the preindustrial age, t equilib is the time for the global equilibrium to be reached, DI(t) 2 DI ter (t) represents the net carbon emissions to the combined atmosphere and ocean, and the buffered atmosphere and ocean carbon inventory I B (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 CO 2 exceeds the equilibrium value CO 2 (t equilib ). Goodwin et al. (2015) extend (6) by taking into account the global ocean undersaturation of carbon to define the transient atmospheric CO 2 response by where I Usat (t) in PgC measures the undersaturation of the global ocean with respect to the instantaneous atmospheric CO 2 (t). The radiative forcing from atmospheric CO 2 is assumed to depend on the change in the logarithm of atmospheric CO 2 (Myhre et al. 1998): where a in W m 22 is the effective radiative forcing coefficient (Forster et al. 2013). The dependence of radiative forcing from atmospheric CO 2 on fossil-fuel cumulative carbon emissions DR CO2 /DI is then defined by combining (7) and (8): which reveals the connection to global ocean undersaturation of carbon I Usat (t) and the increase in the terrestrial carbon inventory DI ter (t), both scaled by the cumulative carbon emission DI(t).

Surface warming as a function of cumulative emissions
Our theory highlights how the ratio of surface warming to cumulative carbon emissions DT/DI 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 fossilfuel carbon emissions (Jones et al. 2013), radiative forcing coefficients (Forster et al. 2013), and some ocean variables needed for our diagnostics.
The Earth system models are forced by a rise in atmospheric CO 2 , rather than by carbon emissions. The models couple the carbon cycle and climate together by sources and sinks of CO 2 being affected by atmospheric CO 2 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 CO 2 , DI(t) has been diagnosed from the imposed rate of change in atmospheric CO 2 plus the globally integrated air-sea and air-land carbon fluxes. For the RCP scenarios, DI(t) has been diagnosed from the difference between the imposed atmospheric CO 2 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 DR(t) is provided by Forster et al. (2013). The radiative forcing DR CO2 (t) is diagnosed from atmospheric CO 2 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-CO 2 greenhouse gases, anthropogenic aerosols, and land-use changes.
The global ocean heat uptake N(t) is evaluated from the tendency in global ocean heat content. The climate 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) 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 CO 2 a; the climate parameter l; and preindustrial values for the global-mean dissolved inorganic carbon in the ocean DIC, the saturated value DIC sat , the buffer factor B, and the buffered atmosphere and ocean carbon inventory I B ; and the implied ECRE a/(lI B ) (Williams et al. 2012 feedback parameter l is diagnosed from a regression between DT(t), DR(t), and N(t) using (4) following Gregory and Forster (2008) with an initial estimate of « 5 1.1 ( Table 2). The nondimensional weighting of ocean heat uptake «(t) is then evaluated from DT(t), DR(t), N(t), and l using (4).
The atmospheric and ocean carbon inventories are diagnosed from atmospheric CO 2 and the dissolved inorganic carbon (DIC). The saturated dissolved inorganic carbon DIC sat 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 DIC sat is evaluated for the instantaneous atmospheric CO 2 and ocean potential temperature, salinity, alkalinity, and phosphate ( Table 2). The global ocean undersaturation in carbon I Usat (t) is diagnosed from the difference in the saturated dissolved inorganic carbon DIC sat and the actual DIC anomaly over the global ocean, given by I Usat (t) 5 V[DDIC sat (t) 2 DDIC(t)], where V is the ocean volume. The buffered carbon inventory for the atmosphere and ocean, given by I B 5 I atmos (t 0 ) 1 VDIC sat (t o )/B, at the preindustrial age t 0 is diagnosed from the sum of the atmospheric inventory I atmos plus the saturated ocean carbon inventory VDIC sat divided by the buffer factor, given by B 5 (DCO 2 /CO 2 )/(DDIC/DIC). The I B and B are evaluated for the preindustrial age at year 1861 with an atmospheric CO 2 of 286 ppm and an atmospheric carbon inventory of 607 PgC (Table 2).

b. For an idealized increase in atmospheric CO 2
For the set of Earth system models forced by an annual 1% rise in atmospheric CO 2 , 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 22 (Fig. 2b) associated with a cumulative carbon emission typically ranging from 1500 to 2000 PgC (Fig. 2c).
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 CO 2 scenario by Gillett et al. (2013).
The near constancy of DT/DI is achieved by two nearly compensating changes in (1), involving the product of the sensitivities of surface warming to radiative forcing DT/DR and radiative forcing to cumulative carbon emissions DR/DI. First, the ratio of the surface warming to radiative forcing DT/DR increases in time for all the models, typically rising from 0.5 to 0.7 K (W m 22 ) 21 ( Fig. 2e; Table 3). Second, the ratio of the radiative forcing on cumulative carbon emissions DR/DI decreases in time for all the models, typically from 3.7 to 2.5 W m 22 (10 3 PgC) 21 ( Fig. 2f; Table 3). This partial compensation leads to a near constancy in DT/DI 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 28C warming is reached after a fossil-fuel cumulative carbon emission ranging from 445 to 911 PgC (Table 4;  There is a limited temporal trend in the sensitivity of the surface warming to cumulative carbon emissions DT/DI 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 DT/DI from 3 to 2.5 K (10 3 PgC) 21 from years 2010 to 2095 (Table 4). The evolution of DT/DI 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.
The temporal trends in DT/DI may be understood by considering the product of DT/DR, DR CO2 /DI, and DR/DR CO2 from (2). The ratio of the surface warming on radiative forcing DT/DR generally increases in time  (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 CO 2 on carbon emissions DR CO2 /DI decreases in time across all RCP scenarios (Fig. 3e); the model mean for RCP8.5 decreasing from 4.8 to 3.0 W m 22 (10 3 PgC) 21 over this century (Table 4). This dependence DR CO2 /DI decreases as a result of the effect of ocean and terrestrial uptake of carbon. This decrease is primarily due to how , and (f) the ratio of the radiative forcing and the radiative forcing from atmospheric CO 2 , DR/DR CO2 . 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-CO 2 greenhouse gases and aerosols. 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 CO 2 DR/DR CO2 is nearly constant for RCP8.5 but slightly declines for the other RCPs (Fig. 3f). This term depends on the effects of non-CO 2 greenhouse gases and aerosols. The trends in this term are relatively small compared with its large intermodel scatter.

d. Evolution of DT/DI toward an equilibrium value
The different intermodel evolution of DT/DI 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 DR/DR CO2 5 1], the surface temperature change asymptotically approaches where DI(t) 2 DI ter (t) represents the net carbon emissions to the combined atmosphere and ocean. The climate parameters a/(lI B ) determining the equilibrium state (Williams et al. 2012) vary in the model ensemble from 1.2 to 2.8 K (10 3 PgC) 21 due to variations in a from 4.2 to 6.2 W m 22 , I B from 3360 to 3550 PgC, and l from 0.4 to 1.1 W m 22 K 21 (Table 2 for RCP8.5); our diagnostics of l for RCP8.5 are slightly lower than the corresponding values for CO 2 forcing alone, which range from 0.6 to 1.4 W m 22 K 21 (Forster et al. 2013).
For most of the integrations of the Earth system models, DT/DI evolves toward the equilibrium value a/(lI B ) 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 DT/DI from 3.1 to 1.8 K (10 3 PgC) 21 over this century, which is toward the equilibrium value a/(lI B ) of 1.3 K (10 3 PgC) 21 . For HadGEM2-ES for RCP8.5, there is an increase in DT/DI from 2.1 to 2.4 K (10 3 PgC) 21 over this century, which is the same as the equilibrium value a/(lI B ) of 2.4 K (10 3 PgC) 21 .
Hence, the climate parameters a/(lI B ) affect the evolution of DT/DI for each Earth system model over this century and provide a measure of the ECRE .

Mechanisms controlling the sensitivity of surface warming on carbon emissions
Our focus is now to highlight the underlying mechanisms controlling the evolution of DT/DI, 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 DT/DI 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.
The dependence of the surface warming on radiative forcing DT/DR varies with the ratio of the ocean heat uptake to the radiative forcing N(t)/DR(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, DT/DR 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 CO 2 on carbon emissions DR CO2 /DI varies with the ratio of the global ocean undersaturation and the carbon emissions I Usat /DI(t) from (9). The rise in atmospheric CO 2 leads to a faster increase in saturated dissolved inorganic carbon in the ocean DIC sat (t) than the actual DIC(t) (Table 2), so that the global ocean undersaturation in carbon, given by I Usat (t) 5 V[DDIC sat (t) 2 DDIC(t)], increases in time. The rate of increase in the ocean carbon undersaturation I Usat (t) is, though, less than the increase in cumulative carbon emission DI(t), so that the normalized ocean carbon undersaturation I Usat (t)/DI(t), progressively decreases in time (Fig. 5d) for all depth ranges. Hence, DR CO2 /DI 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 DR(t); for RCP8.5, the model mean of DR(t) increases from 1.9 to 7.2 W m 22 from years 2010  Table 5). The increase in DR(t) is primarily due to the atmospheric CO 2 contribution DR CO2 (t), rising from 1.6 to 5.7 W m 22 from years 2010 to 2095 (Fig. 6, gray dashed line; Table 6). The increase in radiative forcing DR(t) drives both the expected surface warming response lDT(t) and the scaled ocean heat uptake «(t)N(t) (Fig. 6, red and blue lines) from (4). For RCP8.5, the model means of lDT are 0.8 and 3.5 W m 22 at years 2010 and 2095, respectively, while for «(t)N(t) are 1.1 and 3.7 W m 22 at years 2010 and 2095 (Table 5). The surface climate response lDT(t) is less than the scaled ocean heat uptake, «(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 DT/DR typically ranges from 0.3 to nearly 0.7 K (W m 22 ) 21 by year 2010 and increases to 0.5-0.9 K (W m 22 ) 21 by year 2095 (Fig. 7; Table 5). In nearly all the Earth system models, the scaled ocean heat uptake «(t)N(t) takes up a slightly smaller fraction with time of the radiative forcing DR(t) so that the model mean of their ratio «(t)N(t)/DR(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 DT/DR from (5) and only does not occur for MIROC-ESM (Fig. 7). These changes from «(t)N(t) are mainly due to changes in ocean heat uptake but also involve temporal changes in «(t) over the first few decades then with «(t) approaching a value close to 1 in the latter half of the century.
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 DT/DR: 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 CO 2 on carbon emissions The cumulative carbon emissions from fossil fuels DI(t) drives an increase in the sum of the atmospheric, ocean, and terrestrial inventories of carbon, referred to as DI atmos (t), DI ocean (t), and DI ter (t), respectively: DI(t) 5 DI atmos (t) 1 DI ocean (t) 1 DI ter (t) .
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 DI ter is partly due to including opposing anthropogenic contributions from land-use changes and increases in terrestrial storage.
There is a more striking increase in the carbon undersaturation of the ocean I Usat (t) with respect to the instantaneous atmospheric CO 2 (t) for all the Earth system models for RCP8.5 (Fig. 8, gray dashed line): I Usat (t) increases typically from 700 to over 2000 PgC from 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 DR(t) (W m 22 ; black line), drives an increase in ocean heat uptake «(t)N(t) (blue line) and a surface climate response lDT(t) (red line); l is taken from a regression (Gregory and Forster 2008), and «(t) is diagnosed to ensure that (4) holds. The increase in DR(t) is dominated by the increase in DR CO2 (t), rather than DR nonCO2 (t) (dashed and dotted gray lines, respectively). The ratio of ocean heat uptake and increase in radiative forcing «(t)N(t)/DR(t) decreases in time for all cases.  (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 CO 2 , 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): There is a decrease in the airborne fraction DI atmos /DI in GFDL-ESM2G, GFDL-ESM2M, MIROC-ESM, and MPI-ESM-LR, which is mainly offset by an increase in the landborne fraction DI ter /DI (Fig. 9). On the other hand, there are relatively small changes in the airborne fraction DI atmos /DI in CanESM2, HadGEM2-CC, HadGEM2-ES, and IPSL-CM5A-LR, where an increase in the landborne fraction DI ter /DI is offset by a decrease in the oceanborne fraction DI ocean /DI (Fig. 9). The ratio of the ocean carbon undersaturation and the cumulative carbon emissions I Usat (t)/DI(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 I Usat (t)/ DI(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 CO 2 on cumulative carbon emissions DR CO2 /DI TABLE 5. Radiative forcing DR(t), climate response lDT(t), scaled ocean heat uptake «N(t), and the dependence of surface warming on radiative forcing DT/DR, diagnosed from nine Earth system models for two 10-yr periods centered on years 2010 and 2095, including the nondimensional ratios of «N/DR; all diagnostics are for RCP8.5.  TABLE 6. Radiative forcing from atmospheric CO 2 DR CO2 (t) and its dependence on fossil-fuel cumulative carbon emissions DI(t), together with the nondimensional ratios of ocean undersaturation of carbon and change in the terrestrial store of carbon divided by cumulative carbon emissions, I Usat /DI and DI ter /DI, respectively, from nine Earth system models for two 10-yr periods centered on years 2010 and 2095; all diagnostics are for RCP8.5.  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 I Usat (t)/DI(t) from (9). There are smaller variations in the landborne fraction DI ter (t)/DI(t), which are more significant for GFDL-ESM2G and GFDL-ESM2M (Figs. 9b,c, green line).

d. Interpretation of changes in DT/DI
The contrasting trends of DT/DI for different Earth system models may be understood via its separate dependences in (2). The larger changes in DT/DI correspond to changes in either the dependence of surface warming on radiative forcing DT/DR or the dependence of radiative forcing from atmospheric CO 2 on carbon emissions DR CO2 /DI. For example, the significant drop in DT/DI for GFDL-ESM2M is achieved by a reduction in DR CO2 /DI involving a substantial decrease in the fractional ocean undersaturation of carbon I Usat (t)/DI(t) ( Table 6; Figs. 4b, 9b, and 10b). In contrast, the significant increase in DT/DI for HadGEM2-CC is due to an increase in DT/DR involving a reduction in ocean heat uptake divided by radiative forcing «N(t)/DR(t) ( Table  5; Figs. 4d and 7d). For the other models, the changes in DT/DI are relatively small, involving a partial compensation from the DT/DR and DR CO2 /DI contributions.
The additional factor affecting DT/DI is the non-CO 2 radiative forcing, involving both effects from non-CO 2 greenhouse gases and aerosols (Boucher et al. 2013;Pierrehumbert 2014). The relative importance of this non-CO 2 radiative forcing on DT/DI is approximately represented by the ratio of DR/DR CO2 , equivalent to 1 1 DR nonCO2 /DR CO2 (Fig. 11). The non-CO 2 radiative forcing contribution is relatively small and positive, DR nonCO2 (t), typically ranging from less than 1 to FIG. 7. Dependence of surface warming on radiative forcing DT/DR [K (W m 22 ) 21 ], for this century for nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The DT/DR 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 «(t)N(t)/DR(t) fsince DT/DR is equivalent to l 21 [1 2 «(t)N(t)/DR(t)]g.
2 W m 22 (Fig. 6, gray dotted line), implying that the positive radiative contribution from non-CO 2 greenhouse gases exceeds the negative radiative contribution from aerosols. The model mean for DR/DR CO2 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).

Intermodel uncertainty in the surface climate response
Our framework is now applied to understand the sources of intermodel variability of DT/DI 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 DT/DI and its contributions across the ensemble of Earth system models.
a. Uncertainty in the sensitivity of surface warming to emissions DT/DI The model uncertainty in DT/DI 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 CO 2 , the uncertainty in DT/DI decreases from 0.23 to 0.15 from years 45 and 135 (Table 3). For RCP8.5, the uncertainty in DT/DI likewise decreases from 0.31 to 0.22 for years 2010 and 2095 (Table 4). The uncertainty for the RCP8.5 scenario remaining high 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 DI(t) (PgC; red line) drives an increase in the carbon inventories in the atmosphere DI atmos (t) (black line), the ocean DI ocean (t) (blue line), and terrestrial system DI ter (t) (green line); the terrestrial response is of variable magnitude and sometimes positive or weakly negative. The undersaturation of carbon in the ocean I Usat (t) (PgC; dashed line) increases due to the fast pace of emissions.
until year 2020 may possibly be due to decadal variability being relatively important close to the start of the RCP forcing scenario.
If the uncertainties from each dependence, DT/DR, DR/DR CO2 , and DR CO2 /DI, are assumed to combine in the same manner as random errors, then the uncertainty squared for DT/DI should be comparable to the sum of the uncertainty squared for each contribution. However, the uncertainty for DT/DI is initially only comparable to the uncertainty for DT/DR but eventually becomes comparable to the contributions of both DT/DR and DR CO2 /DI (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 DT/DI is from the contribution of DT/DR, 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 DR CO2 /DI, 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 DR/DR CO2 is important, reaching 0.25 in year 2010, but then declines to 0.10 by year 2095 (Table 4; Fig. 12b, green line). This 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, DI atmos /DI (black line) and DI ocean /DI (blue line), respectively, as well as a slight increase in the landborne fraction DI ter /DI (green line). The ratio of the ocean undersaturation of carbon and the emitted carbon I Usat (t)/DI(t) (dashed line) declines as the ocean gradually takes up a larger fraction of the emissions. contribution is controlled by variations in radiative forcing from non-CO 2 greenhouse gases and aerosols between the Earth system models.
b. Uncertainties in DT/DR and DR CO2 /DI The uncertainty in the dependence of surface warming on radiative forcing DT/DR based on (5) is determined by the uncertainty for the equilibrium climate sensitivity l 21 and the normalized ocean heat uptake «(t)N(t)/DR(t) (Figs. 12c,d).
For the annual 1% increase in atmospheric CO 2 , the uncertainty in DT/DR is always larger than the contribution for the normalized ocean heat uptake «(t)N(t)/DR(t) (Fig. 12c, red and blue lines) and so is mainly determined by intermodel differences in l 21 .
For the RCP8.5 scenario, the uncertainty for DT/DR is affected by the uncertainty in l 21 , augmented by the normalized ocean heat uptake «(t)N(t)/DR(t) over the first couple of decades (Fig. 12c, red and blue lines). For year 2010, the uncertainty for DT/DR of 0.28 compares with an uncertainty for «(t)N(t)/DR(t) of 0.18 (Table 5), while for year 2095, the uncertainty decreases for DT/DR to 0.20 and for «(t)N(t)/DR(t) to 0.12. Hence, intermodel differences in DT/DR are initially affected by both l 21 and «(t)N(t)/DR(t) but later dominated by l 21 .
The intermodel differences in the dependence of radiative forcing from atmospheric CO 2 on emissions DR CO2 /DI based on (9) are affected by contributions from I Usat (t)/DI(t) and DI ter /DI(t), as well as by intermodel differences in a and I B (Tables 2 and 6). For the fractional ocean undersaturation and change in the terrestrial store of carbon, I Usat (t)/DI(t) and DI ter /DI(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 FIG. 10. Dependence of radiative forcing from atmospheric CO 2 on emissions DR CO2 /DI [W m 22 (10 3 PgC) 21 ] for this century for nine different Earth system models for RCP2.6, RCP4.5, RCP6.0, and RCP8.5. The DR CO2 /DI 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 I Usat (t)/DI(t) decreases in time, since DR CO2 /DI is equivalent to (a/I B )f1 1[I Usat (t) 2 DI ter (t)]/DI (t)g.
for year 2095, respectively (Table 6). Hence, the intermodel differences in DR CO2 /DI are initially mainly affected by differences in the fractional ocean undersaturation I Usat (t)/DI(t) and later also by differences in the landborne fraction DI ter /DI(t). The uncertainties in the landborne fraction DI ter /DI(t), though, are larger than the uncertainties in I Usat /DI(t) ( Table 6).

Conclusions
A framework is applied to understand the sensitivity of surface warming on cumulative carbon emissions DT/DI involving the product of three terms: 1) the dependence of surface warming on radiative forcing, 2) the fractional radiative forcing contribution from atmospheric CO 2 , and 3) the dependence of radiative forcing from atmospheric CO 2 on cumulative carbon emissions (Williams et al. 2016). The metric DT/DI is equivalent to the transient climate response to cumulative carbon emissions (TCRE) for climate model integrations including radiative forcing only from atmospheric CO 2 (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 .
The framework for DT/DI 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-CO 2 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 FIG. 11. The inverse fractional radiative forcing from atmospheric CO 2 DR/DR CO2 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 DR/DR CO2 with different models, an overall decrease in (b), (c), (f), and (h) and conversely an overall increase in (d), (e), and (g). different radiative contributions from other non-CO 2 agents may also be needed (Hansen et al. 1984).
The evolution of DT/DI in our diagnostics of a suite of Earth system model experiments, 10 models for radiative forcing from only atmospheric CO 2 and nine models forced by representative concentration pathways, may be understood in terms of an evolution toward an equilibrium response. Initially DT/DI 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/(lI B ) (Williams et al. 2012), depending on effective radiative forcing from CO 2 , climate sensitivity and the amount of available carbon in the atmosphere and ocean.
The ocean affects the evolution of DT/DI by sequestering both heat and carbon 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 , 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 DT/DI 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 CO 2 , which may itself be altered by other physical and biogeochemical mechanisms.
The intermodel variation in DT/DI is important in affecting the maximum permitted fossil-fuel cumulative FIG. 12. Intermodel uncertainty in the sensitivities of surface warming to cumulative emissions DT/DI and surface warming to radiative forcing DT/DR for (a),(c) 10 Earth system models for an annual 1% increase in atmospheric CO 2 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 DT/DI (black line) decreases in time. In (a), the dominant contribution to the uncertainty is from DT/DR (red line) and there is only a comparable contribution from DR CO2 /DI (blue line) after 80 yr. In (b), there are dominant contributions from DT/DR (red line) over most of the record, from DR/DR CO2 (green line) over the historical record and from DR CO2 /DI (blue line) after year 2020. In (c) and (d), the intermodel uncertainty in DT/DR (dashed red line) is due to variability in the equilibrium climate parameter l 21 (black line) augmented by the fraction of the radiative forcing taken up by ocean heat flux «N/DR (blue line). carbon emissions to avoid dangerous climate, which for a 28C 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 CO 2 , and radiative forcing from atmospheric CO 2 on cumulative carbon emissions. The uncertainty in DT/DI based on intermodel differences decreases in time over the century. At the start of the century, intermodel differences in the DT/DI are determined by differences in the climate sensitivity l 21 , ocean heat and carbon uptake, and non-CO 2 radiative forcing contributions. Toward the end of the century, the intermodel differences in the DT/DI are more affected by differences in l 21 and changes in the terrestrial store of carbon. The larger intermodel spread in DT/DI 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 DT/DI and intermodel differences in its value. The mechanisms by which heat and carbon are sequestered in the climate system help determine DT/DI by altering how surface warming depends on radiative forcing and how radiative forcing from atmospheric CO 2 depends on emissions. Intermodel differences in DT/DI 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.