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    The evolution of changes, with respect to 1850 control run averages, in the atmospheric CO2 concentration C (ppm), emissions E (PgC yr−1), temperature T (°C), and CO2 fluxes into the atmosphere FA, land FL, and ocean FO (PgC yr−1). Values are smoothed with an 11-yr moving average. Results for RCP2.6, RCP4.5, and RCP8.5 are plotted in blue, green, and orange, respectively, and emission- and concentration-driven results are plotted as solid and dashed lines.

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    The integrated atmospheric carbon budget terms (PgC) from 2005. The range plotted for the RCP4.5 and RCP8.5 cases increases by a factor of 3 and 6, respectively, compared to the RCP2.6 case. The emission-driven results are plotted as solid lines, with the concentration-driven results as dashed lines.

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    The evolution of the atmospheric CO2 loading H (PgC), CO2 flux into the atmosphere FA (PgC yr−1), emissions E (PgC yr−1), and temperature change T (°C) for the RCP4.5 scenario. Values are smoothed with an 11-yr moving average. Results for the biogeochemically (asterisk), radiatively (plus sign), and fully coupled (prime) simulations are in blue, green, and orange, respectively, and emission- and concentration-driven values are plotted as solid and dashed lines, with differences indicated as δX.

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    The atmosphere, land, and ocean carbon–climate feedback parameters (ΓA, ΓL, and ΓO) calculated from the radiatively coupled simulations [PgC (yr °C)−1] and plotted against T+ and the carbon–concentration feedback parameters (BA, BL, and BO) calculated from the biogeochemically coupled simulations [PgC (yr ppm)−1] plotted against concentration change C* for the emission-driven and C′ for the concentration-driven cases. Values entering the calculations are smoothed with a 30-yr moving average. Results for RCP4.5 and RCP8.5 are plotted in green and orange, respectively, and emission- and concentration-driven results are plotted as solid and dashed lines.

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    CO2 flux F′ (PgC yr−1) into the atmosphere (solid) and the approximation ΓT′ + BC′ (dashed) for emission- and concentration-driven cases. Values for RCP4.5 and RCP8.5 are plotted in green and orange, respectively.

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    Atmospheric CO2 deficit factors as percentages. Values for RCP4.5 and RCP8.5 are plotted in green and orange, respectively, and emission- and concentration-driven results are plotted as solid and dashed lines, respectively. Values entering the calculation are smoothed with a 30-yr running mean.

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Feedbacks in Emission-Driven and Concentration-Driven Global Carbon Budgets

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  • 1 Canadian Centre for Climate Modelling and Analysis, Victoria, British Columbia, Canada
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Abstract

Emissions of CO2 into the atmosphere affect the carbon budgets of the land and ocean as biogeochemical processes react to increased CO2 concentrations. Biogeochemical processes also react to changes in temperature and other climate parameters. This behavior is characterized in terms of carbon–concentration and carbon–climate feedback parameters. The results of this study include 1) the extension of the direct carbon feedback formalism of Boer and Arora to include results from radiatively coupled simulations, as well as those from the biogeochemically coupled and fully coupled simulations used in earlier analyses; 2) a brief analysis of the relationship between this formalism and the integrated feedback formalism of Friedlingstein et al.; 3) the feedback analysis of simulations based on each of the representative concentration pathways (RCPs) RCP2.6, RCP4.5, and RCP8.5; 4) a comparison of the effects of specifying atmospheric CO2 concentrations or CO2 emissions; and 5) the quantification of the relative importance of the two feedback mechanisms in terms of their cumulative contribution to the change in atmospheric CO2.

Feedback results are broadly in agreement with earlier studies in that carbon–concentration feedback is negative for the atmosphere and carbon–climate feedback is positive. However, the magnitude and evolution of feedback behavior depends on the formalism employed, the scenario considered, and the specification of CO2 from emissions or as atmospheric concentrations. Both feedback parameters can differ by factors of two or more, depending on the scenario and on the specification of CO2 emissions or concentrations. While feedback results are qualitatively useful and illustrative of carbon budget behavior, they apply quantitatively to particular scenarios and cases.

Corresponding author address: V. K. Arora, Canadian Centre for Climate Modelling and Analysis, P.O. Box 1700, STN CSC, Victoria, BC V8W 3V6, Canada. E-mail: vivek.arora@ec.gc.ca

This article is included in the (C4MIP) Climate–Carbon Interactions in the CMIP5 Earth System Models special collection.

Abstract

Emissions of CO2 into the atmosphere affect the carbon budgets of the land and ocean as biogeochemical processes react to increased CO2 concentrations. Biogeochemical processes also react to changes in temperature and other climate parameters. This behavior is characterized in terms of carbon–concentration and carbon–climate feedback parameters. The results of this study include 1) the extension of the direct carbon feedback formalism of Boer and Arora to include results from radiatively coupled simulations, as well as those from the biogeochemically coupled and fully coupled simulations used in earlier analyses; 2) a brief analysis of the relationship between this formalism and the integrated feedback formalism of Friedlingstein et al.; 3) the feedback analysis of simulations based on each of the representative concentration pathways (RCPs) RCP2.6, RCP4.5, and RCP8.5; 4) a comparison of the effects of specifying atmospheric CO2 concentrations or CO2 emissions; and 5) the quantification of the relative importance of the two feedback mechanisms in terms of their cumulative contribution to the change in atmospheric CO2.

Feedback results are broadly in agreement with earlier studies in that carbon–concentration feedback is negative for the atmosphere and carbon–climate feedback is positive. However, the magnitude and evolution of feedback behavior depends on the formalism employed, the scenario considered, and the specification of CO2 from emissions or as atmospheric concentrations. Both feedback parameters can differ by factors of two or more, depending on the scenario and on the specification of CO2 emissions or concentrations. While feedback results are qualitatively useful and illustrative of carbon budget behavior, they apply quantitatively to particular scenarios and cases.

Corresponding author address: V. K. Arora, Canadian Centre for Climate Modelling and Analysis, P.O. Box 1700, STN CSC, Victoria, BC V8W 3V6, Canada. E-mail: vivek.arora@ec.gc.ca

This article is included in the (C4MIP) Climate–Carbon Interactions in the CMIP5 Earth System Models special collection.

Keywords: Carbon cycle

1. Introduction

The distribution of carbon in the atmosphere, land, and ocean is changing as a consequence of the anthropogenic emission of CO2. Biogeochemical processes in the carbon cycle are directly affected by an increase in atmospheric CO2, which alters the flux of carbon between the atmosphere and the underlying surface. An increase in atmospheric CO2 also affects the energy budget, resulting in warmer temperatures and other changes in climate that, in turn, affect the carbon budget. The direct carbon budget response to an increase in CO2 is termed the carbon–concentration feedback, and the indirect response via changes in temperature and other climate quantities is termed the carbon–climate feedback.

Feedbacks in the climate system are investigated using full earth system models (ESMs) by means of special simulations in which the atmospheric CO2 concentration and surface temperature are controlled as they affect the flux of carbon between the atmosphere and the underlying surface. Results from a total of 18 simulations are analyzed in order to isolate, quantify, and compare feedbacks. Three Integrated Assessment Modeling Consortium (IAMC) representative concentration pathways (RCPs), namely, RCP2.6, RCP4.5, and RCP8.5, are used, for which different CO2 concentrations, non-CO2 greenhouse gases, and aerosols all affect the climate. For each of these RCPs, “fully coupled” simulations are performed together with “biogeochemically coupled” and “radiatively coupled” simulations, where terms in the energy and carbon budgets are controlled. Finally, for each of these nine cases, scenarios where the rate of CO2 emission into the atmosphere is specified (emission driven) and where the atmospheric CO2 itself is specified (concentration driven) are performed for a total of 18 simulations (in addition to a control simulation).

The Coupled Carbon Cycle Climate Model Intercomparison Project (C4MIP) of Friedlingstein et al. (2006) surveys earlier studies of the effect of changing climate on the carbon budget and characterizes carbon cycle feedbacks in terms of the feedback parameters (γ, β) obtained by linearly relating cumulative CO2 surface flux changes F′ to temperature T′ and CO2 concentration C′ changes. Boer and Arora (2009) argue for a more direct connection, paralleling that for feedbacks in the energy budget, by relating flux changes directly to concurrent temperature and CO2 concentration changes, giving feedback parameters (Γ, B). Some recent studies of carbon feedback processes are listed in Table 1. They use CO2 emission and/or concentration information based on the B1 and A2 scenarios from the Special Report on Emissions Scenarios (SRES) or the special case of CO2 concentrations increasing by 1% yr−1. Only one column is completely populated, since all of the analyses include results from a biogeochemically coupled simulation where CO2 changes affect the carbon budget but not the radiative budget (all models also perform a fully coupled simulation, of course). More recent analyses also make use of results from radiatively coupled simulations where CO2 radiative effects are included, but not biogeochemical effects. Feedbacks may be calculated using full ESMs and/or less comprehensive earth system models of intermediate complexity (EMICs), and the models may be forced by specifying emissions or concentrations of CO2 and may include forcings from other greenhouse gases, aerosols, etc.

Table 1.

Studies involving carbon feedbacks. The direct feedback formalism follows the approach of Boer and Arora (2009), while the time-integrated formalism follows Friedlingstein et al. (2006). Special simulations are radiatively or biogeochemically coupled. RCP forcings include non-CO2 greenhouse gas, aerosol, and land use change, volcano, and solar forcing, as opposed to CO2-only radiative forcing changes in the remaining cases. The unit 1% yr−1 indicates that the CO2 concentration is specified, increasing by 1% yr−1. The results with an x*, using the direct feedback formalism applied to globally averaged quantities, are mentioned briefly in the text. Special Report on Emissions Scenarios (SRES)/SP indicates that a range of SRES scenarios and specified CO2 “stabilization profiles” were used.

Table 1.

The studies listed in Table 1 are in broad agreement that carbon–climate feedback is positive and carbon–concentration feedback is negative for the atmospheric carbon budget, with the opposite sign of the feedbacks for the land and ocean carbon budgets. A number of the studies also note the nonlinearity of the carbon–climate system and the dependence of the feedback parameters on model formulation, feedback formalism, the nature of the CO2 specification, and the scenario employed.

One of the purposes of the study is the extension of the “direct” carbon feedback formalism of Boer and Arora (2009) to include results from radiatively coupled simulations, as well as those from the biogeochemically coupled and fully coupled simulations. The resulting equations are compared with those arising following the “integrated” feedback formalism of Friedlingstein et al. (2006) applied to these cases. Another purpose of the study is to evaluate and characterize differences between feedback parameters in emission- and concentration-driven cases and the implications for the carbon budget across the newer RCP scenarios. Finally, the importance of the carbon–climate and carbon–concentrations feedback processes for the carbon budget is not obvious from the magnitudes and signs of the feedback parameters themselves, which have different units. The relative importance of the feedback processes is explicitly measured in terms of “deficit factors,” which quantify the fraction of the CO2 emitted during the twenty-first century that is lost to the land and ocean as a consequence of the different feedback mechanisms.

2. The earth system model and scenarios

a. CanESM2

CanESM2 is the second generation Canadian Earth System Model (Arora et al. 2009; Christian et al. 2010), as described in Arora et al. (2011; see also their auxiliary material). The physical atmospheric component of CanESM2 [Canadian Atmospheric Model, version 4 (CanAM4)] extends to 1 hPa with 35 layers of increasing thickness, a horizontal resolution of T63, and a range of improved physical parameterizations. The physical ocean component has 40 levels, compared to 29 levels in the first generation (CanESM1), with more levels in the euphotic zone and a resolution of 10 m in the upper ocean. The horizontal resolution of the ocean is 1.41° (longitude) × 0.94° (latitude) and incorporates improved physical parameterizations.

The Canadian Terrestrial Ecosystem Model (CTEM) simulates terrestrial ecosystem processes and their interaction with the atmosphere. The budgets and interactions of three live vegetation pools (leaves, stem, and root) and two dead pools (litter and soil organic carbon) are simulated for nine plant functional types (PFTs), namely, needleleaf evergreen and deciduous trees, broadleaf evergreen and cold and dry deciduous trees, and C3 and C4 crops and grasses (Arora and Boer 2010). CTEM treats land use change interactively as a contribution to the flux between land and ocean, rather than as an external emission component.

The Canadian Model of Ocean Carbon (CMOC) simulates the ocean carbon budget and its interaction with the atmosphere. CMOC includes an inorganic chemistry module (solubility pump) and a nitrogen–phytoplankton–zooplankton–detritus (NPZD) ecosystem model (organic and carbonate pumps) for this purpose (Zahariev et al. 2008). The ocean ecosystem model includes photoacclimation of chlorophyll concentration to varying irradiance, which, in turn, affects penetrating shortwave radiation and, thus, subsurface heating.

b. Representative concentration pathways

The simulations analyzed here make use of RCPs of radiatively important greenhouse gases (GHGs) (see http://www.pik-potsdam.de/~mmalte/rcps/). The RCPs are based on results from a number of integrated assessment models (IAMs), which develop atmospheric concentrations of CO2, other greenhouse gases, and aerosols based on historical and projected emissions. ESM-based simulations may be either emission driven, where evolving emissions of CO2 are specified and the concentrations simulated, or concentration driven, where the evolving concentration of CO2 is specified directly and emissions are implicit. Specifying CO2 emissions for a particular RCP in the ESM need not lead to the CO2 concentrations associated with that RCP from the IAM. The ESM’s carbon budget is the result of complex interactions between the atmosphere, ocean, and land, and the ESM need not behave in the same way as the IAM. Similarly, the implicit emissions that result when CO2 concentrations are specified in the ESM need not match the emissions from the IAM for that RCP. The differences in the carbon cycle behavior of an ESM when an RCP is implemented in terms of emissions or concentrations are one of the interests of the study reported here. The RCPs also include information on radiatively important GHGs other than CO2, on aerosol species and their precursors, and on land cover change (http://luh.umd.edu/) treated interactively. By contrast, only CO2 changes contribute to the radiative forcing in the SRES and 1% yr−1 listed in Table 1.

Historical (1850–2005) and future (2006–2100) simulations based on the RCPs are an important component of phase five of the Coupled Modeled Intercomparison Project (CMIP5; http://cmip-pcmdi.llnl.gov/cmip5/forcing.html), and simulation results will inform the Fifth Assessment Report (AR5) of the Intergovernmental Panel on Climate Change (IPCC). The simulations considered here begin from an 1850 control run and extend to the year 2100. Results for the 2006–2100 period are analyzed where changes in climate parameters are calculated with respect to the average over an extension of the 1850 control run for that period.

c. Controlled experiments

Feedbacks in the carbon–climate system are investigated using results from biogeochemically coupled and radiatively coupled simulations, in which the variations in CO2 and temperature are partially controlled. For the biogeochemically coupled simulations, all biogeochemical processes depending on CO2 are active, with the exception of the radiation code for which the CO2 control value is used. The simulated temperature change in this case is comparatively small since there is no radiative forcing due to the increasing CO2 concentration. Biogeochemical processes nevertheless respond to the CO2 so as to alter the surface energy balance through changes in vegetation structure and spatial distribution, which affects evapotranspiration, etc., so that the temperature change is not zero. In the simulations analyzed here, the effects of aerosols, non-CO2 greenhouse gases, land surface albedo change, etc., are all included in the simulations so that the resulting temperature change is not negligible.

For the radiatively coupled simulations, the radiation code reacts to changes in atmospheric CO2 to provide a radiative forcing that warms the system as well as changing the climate in other ways. In this case, however, the biogeochemical processes on land and in the ocean see only the preindustrial control run value of CO2 while temperature changes and other climate effects act on the carbon cycle. In the fully coupled case, of course, all processes are acting simultaneously.

3. Feedback analysis for carbon and energy

The carbon feedback analysis follows the approach of Boer and Arora (2009, 2010), which considers the globally averaged and vertically integrated carbon budget in the form
eq1
where HG is the amount of carbon in the global system and E is the anthropogenic emission of CO2 into the atmosphere. Anthropogenic emissions include those associated with land use change which, however, are not particularly well known (Arora and Boer 2010). The equations for the atmospheric, land, and oceanic carbon pools are written as
e1
with HG = HA + HL + HO representing the carbon in the atmosphere, land, and ocean, respectively, and with F representing the fluxes between them, taken to be positive into the respective pools so that FA + FL + FO = 0. The fluxes F(T, C, …) are expressed as functions of the surface temperature T and the atmospheric CO2 concentration C, which the underlying surfaces “see” directly, and perhaps other climate quantities that affect the underlying surface. The implicit assumption is that these other quantities are themselves functions of T and C, which are the dominant variables in the expressions.
The globally averaged, vertically integrated atmospheric carbon budget in (1), dropping the A subscript, is considered. The control simulation has zero anthropogenic emissions and is in climatic balance with dHo/dt = F(To, Co) = Fo = 0, where To, etc., are control run values of the variables, and where Ho = mCo (m is the mass of the atmosphere), presuming that CO2 is reasonably well mixed. The dependence on variables other than T and C is suppressed for notational simplicity. The equation for the emission-forced change in the atmospheric carbon is
eq2
where emissions E force changes in the atmospheric carbon budget and the system responds or “feeds back” by altering the flux
eq3
between the atmosphere and the underlying surface. Expanding in Taylor series,
e2
and the equality in (2) serves to define the carbon–climate and carbon–concentration feedback parameters Γ and B for the carbon budget. If the system is linear in T′ and C′, that is, if the higher-order terms in the expansion (2) are small, the feedback parameters will be constants and represent the rates of change or partial derivatives of F for perturbations of CO2 and temperature about the control values. In the more general case, the feedback parameters may depend on other variables and the evolving state of the system. They are evaluated by means of special biogeochemically and radiatively coupled simulations in which temperature and concentration are controlled in order to allow the estimation of partial derivatives related to those in (2).
As discussed in Boer and Yu (2003a,b,c), Boer et al. (2005), Boer (2011), Boer and Arora (2009, 2010), and elsewhere, the globally averaged and vertically integrated energy budget may be represented in a manner similar to (2), with
eq4
where dh′/dt, the rate of change of the heat content of the system (mainly storage in the ocean), is equal to the change R′ in the radiative flux across the top of the atmosphere. The radiative flux change R′ reflects the balance between the radiative forcing f(C′, Y′) due to changes in the CO2 concentration C and other quantities Y that affect the radiative flux (such as non-CO2 greenhouse gases, aerosols, land use changes, and changes in vegetation structure and distribution) and the climate feedback −λT′, which characterizes the system’s energetic response to the radiative forcing.
This representation of the energy budget takes a particularly simple form if the storage term is proportional to the temperature change with dh′/dt = αT′. This cannot be the case as the system approaches a new equilibrium under constant forcing, but it is plausible while the system is evolving. With this approximation the energy equation becomes particularly simple, with temperature changes that are proportional to the forcing:
eq5

a. Emission-driven simulations

In the emission-driven case, the rate E of anthropogenic emission of CO2 into the atmosphere is specified. The atmospheric carbon budget equations for the biogeochemically, radiatively, and fully coupled simulations take the form
eq6
where T and C are expressed in terms of their changes from the control run values as Tb = To + T*, Tr = To + T+, and T = To + T′ (similarly for C). Taking differences from the control run as in (2) gives
e3
for the carbon budget, while for the approximate energy budget
e4
The values of the feedback parameters Γ and B are calculated from the first two equations in (3), that is, from the results of the biogeochemically and radiatively coupled simulations as
eq7

These values of the feedback parameters are not consistent with the fully coupled case (the third equation) unless flux, temperature, and concentration changes add linearly, as discussed in an accompanying paper (Arora et al. 2013). In the earlier analyses in Table 1, there is no separate radiatively coupled simulation, and the feedback parameters are estimated from biogeochemically and fully coupled results, that is, from the first and third equations in (3).

b. Concentration-driven simulations

In the concentration-driven case, the evolution in time of the atmospheric CO2 concentration is specified with C = Co + C′. The atmospheric budget equation remains the same, but in this case, the flux F is calculated for a time step of the model using the specified value of C appropriate to that time together with the other values of the variables that arise during the simulation. The calculated flux F does not affect C itself, which is specified. The result is equivalent to adding a source/sink term E to satisfy the local budget.

Equations paralleling (3) and (4) in this case are
e5
and
e6
The same symbols are used in (3) and (4) and in (5) and (6), although, of course, the values need not be the same in the two cases. The parallel between (3) and (5) is perhaps not immediately obvious, since in (5) the specified concentration change dH′/dt = mdC′/dt takes the role of the specified emission in (3), while the emission term E in (5) is not specified but is “implicit” in the budget calculation. It is analogous to dH/dt in (3) and need not be (and is not) the same in the differently coupled simulations (i.e., E* ≠ E+E′).
For the concentration-driven case, the form of (5) parallels (3), with the exception that the specified value C′ replaces C* in the biogeochemically coupled case. The feedback parameters in this case are
eq8
One purpose of this study is to evaluate and characterize differences between feedbacks in emission- and concentration-driven cases and their implications for understanding the carbon budget.

c. Feedback equations for land and ocean

The feedback parameters for the atmospheric carbon budget (ΓA, BA) (reintroducing the A, L, and O subscripts for the moment) are obtained from the CO2 flux FA into the atmosphere in (3) and (5). The land (ΓL, BL) and ocean (ΓO, BO) carbon feedback parameters are obtained from analogous calculations replacing fluxes into the atmosphere with fluxes into the land and ocean, respectively. Since FA + FL + FO = 0, it follows that this is also the case for the feedback parameters, that is, ΓA + ΓL + ΓO = 0 and BA + BL + BO = 0. The feedbacks apply to their respective carbon pools, so that an increase in land carbon is characterized as a positive land feedback. Since a gain in land carbon comes at the expense of the atmosphere, it corresponds to a negative atmospheric feedback, and so on.

There is a complication in the analysis performed here because of the treatment of the emissions associated with land use change. In CanESM2 the emissions due to land use changes are not specified as a component of the anthropogenic emissions. They are treated as an interactive part of the carbon budget, as discussed in Arora and Boer (2010), and are not readily available as a separate term. Land use change emissions appear in the model as part of FL, the flux of carbon between land and atmosphere, and also as part of the overall flux FA into the atmosphere. The evaluation of the feedback parameters from the fluxes in (4) and (6) will be compromised if land use change emissions are a non-negligible part of these fluxes. The calculations in Arora et al. (2011) indicate that land use change emissions cannot be neglected for RCP2.6 but are a small fraction of fossil fuel emissions for RCP4.5 and RCP8.6. The feedback results do not, therefore, include calculations for RCP2.6.

d. Interpretation and comparison of feedback parameters

The feedback parameters (Γ, B) represent averaged rates of change of the CO2 flux F with respect to temperature and concentration. From (3),
eq9
where Fr is the flux in the radiatively coupled simulation and is the flux in the biogeochemically coupled simulation, excluding, at least approximately, the temperature feedback effect. In the simpler case where there are no non-CO2 forcings, that is, where f(Y′) is small in (4), then and . A finite difference estimate of the local values of the partial derivatives themselves is nominally possible, but the calculation is unstable because of the natural variability of the denominator.
This representation of feedbacks differs from the integrated flux approach of Friedlingstein et al. (2006) and others (Table 1), where time-integrated flux changes are linearly related to temperature and CO2 concentration changes. Those equations, in the notation used here, relate to (3) with
eq10
and
eq11
There is a fairly complex relationship between (γ, β) and (Γ, B) involving weighted averages over time, and a number of consequences follow from the different approaches.

The approach to the carbon budget adopted here parallels that adopted for the energy budget in “climate sensitivity” studies, as previously noted. The (Γ, B) approach relates the CO2 flux change to the current values of the changes in temperature and concentration while the (γ, β) approach involves the integrated value of the flux. The evolution of the feedback parameters will be rather different in the two cases. For the RCP4.5 radiatively coupled simulations for instance (the green lines in Fig. 3), the flux changes do not vary much, while the temperatures T+ are increasing strongly, although somewhat less than linearly with time. The result is that that decreases with time and temperature (see Fig. 4). In other words, the fluxes react less strongly to changes in temperature as temperatures increase, implying a decline in the strength of the feedback. By contrast, γA will be more or less constant or even increasing since the temperature increase is less than linear in time. Results for B and β differ in an analogous way. A quantitative comparison of the two feedback characterizations for the special case of a concentration-forced (1% yr−1 CO2 increase) simulation is discussed in an accompanying paper (Arora et al. 2013).

e. Feedback contributions to the atmospheric carbon budget

The global feedback parameters (Γ, B) for the atmosphere have different units, and the relative importance of the carbon–climate and carbon–concentration feedbacks for the atmospheric carbon budget is not immediately obvious by comparing their numerical values. One way of quantifying the importance of the two feedback processes is in terms of their cumulative contribution to the change in atmospheric CO2. Integrating (2), the change in the atmospheric carbon loading with time is
e7
where is the change in the atmospheric CO2 that would have resulted from the emissions E if F′ were zero and all of the emitted CO2 remained in the atmosphere. Then,
eq12
where a = (1 + d) = ΔHHE is the fraction of the emitted CO2 that remains in the atmosphere while is the fraction of the emitted CO2 that is taken up by the underlying land and ocean, that is, the loss fraction or the “deficit factor” as far as the atmosphere is concerned. If F′ for the atmosphere can be decomposed into components associated with climate and concentration feedbacks as F′ = FT + FC, then
eq13
and the fractions dT and dC characterize the effects of the climate and concentration feedbacks on atmospheric CO2 in a comparable way.
For F′ = FT + FC ≈ ΓT′ + BC′ with Γ and B from (3) or (5),
e8
These factors can be characterized further in terms of land and ocean components since ΓA = −(ΓL + ΓO) and BA = −(BL + BO). This results in the decomposition of the atmospheric feedback contribution into nine components:
e9
which indicate how the various feedbacks contribute to the atmospheric carbon budget.

4. Results

Although global and annual average quantities are treated in what follows, some of the variables, such as fluxes and derived feedback quantities, still display considerable year-to-year variability and so are first smoothed before being plotted or used in calculations.

a. The projected evolution of global carbon budget quantities

Figure 1 plots the basic variables in the atmospheric carbon budget for the emission-driven and concentration-driven cases for the three RCPs, namely, RCP2.6, RCP4.5, and RCP8.5. Changes are calculated with respect to the average over the same 2005–2100 period of a continuation of the control run. Figure 2 plots the integrated atmospheric carbon budget terms in (2). Consistent differences are seen between emission- and concentration-forced cases in Figs. 1 and 2, with larger values of atmospheric CO2, emissions, and temperature in the emission-forced case. The IAMs used to develop the RCPs have the same overall emissions as those in the emission-forced CanESM2 simulations. The resulting IAM atmospheric CO2 concentrations are, however, smaller than those of the ESM. Less CO2 in the atmosphere for the same emissions means a greater loss of atmospheric CO2, hence uptake of CO2 by the surface in the IAM simulations compared to the CanESM2 simulations.

Fig. 1.
Fig. 1.

The evolution of changes, with respect to 1850 control run averages, in the atmospheric CO2 concentration C (ppm), emissions E (PgC yr−1), temperature T (°C), and CO2 fluxes into the atmosphere FA, land FL, and ocean FO (PgC yr−1). Values are smoothed with an 11-yr moving average. Results for RCP2.6, RCP4.5, and RCP8.5 are plotted in blue, green, and orange, respectively, and emission- and concentration-driven results are plotted as solid and dashed lines.

Citation: Journal of Climate 26, 10; 10.1175/JCLI-D-12-00365.1

Fig. 2.
Fig. 2.

The integrated atmospheric carbon budget terms (PgC) from 2005. The range plotted for the RCP4.5 and RCP8.5 cases increases by a factor of 3 and 6, respectively, compared to the RCP2.6 case. The emission-driven results are plotted as solid lines, with the concentration-driven results as dashed lines.

Citation: Journal of Climate 26, 10; 10.1175/JCLI-D-12-00365.1

b. Biogeochemically and radiatively coupled simulations

Figure 3 displays results for the biogeochemically and radiatively coupled simulations for the RCP4.5 case together with results for the fully coupled simulations. The respective roles of dH/dt and E in the emission-driven and concentration-driven cases are apparent in Fig. 3. The single specified emission curve E for the emission-driven case (solid orange) results in three different CO2 loadings (H*, H+, and H′) for the biogeochemically, radiatively, and fully coupled cases. The single specified concentration curve H′ = mC′ in the concentration-driven case (dashed orange) results in three “implicit” emission curves (E*, E+, and E′) for the differently coupled scenarios.

Fig. 3.
Fig. 3.

The evolution of the atmospheric CO2 loading H (PgC), CO2 flux into the atmosphere FA (PgC yr−1), emissions E (PgC yr−1), and temperature change T (°C) for the RCP4.5 scenario. Values are smoothed with an 11-yr moving average. Results for the biogeochemically (asterisk), radiatively (plus sign), and fully coupled (prime) simulations are in blue, green, and orange, respectively, and emission- and concentration-driven values are plotted as solid and dashed lines, with differences indicated as δX.

Citation: Journal of Climate 26, 10; 10.1175/JCLI-D-12-00365.1

For δX, the emission-driven minus concentration-driven values, the difference δH+ in atmospheric CO2 loadings for the radiatively coupled case is large, as is the temperature difference δT+ and the atmospheric flux difference δF+. In the radiatively coupled case, the biogeochemical components do not react to the increase in atmospheric CO2 directly, but they do react indirectly via the radiatively forced temperature increases and other climate changes that affect the carbon balance. This process can reinforce itself in the emission-driven case, as is apparent from Fig. 3, but is restricted by the specified concentrations in the concentration-driven case.

Equations (4) and (6) indicate that the temperature changes for the biogeochemically coupled simulations, T* ≈ f(Y′)/(α + λ), should be nearly the same in the emission- and concentration-forced cases (solid and dashed blue curves, respectively), where f(Y′), the radiative forcing other than that due to CO2, is the same in both cases. The expectation is also that the temperature changes for the fully coupled simulations, T′ ≈ f(C′, Y′)/(α + λ), will be similar for the emission- and concentration-forced results although C′ does differ somewhat in the two cases (Fig. 1). For the emission-driven case, however, T+f(C+, Y′)/(α + λ) (solid green) is larger than T+f(C′, Y′)/(α + λ) (dashed green) in the concentration-driven case as a consequence of the larger CO2 loadings and, hence, radiative forcing. The arrows in the diagram show that δH+ = mδC+ > 0 and δT+ ≈ [f(C+, Y′) − f(C′,Y′)]/(α + λ) > 0, with the most notable difference in the fluxes into the atmosphere seen for the radiatively coupled case as a consequence of δT+ > 0 and δF+ ≈ ΓδT+ > 0.

c. Feedback parameters

The feedback parameters calculated from (3) and (5) are displayed in Fig. 4 for the emission-driven (solid lines) and concentration-driven (dashed lines) cases. The carbon–climate feedback parameter is calculated from the radiatively coupled simulation and the carbon–concentration feedback from biogeochemically coupled simulation. Values of Γ are plotted against T+, and those of B are plotted against C* or C′ for the different RCPs and cases in Fig. 4. The behavior of the carbon–climate system under different conditions is appropriately compared in terms of suitably scaled quantities such as feedback parameters and feedback contributions.

Fig. 4.
Fig. 4.

The atmosphere, land, and ocean carbon–climate feedback parameters (ΓA, ΓL, and ΓO) calculated from the radiatively coupled simulations [PgC (yr °C)−1] and plotted against T+ and the carbon–concentration feedback parameters (BA, BL, and BO) calculated from the biogeochemically coupled simulations [PgC (yr ppm)−1] plotted against concentration change C* for the emission-driven and C′ for the concentration-driven cases. Values entering the calculations are smoothed with a 30-yr moving average. Results for RCP4.5 and RCP8.5 are plotted in green and orange, respectively, and emission- and concentration-driven results are plotted as solid and dashed lines.

Citation: Journal of Climate 26, 10; 10.1175/JCLI-D-12-00365.1

For an increase in temperature, atmospheric carbon–climate feedback is positive (ΓA > 0) and land carbon–climate feedback is negative (ΓL < 0), as the atmosphere gains carbon from the land as warmer temperatures result in increased ecosystem respiration and reduced primary production (e.g., Cox et al. 2000; Friedlingstein et al. 2006). The land carbon–climate feedback parameter ΓL is larger in magnitude than the ocean carbon–climate feedback parameter ΓO in Fig. 4, indicating the stronger carbon response of the land compared to the ocean for a unit increase in global temperature.

The carbon–concentration feedback BA is negative, indicating that processes depending on CO2 concentration act to reduce the amount of atmospheric CO2 while increasing carbon in land and ocean pools for which feedback parameters (BL, BO) are positive. By contrast to the carbon–climate feedbacks, where ΓL is considerably larger than ΓO in magnitude, the values of BL and BO are closer to one another.

The signs of the feedback parameters (γ, β) in the studies listed in Table 1 are similar to those of (Γ, B). The magnitudes, units, and relative sizes of the components are not directly comparable, however, as discussed in section 3.4, where it is also noted that the time evolution of the feedback parameters will differ between the two approaches. Other quantities, such as the ratio of the land-to-ocean carbon–climate feedback parameters, may be compared where ΓLo ≈ 10 for RCP8.5 but only about half of that for RCP4.5 at the end of the simulations at 2100. The corresponding ratios for γL/γO in the studies in Table 1 range from about 0.6 for the model with the lowest value in Friedlingstein et al. (2006), but with an 11-model average of about 2.7, to about 10.6 for Zickfeld et al. (2011) based on the SRES A2 scenario.

Both Γ and B depend on the state of the system. The strengths of the feedbacks ΓA and ΓL decline as temperature increases, as do the magnitudes of all components of B as CO2 increases. For a given value of temperature change, the magnitudes of ΓA and ΓL increase with scenario from RCP4.5 to RCP8.5, as do the magnitudes of the components of B for a given CO2 change. The feedbacks inferred for the emission-forced (solid lines) and concentration-forced (dashed lines) cases differ nontrivially for Γ but are more consistent for B. The feedback parameters exhibit generic behavior across the various cases, but there are differences in their magnitude and evolution depending on the nature and magnitude of the forcing and the state of the system.

In Boer and Arora (2009, 2010) the carbon feedback parameters are estimated from the fully coupled and biogeochemically coupled simulations, that is, the first and third equations in (3), in the absence of a radiatively coupled simulation. The values of Γ in that study are reasonably independent of temperature change, while those of B depend on CO2 change. The more or less linear dependence of B on CO2 change, especially for the more strongly forced cases, is similar in both studies. The independence of Γ as a function of T+ is not apparent in Fig. 4 for RCP4.5, although for the RCP8.5 emission-driven case values become approximately constant as T+ increases.

d. Emission and concentration forcing

The carbon budget quantities in Figs. 1 and 2 are consistently smaller in magnitude for the concentration-forced case compared to the emission-forced case. These differences arise from the mismatch between the behavior of CanESM2 and the IAMs used in scenario development. In particular, for the same specified emissions, the IAM produces smaller atmospheric CO2 concentrations than the ESM as a consequence of stronger carbon uptake by the surface. When these smaller CO2 values are used in concentration-forced simulations with the ESM, the magnitudes of the terms in the carbon budget are also smaller, as seen in the figures. Taking the time-integrated form of the atmospheric CO2 budget from (2) for the IAM, concentration-forced ESM, and emission-forced ESM are:
eq14
where the subscripts i and c refer to the IAM and concentration-forced ESM, respectively, and the unsubscripted values are those for the emission-forced ESM, where the emissions are those used also in the IAM (i.e., E = Ei). Taking the differences,
eq15
where the terms on the left-hand side are greater than zero from Fig. 2. The ESM flux changes are greater than the IAM flux change in both cases and, since in Fig. 2,
eq16
All terms are negative, signifying a loss of atmospheric CO2. In terms of the uptake by the underlying surface, the inequalities reverse and the IAM uptake is greater than the ESM emission-forced uptake, which is greater than the ESM concentration-forced uptake.
Expressing the difference between the concentration-forced ESM and IAM results in terms of differentials
eq17
where the last integral is zero in the concentration-forced case since δC′ = 0 for that case. This negative term is present when the difference is between the emission-forced ESM and the IAM, which is consistent with . In other words, constraining the CO2 concentration in the ESM forces the response even farther away from the IAM balance than by constraining the emissions.

e. Feedback contributions

The loss fractions or deficit factors for the atmospheric carbon budget are obtained following (8) and (9). The analysis depends on being able to estimate the terms in F′ = FT + FC = ΓT′ + BC′ with reasonable accuracy. Here the change in the atmospheric flux F′ for the fully coupled case is approximated by ΓT′ + BC′, where the feedback parameters, calculated from the biogeochemically and radiatively coupled simulations, are multiplied by the temperature and concentration changes of the fully coupled simulation.

It is apparent from Fig. 4 that the feedback parameters depend on the state of the system in a nontrivial way and that Γ and B are functions of temperature, CO2 concentration, and other system parameters. It is not immediately obvious, under the circumstances, that F′(t) ≈ Γ(t)T′(t) + B(t)C′(t). Figure 5 plots F′ and ΓT′ + BC′ separately for RCP4.5 and RCP8.5. The agreement is best for the concentration-forced case for RCP8.5, but agreement is as good or better for the emission-forced case for RCP4.5. The agreement is considered adequate to allow the calculation of the relative contributions of the carbon–climate and carbon–concentration feedbacks to the atmospheric CO2 budget.

Fig. 5.
Fig. 5.

CO2 flux F′ (PgC yr−1) into the atmosphere (solid) and the approximation ΓT′ + BC′ (dashed) for emission- and concentration-driven cases. Values for RCP4.5 and RCP8.5 are plotted in green and orange, respectively.

Citation: Journal of Climate 26, 10; 10.1175/JCLI-D-12-00365.1

Figure 6 displays the loss fractions or deficit factors for the atmospheric CO2 budget associated with carbon–concentration, carbon–climate, and land and ocean feedbacks from (8) and (9). These results apply to the period after 2005, not to the period since 1850. The overall effect of the carbon feedbacks is to remove part of the CO2 emitted into the atmosphere, so that . The flux changes associated with atmosphere, land, and ocean result in the nine feedback contribution components plotted in the figure. Solid lines correspond to emission-forced cases, and dashed lines correspond to concentration-forced cases. The scale of the diagrams is the same except for the top row, where partial cancellation of the subcomponents gives smaller values. Values in the top panels of the diagram are the sum of the two component panels below, and similarly, values in the left panels are the sum of the component panels to the right.

Fig. 6.
Fig. 6.

Atmospheric CO2 deficit factors as percentages. Values for RCP4.5 and RCP8.5 are plotted in green and orange, respectively, and emission- and concentration-driven results are plotted as solid and dashed lines, respectively. Values entering the calculation are smoothed with a 30-yr running mean.

Citation: Journal of Climate 26, 10; 10.1175/JCLI-D-12-00365.1

Measured in this way, the carbon–climate feedback contribution dT is positive but is weaker than the negative carbon–climate feedback dC, resulting in a d that is negative. The atmospheric loss contributions d in Fig. 6 differ by about a factor of 2 as the system evolves under the different forcing scenarios. Values of d are generally similar for the emission- and concentration-forced cases. The results apply to the period after 2005 and the integrated airborne fraction a = 1 + d is about 0.65 at the beginning of this period. This contrasts with a value of about 0.5 for the airborne fraction calculated over the period since 1850 for this model. In other words, the land and ocean are less able, in the twenty-first century compared to the twentieth century, to take up the CO2 emitted into the atmosphere. The magnitude of d weakens slightly (becomes less negative) with time over the twenty-first century for RCP8.5, indicating that the system is tending toward saturation and is starting to lose its ability to absorb the emitted CO2. By contrast, the magnitude of d for RCP4.5 increases with time, indicating that the system is “catching up,” so to speak, in absorbing the emitted CO2 as the rate of emission decreases.

The climate and concentration components of d = dT + dC are of opposite signs, mirroring the signs of the feedback parameters. Results for dT (and dTL) differ between emission- and concentration-forced cases as much as for the difference between RCPs. The value of dT varies comparatively modestly with time or forcing compared to dC, which largely determines the evolution of d. For d = dL + dO, the land contribution dL is weakly negative, with the ocean contribution more negative, and there is modest variation with time and scenario.

For land the two contributions dTL and dCL are of roughly similar magnitude but of opposite sign, so that dL = dTL + dCL is the relatively weak residual of the two competing feedbacks. The two ocean contributions behave quite differently, with dCO comparatively strongly negative as a result of the carbon–concentration feedback while the carbon–climate feedback contribution dTO is weak. As a result, dO = dTO + dCOdCO, and the ocean contribution is associated almost entirely with the carbon–concentration feedback. Overall, ocean uptake is more effective than land uptake in restraining atmospheric CO2 increase.

In this context, the contributions to the atmospheric CO2 budget may be roughly characterized as , where the positive and negative signs in parentheses indicate the sign of the contributions. The overall contribution is negative, with the negative carbon–concentration contribution outweighing the positive contribution associated with the carbon–climate feedback. The carbon–climate contribution is dominated by the land component as a result of enhanced ecosystem respiration and reduced primary production due to global warming. The contributions of carbon–concentration feedbacks for both land and ocean are negative and are of reasonably similar magnitude. These are associated with enhanced primary productivity over land (Luo et al. 1996; Fig. 3b in Arora et al. 2009) and enhanced uptake of CO2 by the ocean associated with increasing atmospheric CO2 concentration.

5. Summary

Emission-forced and concentration-forced simulations of twenty-first-century climate following the RCP2.6, RCP4.5, and RCP8.5 scenarios are undertaken with CanESM2. Additional simulations are performed where aspects of the carbon and energy budgets are controlled to facilitate feedback analyses of the system’s response to temperature and CO2 concentration changes. For biogeochemically coupled simulations, all biogeochemical processes in the model are active, but the resulting CO2 concentration changes are excluded from the radiation code so that the radiative forcing effect of CO2 is not active. For radiatively coupled simulations, the radiation code reacts to changes in atmospheric CO2 and the climate warms, but changes in CO2 concentration do not affect biogeochemical processes. In all, 18 simulations are performed and analyzed (three RCP scenarios; emission- and concentration-forced cases; and fully, biogeochemically, and radiatively coupled simulations for each).

The feedback analysis of Boer and Arora (2009) is applied to obtain carbon–climate Γ [PgC (yr °C)−1] and carbon–concentration B [PgC (yr ppm)−1] feedback parameters for the atmosphere, land, and ocean from biogeochemically and radiatively controlled simulations for the RCP4.5 and RCP8.5 scenarios. The interactive treatment of land use change in CanESM2 means that land use change emissions are not separately available. This is unimportant for RCP4.5 and RCP8.5 but cannot be neglected for RCP2.6, and thus, feedback results are not presented for this case.

The feedback approach applied here expresses the CO2 flux change between the atmosphere and the underlying land and ocean linearly in terms of contemporary temperature and concentration changes. This approach differs from that of Friedlingstein et al. (2006), Gregory et al. (2009), Roy et al. (2011), and others, in which the time-integrated CO2 fluxes are related to changes in temperature and concentration to obtain carbon–climate γ (PgC °C−1) and carbon–concentration β (PgC ppm−1) feedback parameters. The units of the feedback parameters differ between the two approaches, as does the temporal evolution (see also Arora et al. 2013), although the signs of the feedbacks are the same. Other unitless quantities, such as the ratio of land to ocean feedback parameters, also differ between approaches, as well as between scenarios and between models.

The atmospheric carbon–climate feedback ΓA is positive in the sense of acting to enhance atmospheric CO2 concentration (or to decrease implied emissions in the concentration-driven case) as a consequence of temperature increases due to increasing CO2. The land feedback parameter ΓL is negative and larger in magnitude than the ocean feedback parameter ΓO, which is also negative, as the land loses more carbon to the atmosphere than does the ocean for a unit increase in global temperature. The carbon–concentration feedback for the atmosphere BA is negative and counteracts the positive carbon–climate feedback ΓA, while BL is greater than BO. Enhanced atmospheric CO2 concentration promotes carbon uptake through biogeochemical processes at a greater rate than the increased temperature promotes carbon release.

The current results differ from those of Boer and Arora (2009) in a number of ways. They are obtained with a different version of the model and with the newer RCP forcing scenarios, they also include radiative forcings other than CO2 changes, they force the model both with emissions and via concentrations, and they add a radiatively forced case to the set of simulations for the purpose of calculating Γ. Because Γ is now calculated from a radiatively coupled simulation, the associated changes in temperature are considerably larger than for the previous analysis, where this result was not available, and there is little overlap in the temperature changes that would allow comparison between concentration-forced results. The behavior of the components of the carbon–concentration feedback parameter B is broadly similar in the two studies. There are, however, differences of magnitude in the various feedback parameters between the two analyses which, presumably, depend on the differences in the model versions exacerbated by the nonlinearity of the system.

For the current study, there are nontrivial differences between emission- and concentration-forced results in the evolution of the global values of CO2 concentration, temperature change, surface fluxes of CO2, and direct and inferred emissions, as well as in the associated feedback parameters. Specifying the emissions associated with a particular RCP results in larger CO2 concentrations in CanESM2 than in the IAM used in scenario development. There is a mismatch in the behavior of the models, with a weaker land and ocean uptake of carbon in the ESM compared to the IAM. When these smaller CO2 concentrations are used to force the ESM, the result is weaker surface CO2 flux differences, weaker warming, and weaker inferred emissions compared to the emission-forced case. The feedback parameters ΓA and ΓL (but not ΓO) display comparatively large differences between emission- and concentration-forced values for the same temperature change. These differences are larger than those associated with the different RCPs for the same temperature change. The system is not linear, nor is the nature and magnitude of the forcing immaterial.

While the feedback parameters quantify system behavior as it relates to temperature and CO2 changes, they have different units, and their magnitudes are not directly indicative of the effect on the CO2 budget. The evolving importance of the feedback processes on atmospheric CO2 itself is evaluated in terms of loss fractions or deficit factors as they contribute to the change in atmospheric CO2 since 2005. Here d is the fraction of the emitted CO2 that the atmosphere gains from the underlying surface as a consequence of the feedbacks operating. The result may be roughly summarized as , indicating that d is negative as a consequence of the negative carbon–concentration feedback processes removing more CO2 from the atmosphere than the positive carbon–climate feedback processes engender. The contribution of ocean carbon–climate feedback is small. Contributions evolve modestly with time but differ by as much as a factor of 2 between RCP4.5 and RCP8.5 results. Differences between emission- and concentration-forced cases are comparatively modest, although less so in the case of dTL, which also affects dT and dL.

The feedback approach to global carbon budgets considered here depends on performing additional simulations that “control” for the effects of temperature and CO2 concentration in order to derive carbon–climate and carbon–concentration feedback parameters. These parameters characterize broad features of system behavior. They are not, however, independent of system state, forcing scenario, or the manner in which the CO2 source is treated in the model—that is, directly by specifying CO2 emissions or indirectly by specifying evolving atmospheric CO2 concentration. The complexity and nonlinearity of the system and the large changes in CO2 concentration involved preclude simple linear behavior. Thus, while global feedback analyses are useful in characterizing general aspects of system behavior, they apply quantitatively to particular cases. Other approaches, including the study of local and regional feedback mechanisms, are required to deepen understanding of feedback processes as they affect the coupled carbon–climate system.

Acknowledgments

Comments on an earlier version of the paper by J. Christian and N. Gillett are gratefully acknowledged.

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