Before the beginning of the industrial time and considerable land use the ratio of CO2 in the atmosphere and in the oceans had been stationary. At the beginning of the industrial era (AD ∼1750) the atmospheric CO2 concentration was 277 ppm , corresponding to 2.12 × 277 = 587 Gt C with 2.12 GtCppm-1 as the ratio of atmospheric carbon to CO2 concentration . The CO2 content of the ocean is much higher, approximately 37 000 Gt C .

Presently (2013), the atmospheric CO2 concentration has risen to 395 ppm, or to an extra of (395-277)×2.12=250 Gt C, mainly due to fossil fuel burning, slash-and-burn of forests and cement production. The total anthropogenic CO2 production is ∼ 10 GtCyr-1 or ∼ 4.7 ppmyr-1 CO2. About 2.5 ppmyr-1 of this quantity remains in the atmosphere, the rest is absorbed by the ocean and the biosphere in roughly equal amounts .

Since 1959 observations and measurements of atmospheric CO2 contents and fluxes between atmosphere, ocean and biosphere have increased substantially. The first stock of these data was established in the year 2006, the latest as “Global Carbon Budget 14” in the year 2014 . The latter covers the years from 1959–2013. The “Global Carbon Budget 14” is used in the present paper. Historic CO2 data for the preceeding years 1850 until 1959 are also given by . However, no systematic comparison of the extensive CDIAC data with any CO2 global circulation model has been published till now.

Modeling the carbon cycle under the forcing of anthropogenic CO2 emissions has been published among others by and . Most model work before 1970 is cited by . In particular, describes the details of the 15 best known complex carbon circulation models and compares their results on the response to a CO2 impulse of 100 Gt C in the year 2010. Modern models include the details of complex interactions between atmosphere, ocean and biosphere with their pertinent parameters. Among these are saturation of the ocean uptake under increasing atmospheric CO2 concentrations, soil respiration, mixed atmospheric and oceanic multi-layers, divisions of the hemispheres into segments, and more than one time constant for the CO2 exchange of atmosphere, ocean, and biosphere. The model parameters are obtained from observations, measurements and fitting procedures. , for instance, extracts model parameters from 14C concentration measurements.

The results of atmospheric 14CO2 measurements , which showed the interruption of the natural 14CO2 equilibrium by the nuclear bomb test program, yielded new insight in the CO2 exchange between atmosphere and ocean. However, the rapid decrease of 14CO2, of an initial thousandfold concentration compared to the natural level, after the end of the bomb tests, has caused some confusion between the residence time RT and the adjustment time AT of an artificial CO2 excess in the atmosphere. The RT of CO2 has the rather small value of ∼5 years, whereas the AT is more than an order of magnitude higher (RT and AT values in half-life). The carbon exchange between atmosphere and ocean of ∼90 GtCyr-1 compared with the pertinent present carbon net flux of ∼2 GtCyr-1 explains the difference . The abrupt end of the bomb tests left a fast decreasing 14CO2 flux from atmosphere into the ocean without a counterpart of the opposite way. In contrast to this, the 12CO2 fluxes are always in two directions and are similar in magnitude because the CO2 partial pressures of the upper ocean layer and the atmosphere are nearly equal.

In contrast to complex CO2 circulation models, our objective was to model the anthropogenic forced CO2 cycle by a minimum of physical assumptions and approximations. Therefore, we use only a single time constant for the atmospheric-oceanic net flux of CO2 and obtain the model parameters from measurements. The validity of the model results is verified by comparison with . As a model input we use the anthropogenic CO2 emissions from 1850 until present. From the response to a hypothetical CO2 impulse of 100 Gt C in the year 2010, as proposed and used by for the comparison of the 15 circulation models mentioned above, we evaluate the CO2 remaining from the impulse and compare it with the results of .

In the following carbon quantities and fluxes instead of CO2 quantities are preferentially used. The mentioned factor 2.12 GtCppm-1 yields the conversion between both. For clarity, carbon fluxes [GtCyr-1] are written in small and their integrated values [Gt C] in capital letters.

Our model makes only two assumptions: Firstly, the carbon net-flux between atmosphere and ocean ns(t) can be approximated by ns(t)=1/τ⋅Na(t)-N0 with Na(t) the carbon content of the atmosphere in the year t, N0=Na(1750)=587.2 Gt C the pertinent value in the year 1750, and τ the time factor of the process. 587.2 Gt C is equivalent to 277 ppm atmospheric CO2 concentration in the year 1750 due to 587.2=277×2.12.

In a second assumption, biospheric increase nb(t) far from saturation is estimated as proportional to the atmospheric carbon increase na(t) nb(t)=b⋅na(t) with na(t)=dNa(t)/dt the carbon flux into the atmosphere, nb(t)=dNb(t)/dt the carbon flux into the biosphere, and b the parameter of the process. By the basic Eqs. (1, 2) the model is linear.

In the following, bars are used for measured quantities for distinction from model quantities. Together with the anthropogenic carbon emissions n‾tot(t) and Eq. (2) the sum rule n‾tot(t)=na(t)+nb(t)+ns(t)=1+b⋅na(t)+ns(t) and the equivalent sum rule for the integrated quantities N‾tot(t)=Na(t)+Nb(t)+Ns(t) hold. Equations (1)–(3) can be combined to dNa(t)dt=n‾tot(t)-ns(t)/(1+b)=n‾tot(t)-1/τNa(t)-N‾0/(1+b). Na(t) in Eq. (1) has to be completed with a temperature term because the equilibrium between oceanic and atmospheric partial CO2 pressures shifts slightly with sea temperature, S‾a(t)=μ⋅2.12⋅T‾(t)=15.9⋅T‾(t). S‾a(T) [Gt C] is the amount of carbon released into the atmosphere or absorbed by the ocean caused by changing temperatures, T‾(t) [∘C] the average Earth temperature converted to an anomaly around the AD 1850 value, μ the CO2 production coefficient given by as μ = 7.5 ppm ∘C-1, and 2.12 [Gt C ppm-1] the already mentioned ratio of atmospheric carbon to CO2 concentration. This completes the first order differential Eq. (5) to dNa(t)dt=n‾tot(t)-ns(t)/(1+b)=n‾tot(t)-1/τNa(t)+S‾a(t)-N‾0/(1+b). Equation (7) has two parameters, 1/τ of Eq. (1) and b of Eq. (2). The next paragraph shows that both parameters can be evaluated from measurements. Thus, the model has no free parameters.

With N‾a(t0) as initial condition for t=t0, the measured total anthropogenic carbon emissions n‾tot(t), and the temperature term S‾a(t) the differential equation Eq. (7) can be solved numerically, yielding Na(t). By the sum rule Eq. (3), the quantities ns(t) and nb(t) are determined from na(t). Finally, by numerical integration of ns(t), and nb(t) the quantities Ns(t), and Nb(t) are obtained.

The model results can be directly compared with the observed quantities for the period 1959–2013 , such as the atmospheric carbon content N‾a(t), the integrated oceanic uptake N‾s(t), and the integrated uptake of the biosphere N‾b(t).

The following measurements, observations and estimations of carbon fluxes, which cover the years 1959–2013 are given by and and references cited therein: fossil fuel burning and cement production n‾fuel(t), land use change such as deforestation n‾landuse(t), atmospheric accumulation n‾a(t), ocean sink n‾s(t), and transforming organic materials in the biosphere n‾b(t). The latter was estimated from the residual of the other budget terms such as n‾b=(n‾fuel+n‾landuse)-n‾a-n‾s. The total anthropogenic emissions are n‾tot(t)=n‾fuel(t)+n‾landuse(t)=n‾a(t)+n‾b(t)+n‾s(t). The graphs of n‾a(t), n‾b(t), and n‾s(t) are given in the upper right and in both lower panels of Fig. 3 in red.

Data on carbon emissions from fossil fuel burning, cement production, and land use change, such as deforestation, which extend back to 1850 are also available from . For the times before 1959 the uncertainties associated with land use are estimated between 40–100 % . Atmospheric CO2 concentrations as global means are given by reaching back until the year 1000 AD.

The fluxes n‾a(t), n‾s(t), n‾b(t) and the integrated values N‾a(t), from t1=1959 to t2=2013, together with N‾0=N‾a(1750)=587 Gt C yield the model parameters 1/τ and b of the Eqs. (1) and (2) as 1/τ‾=∑t1t2n‾st∑t1t2(N‾a(t)-N‾0)=0.01223 or τ‾ = 81.7 years and b‾=∑t1t2n‾bt∑t1t2n‾a(t)=0.668. used Eq. (1) for the oceanic CO2 uptake as well and estimated τ‾ as 81.4 years in excellent agreement with the result of Eq. (8).

Future global anthropogenic emissions in GtCyr-1 are estimated by as six different scenarios which end in the year 2100 (Fig. 1). The extreme scenario A1FI has an integrated value of ∼2000 Gt C. However, estimates for the coal reserves of the Earth cite numbers of ∼1100 Gt C . Therefore, we assume as tentative data for our model after AD 2100 until 2150 a decrease of anthropogenic carbon emissions. Because no estimates for this future period are available we assume arbitrarily for all six scenarios a linear decrease of n‾tot to half the value of the year 2100.

The integration of a first order differential equation dy/dt=f(t,y(t)) such as Eq. (7) can be simply carried out by the explicit EULER technique . y(t+Δt)=y(t)+Δt⋅f(t,y(t)) The use of more elaborate numerical methods, such as RUNGE-KUTTA, yields no substantially different results for Eq. (7). With a time step of Δt = 1 year and an initial value N‾a(t0), Eqs. (7) and (10) lead to the following iteration (i = 1, 2, ...): na(ti)=11+bn‾totti-1-1/τNati-1+S‾ati-1-N‾0Na(ti)=Na(ti-1)+na(ti)⋅Δt and further to ns(ti)=n‾tot(ti)-(1+b)⋅na(ti)nb(ti)=n‾tot(ti)-na(ti)-ns(ti)Nb(ti)=Nb(ti-1)+nb(ti)⋅ΔtNs(ti)=Ns(ti-1)+ns(ti)⋅Δt. The airborne fraction AF(t)=na(t)/n‾tot(t) is obtained from the result na(t) of Eq. (11). Initial values of N‾s(t0) and N‾b(t0) are not known. Thus, the values are uncertain by an additive constant. Therefore, the starting values in the left upper panel of Fig. 3 are arbitrary and shifted for clarity. For the early period 1850–1959 only n‾tot(t), n‾a(t) and N‾a(t) are available from observations. For the period 1959–2013 these data and additionally n‾a(t), n‾s(t), and n‾b(t) were measured. For the future 2013–2150 only scenarios of n‾tot(t) can be used.

The root of the squared differences between the integrated quantities of observations N‾a(t), N‾s(t), N‾b(t) and their pertinent model counterparts yields a measure of the model accordance with the observations, for the period 1850–1959 as G=∑t=18511959N‾a(t)-Na(t)2 and for the period 1959–2013 as F=∑t=19602013N‾a(t)-Na(t)2+N‾s(t)-Ns(t)2+N‾b(t)-Nb(t)2.

Due to the different data basis, the iteration equations Eqs. (11)–(16) are solved separately for the following periods: For 1851–1959 we apply Eqs. (11) and (12) and the initial value N‾a(1850) = 605.8 Gt C which yields G of Eq. (17). For the period 1960–2150 we use Eqs. (11)–(16) and the initial value N‾a(1959)=670.85 GtC which yields F of Eq.(18). For the future we use Eqs. (11) and (12), the initial value N‾a(2013)=838.1 GtC, and emission scenarios for n‾tot.

The model allows to evaluate the adjustment time AT of reestablishing a stationary state after a CO2 perturbation. We followed by applying a CO2 impulse of 100 Gt C in the year 2010 and evaluated the model response as the time dependent CO2 amount remaining from the impulse. As an alternative test, we assumed the anthropogenic CO2 emission from 2013 on completely stopped and analyzed the decreasing atmospheric CO2 from this time on. The results from both tests are compared below.

Table 1 gives the model results for the parameters τ‾, b‾ from Eqs. (8) and (9) and for τ, b evaluated by nonlinear optimization. The results F(τ,b) and G(τ,b) from nonlinear optimization and F, G from measurements agree well. According to Table 1 also the temperature term does not play a substantial role. Figures 2–4 depict the time series of the observations against their model counterparts. These latter are obtained without the temperature term and with τ, b from nonlinear optimization. In corresponding figures that would use τ‾, b‾ calculated with Eqs. (8) and (9) or include the temperature term, differences to the Figs. 2–4 can hardly be detected by eye.

Figure 2 shows the atmospheric CO2, the anthropogenic CO2 emissions n‾tot(t), and the airborne fraction AF=na(t)/n‾tot(t) from 1850 to 2013. Model and observation of the atmospheric CO2 concentrations are hardly distinguishable by eye (upper panel). However, this agreement of integrated quantities does not occur for the airborne fraction since it is not an integrated quantity (lower panel).

Figure 3 shows the observations N‾a(t), N‾b(t), N‾s(t), n‾a(t), n‾b(t), and n‾s(t) from 1959 to 2013 together with their pertinent model time series. As already mentioned, Ns(t) and Nb(t) are uncertain by an additive constant and were shifted here for clarity.

Figure 4 covers the period 2013 until the future year 2150. The left panel shows the time series of atmospheric CO2 caused by the six emission scenarios given by and a CO2 curve caused by a stop of anthropogenic CO2 emissions in the year 2013. The right panel shows the airborne fractions caused by the six emission scenarios. All scenarios yield a maximum of the atmospheric CO2, the earliest in the year ∼2090, the latest in the year ∼2130. At these times the airborne fraction changes sign. The steadily decreasing CO2 in the left panel (magenta curve) is the result of the stop of anthropogenic CO2 emissions at the year 2013. The CO2 adjustment time AT evaluated from this event is t1/2 = 103 years. At t1/2 the carbon concentration reaches its half value Na(t1/2)=[Na(2013)-Na(∞)]/2.

Figure 5 depicts the atmospheric CO2 concentration and the response for a 100 Gt C impulse charged in the year 2010. After the year 2010 the anthropogenic carbon emissions are kept at the 2010 year-value of 10 Gtyr-1 as proposed by . The response function shows the percentage of carbon remaining from the impulse, obtained from the difference of the atmospheric carbon with and without the impulse and has an adjustment time AT of t1/2 = 100 years. The upper panel shows that during the first 100 years after the impulse the decreasing CO2 exceeds the upper limit of the far more comprehensive models, reported by . After the year 2200 the lower panel shows the model results distinctly below the range of the comprehensive models. In the long run the CO2 concentrations of the model approach zero. The final difference of ∼ 20 % is discussed below.

Our simple linear model gives excellent agreement with all relevant measurements, i.e. N‾a(t), N‾s(t), and N‾b(t). Also the agreement with the rates n‾a(t), n‾s(t), and n‾b(t) is on average excellent. That the fluctuations of the measured rates are larger than those of the model results is apparently due to the non-inclusion of natural variations (such as El Niño etc.). We mention as an important model result a decrease of the atmospheric CO2 concentration from roughly 2100 on. We mention further that the change of the sea surface temperature since the beginning of industrialization had apparently no appreciable influence on the anthropogenically forced carbon cycle.

The perfect agreement of the linear model results with the measurements indicates, that we are far from influences of nonlinearities or the Revelle effect . An increase of the CO2 content of the atmosphere by a factor of 2, as expected for the next 100 years, will therefore probably not cause substantial deviations from the linear model results. The model should therefore at least be suited for predictions during this nearer future period.

We note that on a decade to century scale the more elaborate models predict a faster decrease of the CO2 concentration after the 100 Gt C test impulse than the linear model. On a century to millennium scale the more elaborate models predict a persistent CO2 concentration, while the linear model shows a decrease to zero concentration with an adjustment time of 100 years.

The difference in the long run may stem from the Revelle effect, included in the elaborate models, a resistance to absorbing atmospheric CO2 by the ocean due to bicarbonate chemistry. However, as underlines, there exists so far no evidence for the Revelle effect. Thus, such effects are presently hypothetical.