ESDDEarth System Dynamics DiscussionsESDDEarth Syst. Dynam. Discuss.2190-4995Copernicus GmbHGöttingen, Germany10.5194/esdd-6-2043-2015A simple model of the anthropogenically forced CO2 cycleWeberW.LüdeckeH.-J.moluedecke@t-online.deWeissC. O.Technical University Dortmund, Institute of Physics, Dortmund, GermanyHTW, University of Applied Sciences, Saarbrücken, GermanyPhysikalisch-Technische Bundesanstalt, Braunschweig, Germanyretireddeceased 2014H.-J. Lüdecke (moluedecke@t-online.de)20October2015622043206221September20157October2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://esd.copernicus.org/preprints/6/2043/2015/esdd-6-2043-2015.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/preprints/6/2043/2015/esdd-6-2043-2015.pdf
From basic physical assumptions we derive a simple linear model of the global CO2 cycle without free parameters. It
yields excellent agreement with the observations reported by the carbon dioxide information analysis center (CDIAC) as time
series of atmospheric CO2 growth, of sinks in the ocean and of absorption by the biosphere. The agreement extends from
the year 1850 until present (2013). Based on anthropogenic CO2 emission scenarios until 2150, future atmospheric
CO2 concentrations are calculated. As the model shows, and depending on the emission scenario, the airborne fraction of
CO2 begins to decrease in the year ∼ 2050 and becomes negative at the latest in ∼ 2130. At the same time the
concentration of the atmospheric CO2 will reach a maximum between ∼500 and ∼ 900 ppm. As
a consequence, increasing anthropogenic CO2 emissions will make the ocean and the biosphere the main reservoirs of
anthropogenic CO2 in the long run. Latest in about 150 years, anthropogenic CO2 emission will no longer increase
the CO2 content of the atmosphere.
Introduction
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-1CO2. 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 ∼90GtCyr-1 compared with the pertinent present carbon
net flux of ∼2GtCyr-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 .
The model
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).
Model input and parameters
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.
Solving the model equations
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)=670.85GtC which yields F of Eq.(18). For the future we use Eqs. (11) and (12),
the initial value N‾a(2013)=838.1GtC, 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.
An alternative method of parameter optimization
In order to judge the reliability of the values τ‾, b‾, determined from Eqs. (8) and (9), an independent second method is
helpful. For this purpose, τ and b can be determined by
nonlinear optimization. We restricted the iteration to the period
1959–2013 of the more reliable measurements. The procedure
minimizes the objective function F(τ,b) given by Eq. (18). 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 SIMPLEX method
of with randomly generated starting values.
Results
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.
Summary and discussion
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.
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Results: τ‾ and b‾ are extracted
from Eqs. (8, 9), τ and b from nonlinear optimization; F, G,
F(τ,b), G(τ,b) are the differences between model and
measurements extracted from Eqs. (17, 18). Row 1 is with and Row 2
without the temperature term S‾a(t) of Eq. (6).
Future carbon emission scenarios until AD 2100 given
by . Emissions from 2100 to 2150 are arbitrarily assumed
to decrease linearly to half the 2100 year value.
Results for the period 1850–2013.
Upper panel, left y axis: model (black) and observations (red) of
atmospheric CO2 concentrations; in the period 1850 to 1959 the
observation uncertainties are indicated. Upper panel, right y axis:
Total anthropogenic emissions n‾tot(t) (green). Lower
panel: Airborne fraction AF = na/n‾tot (model in blue,
observations in red).
Results for the period 1959–2013.
Model and observations of Na(t), Nb(t), Ns(t), na(t),
nb(t), and ns(t) (observations in red color). Nb(t) and
Ns(t) in the left upper panel are shifted for clarity (Na in
the left upper panel identical except for the factor 2.12 with
CO2).
Results for the period 2013–2150.
Left panel: Atmospheric CO2 concentrations according to the
scenarios given by , see Fig. 1: A1FI (blue), A2 (red),
A1B (black), B2 (cyan), A1T (green), and B1 (brown). Zero emission
of anthropogenic CO2 from the year 2013 on causes the steadily
decreasing magenta curve with an adjustment time t1/2 of 103 years. Right panel: Airborne fraction AF = na(t)/n‾tot(t) for the six emission scenarios. The horizontal dotted
line indicates AF = 0 for clarity.
Results for the period 2013–3010.
Upper panel, left y axis: CO2 remaining from a 100 Gt C impulse
in the year 2010 until the year 2110 (blue). Upper panel, right y axis: Atmospheric carbon content Na(t) (green). The impulse
applied in 2010 is visible as a step in the green curve. Lower
panel: CO2 remaining as in the upper panel for a period of 1000 years. The adjustment time t1/2 is 100 years. The grey shaded
regions indicate the pertinent impulse responses of 15 models
published by .