The budget equation for atmospheric carbon is dmAdt=e-fO-dmLdt, where mA denotes the atmospheric carbon stored in CO2, e CO2 emissions, fO the flux to the ocean, mL the land biosphere carbon stock, and t time. Here, mL refers to the (potential) natural biosphere. Human impacts on the land biosphere exchange including land use and land use changes are not simulated in the present version and are treated as exogenous emissions (e). These emissions may be prescribed based on results from spatially explicit terrestrial models. An overview of the model variables and parameters is given in Tables  and .

The change in land carbon is given by the balance of NPP and decay of assimilated terrestrial carbon, dmLdt=fNPP-fdecay. Decay includes heterotrophic respiration (RH), fire, and other disturbances due to natural processes.

Carbon is taken up by the ocean through the air–sea interface (fO) and distributed to the mixed surface layer (mS) and the deep ocean interior (fdeep): fO=dmSdt+fdeep. Global NPP (fNPP) is assumed to be a function of the partial pressure of atmospheric CO2 (pCO2) and the SAT deviation from preindustrial equilibrium (functions for the implemented land components are given in Appendix ),

The net flux of carbon into the ocean is proportional to the gas transfer velocity (kg) and the CO2 partial pressure difference between surface air and seawater: fO=kgAOε(pACO2-pSCO2), where AO is ocean surface area and ε the atmospheric mass of C per mixing ratio of CO2.

The global average perturbation in surface water ΔpSCO2 is a function of dissolved inorganic carbon change (ΔDIC) in the surface ocean at constant alkalinity and SAT ; ΔDIC and pACO2 are related to model variables (see Appendix ), ΔDIC=mSHmixAOϱMµmol10-15Gtg-1,pACO2=mAε-1. The carbon cycle equation set is closed by the specification of fdecay and fdeep (Sect. ), as well as ΔT, i.e., the coupling to the climate component (Sect. ).

BernSCM simulates the deviation in global mean SAT from the preindustrial state. SAT is approximated by the temperature perturbation of the surface ocean ΔT, which is calculated from heat uptake by the budget equation dΔTdtcS=fOH-fdeepH, where cS is the heat capacity of the surface layer, fOH is ocean heat uptake, and fdeepH is heat uptake by the deep ocean (and accounts for the bulk of the effective heat capacity of the ocean). Continental heat uptake is neglected due to the much higher heat conductivity of the ocean in comparison to the continent.

fOH is taken to be proportional to RF and the deviation of SAT from radiative equilibrium (ΔT=ΔTeq(RF); see Table  for parameter definitions), fOH=RF1-ΔTΔTeqAOaO. This relation follows from the assumption that feedbacks are linear in ΔT e.g.,. ΔTeq is given by ΔTeq=RFΔT2×RF2×, where ΔT2× is climate sensitivity (defined as the equilibrium temperature change corresponding to twice the preindustrial CO2 concentration). Equation () describes ocean heat uptake as the difference between RF and the climate system's response, λ⋅ΔT, with λ=RF/ΔTeq the climate sensitivity expressed in Wm-2K-1.

Climate sensitivity is an external parameter, as the model does not represent the processes determining equilibrium climate response. RF of CO2 is calculated as RFCO2=ln⁡pACO2pA0CO2RF2×ln⁡(2), where pA0CO2 is the preindustrial reference concentration of atmospheric CO2, and RF2× is the RF at twice the preindustrial CO2 concentration. RF of other greenhouse gases (GHGs), aerosols, etc. can be parametrized in similar expressions involving GHG and pollutant emissions and concentrations . In the provided BernSCM code, non-CO2 RF is treated as an exogenous boundary condition. Total RF is then RF=RFCO2+RFnonCO2. The calculation of fdeepH (Sect. ) completes the climate model.

The response of a time-invariant linear system to a time-dependent forcing f can be expressed by m(t)=∫-∞tf(t′)r(t-t′)dt′. The function r is the system's IRF, as can be shown by evaluating the integral for a Dirac impulse (f(t′)=δ(t′)). The IRF indicates the fraction remaining in the system at time t of a pulse input at a previous time t′. Because of linearity of the integral, any physically meaningful integrand f can be represented as a sequence of such impulses of varying size.

In BernSCM, an IRF is used to calculate the perturbation of heat and carbon in the mixed surface ocean layer (mixed layer IRF; ). For carbon, mS(t)=∫-∞tfO(t′)rO(t-t′)dt′, and similarly, for heat ΔT(t)cS=∫-∞tfOH(t′)rO(t-t′)dt′. This approach has been shown to faithfully reproduce atmospheric CO2 and SAT as simulated with the models from which the IRF is derived . For temperature, the linear approach works since relatively small and homogeneous perturbations of ocean temperatures do not affect the circulation strongly and can be treated as a passive tracer . Note that for compatibility with commonly used units, carbon fluxes are expressed in Gt yr-1, while heat fluxes are expressed in joules per second (watt) in Eqs. () and (), respectively.

Equation () closes the ocean C budget equation (Eq. ), as can be seen by taking the derivative with respect to time (using r(0)=1), dmSdt=fO(t)--∫-∞tfO(t′)drOdt(t-t′)dt′︸fdeep, where fdeep is the flux to the deep ocean. Similarly, Eq. () closes the heat budget equation (Eq. ) for the surface ocean, dΔTdtcS=fOH(t)--∫-∞tfOH(t′)drOdt(t-t′)dt′︸fdeepH. Another IRF is used for the carbon mL in living or dead biomass reservoirs of the terrestrial biosphere, mL(t)=∫-∞tfNPP(t′)rL(t-t′)dt′. Again, Eq. () closes the budget equation for the land biosphere (Eq. ), as shown by the derivative with respect to time, dmLdt=fNPP(t)--∫-∞tfNPP(t′)drLdt(t-t′)dt′︸fdecay. The time derivative of the land IRF is also known as the decay response function e.g.,.

The IRFs above can be expressed as a sum of exponentials, r(t)=a∞+∑kake-t/τk, where the constant term a∞ corresponds to an infinite decay timescale.

The ocean IRF contains a positive constant coefficient a∞, indicating a fraction of the perturbation that will remain indefinitely (implied by carbon conservation in the ocean model). CaCO3 compensation by sediment dissolution and weathering are not considered here, but could be described using analogous elimination processes with timescales on the order of 104 to 105 years . We emphasize that the implementation considering only the partitioning of excess carbon among atmosphere, land, and ocean (hence a∞≠ 0), neglecting ocean sediment interactions and weathering flux perturbations, is only valid for timescales shorter than about 2000 years. In land biosphere models, in contrast, organic carbon is lost to the atmosphere by oxidation to CO2 at nonzero rates, and consequently all timescales are finite (i.e., a∞=0), and the IRF tends to zero (Fig. ).

Inserting the formula (Eq. ) in the pulse response equation (Eq. ) yields (f is a perturbation flux when a∞≠ 0) m(t)=∑k∫-∞tf(t′)ake-(t-t′)/τkdt′+∫-∞tf(t′)a∞dt′. Thus the expression (Eq. ) separates into a set of independent integrals mk corresponding to the number of timescales of the response. Taking the time derivative of the expression (Eq. ) reveals the equivalence to a system of uncoupled first-order ordinary differential equations. dmkdt=f(t)ak-mk/τk;dm∞dt=f(t)a∞m=∑kmk+m∞ The direct numerical evaluation of Eq. () involves integrating over all previous times at each time step. The differential form Eq. () allows a recursive solution, which is much more efficient, especially for long simulations (the recursive solution implemented in BernSCM is described in Appendix ).

The differential equation system (Eq. ) can be considered to consist of several boxes, whereby each box mk receives a fraction ak of the input f and has a characteristic turnover time τk (Fig. ). In the following this is referred to as a box model. For the mixed ocean surface layer the carbon content of box k is given by dmSkdt=fO(t)aOk-mSk/τOk;dmS∞dt=fO(t)aO∞ and the change in total carbon content in the mixed layer is mS=∑kmSk+mS∞. Similar equations describe the heat content in the ocean surface layer, as well as the carbon stored in the land biosphere (Fig. ).

The timescales of an IRF describing a linear system are equivalent to the inverse eigenvalues of the model matrix of that system and may also be interpreted in the context of the Laplace transformation . For example, the timescales of the mixing layer IRF are the inverse eigenvalues of a matrix describing a diffusive multilayer ocean model . A large model matrix yields a spectrum of many eigenvalues and timescales and corresponding model boxes. In practice, IRFs are approximated with fewer fitting parameters and, equivalently, timescales (four to six in the case of BernSCM). used IRFs combined from two or more functions to minimize the number of parameters needed for an accurate representation. In BernSCM, simple IRFs of the form (Eq. ) are used exclusively. This allows adequate accuracy and a consistent interpretation as a multibox model.

Thinking of IRF components as box models is conceptually meaningful. The simple Bern 4-box biosphere model , for example, contains boxes corresponding to ground vegetation, wood, detritus, and soil (Appendix ). The High-Resolution Biosphere Model (HRBM) land component , however, is abstractly defined by an IRF, but corresponds to boxes which correlate with biospheric reservoirs. However, since different box models may show a similar response, in practice the coefficients ak and timescales τk may not be uniquely defined by the IRF and should be interpreted primarily as abstract fitting parameters .

The IRF representation is, strictly speaking, only valid if the described subsystem is linear and the timescales of the system are time invariant. Then, the response function r does not depend on time and on state variables. In the BernSCM, major nonlinearities in the carbon cycle, namely air–sea gas exchange and the nonlinear carbonate chemistry, and changes in NPP in response to changes in environmental conditions are treated by separate nonlinear equations (Eqs. 4 and 5), while surface-to-deep ocean transport of carbon and heat and respiration of carbon in litter and soils are viewed as approximately linear processes using IRFs. However ocean circulation and the respiration of carbon from soil and litter is likely to change under global warming, violating the assumption of linearity. In practice, the IRF representation remains a useful approximation as long as the impact of associated nonlinearities on simulated atmospheric CO2 and temperature remain moderate.

The interpretation of the IRF representation as a box model provides a starting point for considering nonlinearities in the response. To account for nonlinearities, the response timescales τk and the coefficients ak may be gradually adjusted as a function of state variables such as temperature. As the integral form (Eq. ) involves integration over the whole history at each time step, changing parameters along the way would result in inconsistencies. In contrast, the differential or box model form (Eq. ) does not depend on previous time steps. Changing the model parameters from one step to the next thus equates to applying a slightly different model at each time step. Within each time step, the parameters remain constant, and the solution for the linear case applies. As time steps are small compared to the whole simulation, this discretization yields accurate results, which is confirmed by the close agreement between the different time resolutions (Table ).

Varying coefficients have been successfully implemented and tested for the HRBM land component and its decay IRF . In this way, the enhancement of biomass decay by global warming is captured (see the Appendix  and Sect. ). In such a modification, the advantage of the IRF and the equivalent box model representation – the faithful representation of the characteristic response timescale of a model system – is largely maintained, while at the same time the impact of time- and state-dependent system responses on simulated outcomes is approximated.

The carbon cycle–climate uncertainty of simulations with BernSCM can be assessed in two ways. First, to assess structural uncertainty, different substitute models for the ocean and land components can be used. Currently, this approach is quite limited by the set of available substitute models (see Appendix ). Second, parameter uncertainty can be assessed by varying the temperature and CO2 sensitivities of the model, based on a standard set of components that represent the key dependencies as completely as possible (here, the IRF substitutes for the high-latitude exchange/interior diffusion–advection (HILDA) ocean model and for the HRBM land biosphere model are used in the standard setup).

The uncertainties of the global carbon cycle concern the sensitivity of the modeled fluxes of carbon and heat to changing atmospheric CO2 and climate. Key uncertainties strongly affecting the overall climate response are associated with land C storage: the dependency of NPP on CO2 (CO2 fertilization), and the dependency of land C on temperature (fdecay increases with warming). This gives rise to large and opposed carbon flux perturbations which are both very uncertain in magnitude . While all substitute land models available for BernSCM include CO2 fertilization, only the HRBM substitute model represents temperature sensitivity of biomass decay (Appendix ).

As for the ocean, the uncertainty of heat uptake into the surface ocean is treated in terms of climate sensitivity (Eq. ). The efficiency of the uptake of heat (fdeepH) and carbon (fdeep) into the deep ocean is not sensitive to temperature, as the currently available substitute models all represent a fixed circulation pattern (IRF or box model parameters are not temperature dependent; Appendix ). The nonlinear chemistry of CO2 dissolution in the surface ocean (Eq. ), which determines the sensitivity of ocean C uptake to atmospheric CO2, is scientifically well established and is not treated as an uncertainty in BernSCM. The temperature sensitivities of NPP and CO2 dissolution in the surface ocean are treated as uncertain here, but have secondary influence on the climate response.

Similar to previous studies using models from the Bern family , the parameter uncertainty range is assessed using the following setups. Coupled.

All temperature and CO2 sensitivities are set to their standard values.

The climate response of BernSCM is illustrated using idealized simulations with prescribed forcing. One series of simulations (a) was run for CO2 concentration increasing exponentially from the preindustrial value by 1 % yr-1 over 140 years to approximately 4 times the preindustrial concentration, corresponding to a linear increase in RF (Fig. a); in a second series of simulations (b), CO2 was abruptly increased to 4 times the preindustrial concentration (Fig. b).

compare similar simulations of Earth system models (ESMs) performed within the Coupled Model Intercomparison Project Phase 5 (CMIP5), and Earth System Models of Intermediate Complexity (EMICs) . As a model comparison metric sensitive to the long-term climate response, use the fraction of realized warming, defined by the ratio of the temperature response at a given year and the equilibrium temperature for the corresponding RF. They show that the smaller realized warming of ESMs in comparison to EMICs (Fig. ) is connected to a higher long-term warming response; this implies an increase in the coefficient relating global warming to cumulative carbon emissions on multicentennial timescales and suggests a lower quota on allowed emissions for a given global warming target . The realized warming fraction simulated with BernSCM is in good agreement with the responses of the ESMs (and lower on average than that of the EMICs). The validity of the IRF approach has also been shown by using a SCM to reconstruct and interpret atmosphere–ocean general circulation model (AOGCM) projections. For the 150-year timescale of the CMIP5 experiments, show that the climate response of AOGCMs is well captured by a two-layer energy balance model with two effective response timescales.

In BernSCM, the fraction of realized warming depends primarily on the choice of climate sensitivity and is qualitatively similar for the different model setups. Such a clear relationship is not seen in the EMS and EMICs. Thus the structural uncertainty and model differences of complex models are not fully represented in BernSCM. The BernSCM climate response to abrupt warming (Fig. b) is qualitatively similar, especially on multicentennial timescales.

Coupled carbon cycle–climate models can be characterized and compared based on their response to a CO2 emission pulse to the atmosphere . The airborne fraction (AF) denotes the fraction of emissions found in the atmosphere at a given time. In IRFMIP, the AF for a pulse of 100 Gt C, emitted on top of current (i.e., year 2010) atmospheric CO2 concentrations, was simulated by a set of 15 carbon cycle–climate models of different complexity. For three of these models (Bern3D-LPJ, GENIE, MAGICC), ensembles sampling the parameter uncertainty of these models are included in IRFMIP. Thus, IRFMIP captures structural as well as parameter uncertainty.

The IRFMIP pulse experiment was repeated with BernSCM, exploring parameter uncertainty of the carbon cycle (Sect. ), as well as structural uncertainty, using the ocean model IRFs HILDA and Princeton in various combinations with the land biosphere components HRBM and the Bern 4-box model (Fig. ). Simulations were run for equilibrium climate sensitivities of 3 (standard setup), 2, and 4.5 ∘C.

The AF simulated with BernSCM broadly agrees with the set of simulations from IRFMIP. At 100 years after the pulse, the AF is 0.40 (0.34–0.57) for a climate sensitivity of 3 ∘C (for coupled setup with uncertainty range in brackets). Climate sensitivity uncertainty only slightly affects the upper end of this range (Fig. ). For AF simulated with BernSCM, the standard coupled setup is close to the IRFMIP multimodel median. The BernSCM uncertainty range is asymmetric, like the IRFMIP multimodel range. For the MAGICC and GENIE ensembles, the medians also correspond with the BernSCM standard case, while the uncertainty ranges are more symmetric.

The BernSCM SAT response also broadly agrees with IRFMIP. The standard coupled simulation is somewhat lower than the IRFMIP median, which is explained in part by the climate sensitivity (3 ∘C) being slightly lower than the IRFMIP average (3.2 ∘C). The short-term temperature response of BernSCM in particular is on the lower side of the IRFMIP range, suggesting stronger ocean mixing. The quickest initial temperature increase in the BernSCM simulations is obtained with the Princeton ocean model component (dashed lines), which shows a slower initial mixing to the deep ocean than the other implemented components (Fig. ). The comparability of the SAT projections is limited, as the range of climate sensitivities considered in the BernSCM simulations (2–4.5 ∘C) differ somewhat from those of the IRFMIP multimodel set (1.5–4.6 ∘C) and the single model ensembles (1.9–5.7 ∘C) and are compounded with RF differences resulting from the uncertainty in atmospheric CO2.

Figure  shows how the added carbon is redistributed within the Earth system. In the coupled setup, the fraction of the initial pulse sequestered by the land and by the ocean increases over the first century, while the airborne fraction decreases. After 100 years, slightly more than 20 % of the added carbon is stored in the land and about 40 % in the ocean. The ocean continues to sequester excess carbon in the following centuries to become the dominant sink for excess carbon. In contrast, the land returns part of the sequestered carbon back to the atmosphere and ocean as decreasing atmospheric CO2 reduces the modeled CO2 fertilization of the land biosphere. In the T-only setup, in which CO2 fertilization is not operating, the land is a source of carbon to the atmosphere due to accelerated soil turnover in response to warming. The largest land sink is simulated in the C-only setup, in which soil turnover timescales remain invariant and CO2 fertilization is on. The different BernSCM setups span a range of plausible land biosphere and ocean responses to continued anthropogenic CO2 emissions as reflected in the simulated range in the airborne fraction (Figs. a and ).

Climate models with explicit and detailed carbon cycle components exhibit a wide range of responses, as shown in the intercomparison studies of climate models with a detailed carbon cycle, C4MIP and CMIP5 . The authors analyzed the feedback of carbon cycle–climate models using linearized sensitivity measures. These are derived from a simulation with temperature dependence (“coupled”) and one without (“uncoupled”; note that these names have a different meaning in BernSCM). Total CO2 emissions for the coupled (left-hand side) and uncoupled (right-hand side) simulations can be expressed as ΔCAc(ε+βL+βO+α(γL+γO))=ΔCAu(ε+βL+βO), where ΔCA is the cumulative change in atmospheric CO2 (in parts per million) in the coupled (c) or uncoupled (u) cases, and the terms in parentheses represent the total sensitivity of C storage to ΔCA; in particular, β is the sensitivity of carbon storage to atmospheric CO2 (in Gt C ppm-1) on land (βL) or in the ocean (βO). γ is similarly the sensitivity in carbon storage to climate change, and α is the linear transient climate sensitivity to CO2 (∘C ppm-1) as in ; ε converts ppm to Gt C (see Table ; the formula in the original paper implies identical units for atmospheric and stored carbon).

The climate–carbon cycle feedback is measured by the feedback metric g, defined by ΔCAcΔCAu=11-g and is thus estimated by g=-α(γL+γO)ε+βO+βL. Thus the feedback strength scales with the assumed climate sensitivity and the temperature sensitivities and is reduced by CO2-induced sinks.

The C4MIP study used a SRES A2 emission scenario to compare the carbon cycle sensitivities of a range of models. As in the C4MIP exercise, BernSCM was run for SRES A2 without any non-CO2 forcings (Fig. ; prescribed historical and scenario emissions were smoothed with the R smooth.spline function for 41 degrees of freedom for use with different time steps). Land use was treated as an exogenous CO2 emission, while the land model simulates an undisturbed biosphere.

The BernSCM sensitivity setups can be expressed in terms of the C4MIP sensitivity metrics: T-only corresponds to βL=0, C-only to γL=γO=0, and uncoupled to βL=γL=γO=0. This can be used to estimate climate–carbon cycle feedback g captured in BernSCM. The sensitivity metrics for the BernSCM standard simulation (HILDA-HRBM with coupled carbon cycle) lie within the C4MIP range (Table ). The uncertainty range for BernSCM, however, is not congruent with the multimodel range of C4MIP. Maximum and standard sensitivity for BernSCM are practically identical. Notably, this sensitivity is smaller (absolutely) than the C4MIP average for the land carbon response to CO2 increase and warming. The resulting gain g is also smaller, though this results in large part from the lower climate sensitivity in BernSCM (which corresponds to 2.5 ∘C as used for the Bern-CC model contribution to C4MIP). The lower end (in absolute terms) of the BernSCM carbon cycle sensitivity range is, however, zero per definition for all but the ocean-CO2 sensitivity βO (see Sect. ). As a consequence, the climate–carbon cycle feedback range also includes zero. In contrast, the C4MIP range does not include zero for all sensitivity parameters.

The land carbon uptake until 2100, under the different BernSCM configurations, varies over 500 Gt C (Fig. ), more than 3 times the range of ocean uptake (180 Gt C). This partly reflects the limited coverage of the uncertainty in ocean mixing but also the fact that the land carbon sink is, together with the source related to land use, the most uncertain item in the budget .