Simple Model for the Antropogenically Forced CO2 Cycle Tested on Measured Quantities

The carbon dioxide information analysis center ( CDIAC ) provides a remarkable 163 years of data on atmospheric CO2 concentrations, man-made CO2 emissions, and from 1959 onwards CO2 net-fluxes into oceans and biosphere. Currently, half of the anthropogenic CO2 emissions remain in the atmosphere. Predominantly the ocean and the biosphere absorb the second half in about equal parts. We describe the anthropogenically forced CO2 dynamics by a linear model of only two parameters which represent physics and biological laws. Our model reproduces the CDIAC measurements perfectly, and allows thus predictions for the future. It does not deal with the equilibrium exchanges of CO2 between atmosphere, oceans and biosphere, but treats merely the net-fluxes resulting from the perturbation of the equilibrium by the anthropogenic emissions. Details as yielded by tracer measurements or ocean chemistry are not required. We applied the model for a tentative projection of the future CO2 cycle based on prospective anthropogenic emission *Corresponding author: E-mail: moluedecke@t-online.de; Lüdecke and Weiss; JGEESI, 8(4), 1-12, 2016; Article no.JGEESI.30532 scenarios from the literature. As a result, the increase of atmospheric CO2 will gradually come to an end and the ocean as well as the biosphere will be the primary sinks of future CO2 emissions of mankind.


INTRODUCTION
Before the industrialisation 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 ( 1750 AD ) the atmospheric CO2 concentration was 277 ppm [1], corresponding to 2.12 · 277 = 587.2 GtC with 2.120 GtC/ppm as the correspondence of atmospheric carbon with CO2 concentration [2]. The CO2 content of the ocean is much higher, approximately 37 000 GtC [3].
Presently ( 2013 ), the atmospheric CO2 concentration has risen to 395 ppm, or to an extra of (395 ppm -277 ppm)·2.12 GtC/ppm=250 GtC, mainly due to fossil fuel burning, slash-and-burn of forests and cement production. The total anthropogenic CO2 production is 10.7 GtC/yr or roughly 5 ppm/yr CO2. About 2.5 ppm/yr of this quantity remains in the atmosphere, the rest is absorbed by the ocean and the biosphere in roughly equal amounts [4].
Since the year 1959 observations of atmospheric CO2 contents and fluxes between atmosphere, ocean and biosphere have increased substantially. Most familiar among these is the atmospheric CO2 content published presently in monthly updates by the Carbon Dioxide Information Analysis Center [4]. From the year 2006 on CDIAC established stocks of the major components of the global carbon budget that are the anthropogenic CO2 emissions, the CO2 fluxes into the atmosphere, the ocean and the biosphere. In the present paper we use the "Global Carbon Budget 14" which comprises the years 1959 -2013. For the years before 1959 back to 1850 we use historical data of anthropogenic CO2 emissions [4] and of atmospheric CO2 concentrations [1].
Models of the carbon cycle under the forcing of anthropogenic CO2 emissions have been publis-hed among others by [5,6]. Most model -work before 1970 is cited by [6]. More recently, 15 known carbon cycle models are compared on the response to a CO2 impulse of 100 GtC in the year 2010 [7].
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 multiple time constants for the CO2 exchange between atmosphere, ocean, and biosphere. The model parameters are obtained from observations, measurements -as for instance from the 14 C/ 12 C ratio [6] -and fitting procedures. However, no systematic comparison of the whole entity of the CDIAC observations with any CO2 global circulation model has been published so far.
The results of atmospheric 14 CO2 measurements, which showed the interruption of the natural 14 CO2/ 12 CO2 ratio by the nuclear bomb test program between 1950 and 1963, yielded new insight in the CO2 exchange between atmosphere and ocean. During this program the ∆ 14 C increased from -20 per mil to ∼1000 per mil in the Northern hemisphere [8]. The rapid decrease of the 14 CO2 after the nuclear test ban treaty of the year 1963 has caused some confusion between the Residence Time "RT" and the Adjustment Time "AT" of an artificial CO2 excess in the atmosphere.
The carbon cycle exchanges vast amounts of CO2 between atmosphere, ocean and biosphere -roughly 90 GtC/yr in all directions ( total fluxes ).
Due to the large total fluxes the RT of CO2 of ∼5 years in the atmosphere is relatively short. In contrast to this, the net-fluxes from the atmosphere into the ocean and the biosphere, with only ∼2.5 GtC/yr each, are much smaller and are caused by anthropogenic carbon emissions -except for minor natural fluctuations. The net-fluxes are essential for the AT to a new CO2 equilibrium. As a consequence, the AT is more than an order of magnitude higher than the RT and independent of it [9], [10]. Both quantities RT and AT are generally in the literature given as "half-life".
In contrast to the many complex CO2 circulation models which consider all CO2 fluxes our objective was to model the anthropogenically forced CO2 cycle by a minimum of physical assumptions considering exclusively the carbon net-fluxes. We do not model the detailed processes of the total fluxes but simply determine the only two relevant parameters of our linear Ansatz from the CDIAC data. Details of the cycle such as tracer results, biogenic ocean dynamics and chemistry as explored by complex and extensive models and belonging to the equilibrium state of the total CO2 fluxes are beyond the scope of this work and unnecessary for such a simple model.
Following the aim of simplification, we use only one parameter for the atmospheric-oceanic netflux of CO2 and another one for the complex interaction of the biosphere with the atmospheric CO2. The validity of the model is verified by comparison with the data [4]. As a model input we use only the anthropogenic CO2 emissions from 1850 until present.
We evaluate and compare the CO2 remaining from a hypothetical CO2 impulse of 100 GtC in the year 2010 of our model with [7].

THE MODEL
In the following we generally use carbon quantities and fluxes instead of CO2 quantities. The already mentioned factor 2.12 GtC / ppm yields the conversion between both. For clarity, carbon fluxes n(t) are generally written in small and their integrated values N(t) in capital letters. All carbon fluxes in our model are net-fluxes.
Our model makes only two assumptions that are based on the linear chemical law of mass action: Firstly, the carbon flux between atmosphere and ocean ns(t) is proportional to the difference of the actual and the preindustrial atmospheric CO2 concentration yielding with Na(t) the carbon content of the atmosphere in the year t, the constant carbon content N0 = Na(1750) = 587.2 GtC as the already mentioned value in the year 1750, and 1/τ [yr −1 ] as parameter of the net CO2 uptake process by the ocean. Our assumption of a constant N0 is based on a negligible change of the oceanic carbon amount because the ocean contains roughly the fortyfold dissolved carbon of the atmosphere. We note that the model neither contains the much higher mutual CO2 exchange between ocean, atmosphere and biosphere nor the complex biogenic activity in the oceans nor the ocean chemistry. Similar simple assumptions as Eq. (2.1) have already be given by [11], [12], [13].
In a second assumption, biospheric increase n b (t) far from saturation is estimated as proportional to the atmospheric carbon increase with na(t) = dNa(t)/dt the carbon flux into the atmosphere, n b (t) = dN b (t)/dt the carbon flux into the biosphere, and b the parameter of the process. This simple approximation corresponds to the known effect of better plant growth due to higher atmospheric CO2 concentrations [14], [15], [16]. Our model of the Eqs. (2.1, 2.2) is linear.
In the following, bars are used for observed quantities for distinction from model quantities.
Together with the anthropogenic carbon emissionsntot(t) and Eq. (2.2) the sum rulē and the equivalent sum rule for the integrated quantities (2.5) Na(t) in Eq. (2.1) has to be completed with a temperature term because the equilibrium between oceanic and atmospheric partial CO2 pressures shifts slightly with sea temperature, is the amount of carbon released into the atmosphere or absorbed by the ocean caused by changing temperatures,T (t) [ 0 C] the average Earth temperature [17] converted to an anomaly around the 1850 AD value, µ the CO2 production coefficient given as µ = 7.5 ppm / 0 C [1], and 2.12 GtC/ppm the already mentioned ratio of atmospheric carbon to CO2 concentration. This completes the first order differential equation Eq. (2.5) to excluding biogenic activity in the ocean and oceanic chemistry. Eq. (2.7) has two free parameters, 1/τ and b that can be evaluated by nonlinear optimization. We note that the input of Eq. (2.7) is a) the anthropogenic CO2 emissionsntot(t), b) the sea surface temperature termSa(t), and c) the constantN0.
The differential equation Eq. (2.7) can be solved numerically yielding Na(t).Na(t0) is used as initial condition for t = t0. By the sum rule Eq. (2.3) the fluxes ns(t) and n b (t) are determined from na(t) = dNa(t)/dt. Finally, by numerical integration of ns(t), and n b (t) the quantities Ns(t) and N b (t) are obtained. The model results Ns(t), and N b (t) can be directly compared with the observations for the period 1959 -2013 [4]. For the earlier period 1850 -1959 onlyna(t) andNa(t) exist for a comparison.

THE DATA
Data on the carbon cycle are subdivided here into the periods 1850 to 1959, 1959 to 2013, and 2013 to 2150. For the period 1850 to 1959 only the estimations for anthropogenic CO2 emissions and proxy data on the CO2 content of the atmosphere are available [4], [18], [19], [1].
The year 1959 can be considered as the start of more systematic data acquisition on the carbon cycle. From this year on measurements, observations and estimations of carbon fluxes are reported [4], [2] as follows: Rate growth of atmospheric CO2 concentrationna(t), fossil fuel burning and cement productionn f uel (t), land use change such as deforestationn landuse (t), ocean sinkns(t), and transforming organic materials in the biospheren b (t).na(t) are obtained by direct measurements,n f uel (t) from energy statistics, n landuse (t) from data of land cover change, deforestation and from models.ns(t) are based on the mean ocean sink obtained by observations in the 1990 decade and ocean models.n b (t) is the residual of the other budget terms such as In the upper panel of Fig. 1

METHODS
The integration of a first order differential equation dy/dt = f (t, y(t)) such as Eq. (2.7) can be simply carried out by the explicit EULER technique [22].
The use of more elaborate numerical methods, such as RUNGE-KUTTA, yields no substantially different results for Eq. and for the period 1959 -2013 as The model allows to evaluate the AT for the restoration of a stationary state after a CO2 perturbation. We followed [7] by applying a CO2 impulse of 100 GtC in the year 2010 and evaluated the model response as the time dependent CO2 amount remaining from the impulse. The results of [7] and of our model are compared in Fig. 4. As an alternative evaluation of the AT, we assumed the anthropogenic CO2 emission from 2013 on completely stopped and analyzed the decreasing atmospheric CO2 from this time on. The result of this numerical experiment is shown in the left lower panel of Fig. 3.

PARAMETER ESTIMATION
The procedure of nonlinear optimization for estimation of the parameters τ and b in Eqs. (2.1, 2.2) minimizes the objective function F (τ, b) given by Eq. (4.9). This means that we restricted the nonlinear optimization to the period 1959 -2013 of the more reliable measurements. We note that we could also have used a related objective function of the fluxes, instead of the integrated fluxes, as direct data. However, the large scatter of the fluxes leads the minimization very often to local minima. In contrast, minimizing the objective function for the integrated fluxes always yields unique global minima. For the minimizing procedure we applied the nonlinear SIMPLEX method [23].
The results of the optimization are depicted in Table 1 and show no essential differences by including or excluding the temperature term of Eq. (2.6).
In the following we therefore refer generally to the parameter values of the optimization that exclude the temperature term.
The time series of the model together with the pertinent CDIAC observations are shown in Fig.  1 for the years 1850 to 1959 and in Fig. 2 for the years 1959 to 2013. We mention that the values of the parameter τ in Table 1 agree well with an estimation of 81.4 years given in [13] for the oceanic carbon uptake. The value of parameter b of Eq. (2.2) goes well together with measurements of increases in yield for food plants under rising atmospheric CO2 concentrations [16] (see also 6. Results). Table 1

RESULTS
The agreement of the integrated quantities Na(t), Ns(t), and N b (t) with the observations is nearly perfect, differences are hardly detectable by eye ( see Fig. 1 and Fig. 2 ). The yearly fluctuations of AF (see lower panel of Fig. 1) and the fluxes n(t) ( see Fig. 2 ) are not simulated as perfectly as the integrated quantities, only the average courses over several years show good agreement. These fluctuations are probably caused by natural variations, as for instance varying intensity of the photosynthesis in the seasons or El Niño, which can not be considered in our simple model. Noise in the observations data could be an additional cause. As a main conclusion the overall perfect agreement of the model with the observational data indicates that model linearity is a valid approximation at least for the long period of the last 163 years of observations. The following details confirm this: (a) The two parameters of our model are evaluated by minimizing the objective function F (τ, g) of Eq. (4.9) for the period 1959 -2013. Hence, one could expect that these parameters would not agree as well with the observations of the longer early period 1850 -1959. However, this is not the case ( see Fig. 1 ), which indicates the robustness of our linear model over periods not considered by the parameter estimation. (c) Parry et al. [16] report that a wide variety of food plants show nearly linear increases in yield from 0.02% to 0.06% per 1 ppm rising atmospheric CO2 until an atmospheric CO2 concentration of 750 ppm. This fits well with our model: From the total anthropogenic CO2 emissions of ∼10.5 GtC/yr in the year 2013 our model yields ∼4.5 GtC/yr or 4.5/2.12 = 2.2 ppm/yr atmospheric CO2 increase and ∼3 GtC/yr absorbed in roughly equal amounts by ocean and biosphere (see Fig. 2). The total carbon content of the biosphere is roughly 4000 GtC [24]. If one assumes roughly equal increase for the total biosphere as for food plants by rising atmospheric CO2, as reported in [16] The discussion about man-made global warming in mind we think that an estimation of the future increase of atmospheric CO2 by our linear model could be useful and valid. For this purpose we postulate that the linearity of our model remains at least for the next 100 years. This seems reasonable because in spite of an atmospheric CO2 increase of about 40% from the year 1850 until present an increase of the AF cannot be seen in the CDIAC data ( see lower panel of Fig. 1, red curve and the abstract of [13] ).
Under the assumption of future model validity and depending on the emission scenarios given in [20] until 2100 AD as well as the assumed emission scenarios after 2100 AD until 2150 AD ( see Fig. 3, upper panel ), our model gives the results depicted in the lower panels of Fig. 3: The atmospheric CO2 content will not exceed a maximum between ∼500 to ∼800 ppm ( see left lower panel of Fig. 3 ). The AF remains constant until 2050 AD and does decrease later steadily for all scenarios. At maximum atmospheric CO2 concentration the atmospheric net-flux na and with it the AF = na(t) /ntot(t) change sign because the sum of the net-fluxes of atmospheric CO2 into the ocean and the biosphere is smaller than the emitted anthropogenic CO2 ( see right lower panel of Fig. 3 ).
The AT of the atmospheric CO2 has been calculated with our model based on two different numerical experiments. In the first experiment we stopped the anthropogenic CO2 emissions in the year 2013. As a consequence, the atmospheric CO2 concentration begins to decrease until the natural equilibration will be accomplished.  [7]. During the first 100 years after the impulse the decrease of the response of our model lies above the upper limit of the 15 comprehensive models but after 2200 years distinctly below their lower limits. In a far distant future the impulse-response of our model approaches zero whereas all responses of the comprehensive models do not fall below ∼20%. The difference 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, there exists so far no evidence for the Revelle effect in the CDIAC data. Gloor et al. [13] emphasize similarly in the abstract as "claims for a decreasing long-term trend in the carbon sink efficiency over the last few decades are currently not supported by atmospheric CO2 data and anthropogenic emissions estimates". Thus, such effects are at present hypothetical. This could be related to (1) uncertainties in current estimates of emissions, (2) noise in the AF due to natural climate variability, or (3) still unknown biogenic effects, for instance a higher downward flow of organic carbon in the deeper ocean over many decades or centuries [25].

CONCLUSIONS
Our linear model with only two parameters gives perfect agreement with the major observation components of the global carbon cycle provided by [4] as are the integrated carbon rates of the atmosphereNa(t), the oceanNs(t), and the biosphereN b (t). Also the agreement with the observed yearly carbon net-fluxesna(t), ns(t), andn b (t) is on average perfect. That the fluctuations of these fluxes are larger than those of the model is apparently due to the noninclusion of natural variations ( such as different strength of the photosynthesis in the seasons and El Niño etc. ). Our model gives no indication that the change of sea surface temperatures since the beginning of industrialization has an appreciable influence on the anthropogenically forced carbon cycle.
The linearity of our model and increases in yields of food plants due to rising atmospheric CO2 concentrations are confirmed by further authors [13], [16], [11], [12]. The agreement of our simple model with the observations [9], [26] indicates that we are far from the influences of nonlinearities or the Revelle effect. As a consequence, we expect no substantial deviations from the linear model results at least for the next 100 years.
With regard to the actual discussion about an assumed dangerous climate change by anthropo-genic CO2 emissions in future, our most important model result yields the impact of anthropogenic CO2 emission scenarios given in [20] until the year 2100 AD ( completed by own tentative continuations from 2100 to 2150 AD ). For all scenarios, between 2080 to 2140 AD atmospheric CO2 concentrations up to at most ∼500 until ∼800 ppm occur. These future turning points of the atmospheric CO2 content take place 30 years after the pertinent emissions maxima. The maximum of 800 ppm atmospheric CO2 concentration seems to be safe because the pertinent emission scenario assumes burning a twofold of the actual known coal reserves worldwide. After the apogee of atmospheric CO2 concentration predominantly the ocean and the biosphere will be the sinks of future CO2 emissions of mankind.