global warming and human activity: a model for studying...

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e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 243–256

avai lab le at www.sc iencedi rec t .com

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

lobal warming and human activity: A model fortudying the potential instability of the carbonioxide/temperature feedback mechanism

lessio AlexiadisCY-CompSci, European Marie Curie Transfer of Knowledge Center (TOK-DEV) for the Computational Sciences, Department ofechanical and Manufacturing Engineering , University of Cyprus, 75 Kallipoleos St, PO Box 20537, 1678 Nicosia, CYPRUS

r t i c l e i n f o

rticle history:

eceived 27 March 2006

eceived in revised form

7 November 2006

ccepted 20 November 2006

ublished on line 8 January 2007

a b s t r a c t

In this paper, control theory is used to study the connection between human activities and

global warming. A feedback model is proposed and tested against temperature and carbon

dioxide concentration historical data. Four scenarios are taken into account and simulated

by the model; stability analysis is also discussed. The model proposed here simulates the

historical data correctly and the scenarios show that, even in the case of dramatic reduc-

tion of the anthropogenic carbon dioxide emission, the temperature will not decrease for a

eywords:

lobal warming

limate change

reenhouse effect

certain time. Stability analysis reveals a complex pole near the unit circle. This means that,

although the system at the moment is stable, it is very close to becoming unstable with

unpredictable consequences on climate change.

© 2006 Elsevier B.V. All rights reserved.

. Introduction

reenhouse effect and global warming are controversial is-ues in the present scientific and political debate. For thiseason, researchers are studying models able to describe theonsequences of human activity on the temperature of thelanet. General circulation models (GCM) and model-basedethods (MBM) dealing with historical data are the most com-on approaches. In the first case, the fundamental equations

escribing the conservation of mass, energy and momen-um are approximated and solved numerically. In the secondase, the relationship between temperature and forcing vari-
bles is studied by analyzing time-series with various math-matical and statistical tools (e.g. Kaufmann and Stern, 1997;homson, 1997; Liu and Rodriguez, 2005; Loehle, 2004; Krivovand Solanki, 2004).
E-mail address: [email protected]/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2006.11.020

The GCMs (e.g. Friedlingstein et al., 2003; Manabe andStouffer, 1993; Joos et al., 2001; Plattner et al., 2001; Cox etal., 2000) are more ambitious than the MBMs since they at-tempt to describe the actual physics of the process. Thesemethods, however, clash with the objective difficulties in ap-proximating the complex dynamics of the planet and somereviews (e.g. Lindzen, 1997; Rodhe et al., 2000; Soon et al.,2000, 2001; Visser et al., 2000) point out to some uncer-tainties in the GCMs that need clarification. On the otherhand, the MBMs fall into the category of the black-box mod-els. They do not deal with the system’s physics directly butare based on certain model-families whose parameters have
to be determined according to historical data. Beside thesetwo categories, there is a third group of models, based ona simplified description of the planet’s dynamics that fallsbetween the previous two (e.g. Lenton, 2000; Sarmiento et
mailto:[email protected]

dx.doi.org/10.1016/j.ecolmodel.2006.11.020

i n g

by the simplicity of the model. Another reason that makes itdifficult to include these gasses is that the available emissiondata for CH4 and N2O are not as complete as the CO2 data. Asimilar discussion can be done for CFCs and aerosols with the

244 e c o l o g i c a l m o d e l l

al., 1995; Ichii et al., 2003; Wigley, 1993; Kheshgi and Jain,2003).

The method used in this paper is based on control theoryand uses system identification techniques to determine thetransfer functions that approximate the system. Four black-box models are considered together and connected in a feed-back loop structure. In this way, all the fundamental variablesare included in the same structured model: some of them asinput and some of them as internal variables, which play therole of input and output at the same time. The overall modelis composed by four black-box models, whose characteristicsare always to be statistically determined by historical data.However, the way the models are connected imitates the fea-tures of a typical physical phenomenon (feedback) which hasa fundamental importance in the case of global warming.

The particular feature, which makes the model proposedin this paper different from other global warming models, isthe possibility to use control theory in order to analyze thestability of the system. The feedback mechanism between car-bon dioxide and temperature has been working for millennia,showing stable and self-regulating characteristics. It is nowconsidered possible, however, that human activity can inter-fere and destabilize this equilibrium. The question that stabil-ity analysis can answer is whether it is likely that an externalforcing can transform the previous stable and self-regulatingmechanism into an unstable and auto-excited system withtypical run-away behavior.

2. Brief survey on dynamic models

Dynamic models describe relationships between time depen-dant inputs u(t), time depending outputs y(t) and possible dis-turbances �(t). Different types of models are possible depend-ing on the kind of equations used as basis for the model. Aclass of models often used in engineering and physics is thetime-invariant linear systems.

y(t) + a1y(t − 1) + · · · + ana y(t − na)

= b0u(t) + · · · + bnbu(t − nb) + �(t) (1)

Eq. (1) is limited to linear systems but, usually, this is not reallya limit because many non-linear systems can be linearized.1

Eq. (1) is sometimes written as

A(q)y(t) = B(q)u(t) + C(q)e(t) (2)

or

y(t) = G(q)u(t) + H(q)e(t) (3)

where A(q), B(q) and C(q) are polynomial of the backward shiftoperator q−kx(t) = x(t − k); G(q) = B(q)/A(q) the transfer func-tion, H(q) = C(q)/A(q) the disturbance filter and the distur-
bance is modelled as function of white noise e(t) with zeromean and � variance. The simplest way to take into accountthe white noise term is obtained when e(t) enters as a direct
1 Linear, in this case, means that the parameters a1,...,na and b0,...,nb

do not depend on y or u.

2 0 3 ( 2 0 0 7 ) 243–256

error in the difference equation.

A(q)y(t) = B(q)u(t) + e(t) (4)

This family of models is often called ARX. AR refers to theautoregressive part A(q)y(t) and X to the extra input B(q)u(t).According to this model, the white noise goes through theA(q)−1 dynamics before adding to the output. Although thisassumption is not always the most natural from a physi-cal point of view, it has nonetheless very important proper-ties that make it the first choice in many applications (seeLjung 1997).

3. Historical data

Temperature, carbon dioxide atmospheric concentration, an-thropogenic carbon dioxide emission and solar irradiancetime-series are used in this paper. Two types of data are takeninto account: instrumental (from 1857 to 2004) and recon-structed (from 1610 to 1856) data. The first series are based ondirect measurements, while the second ones depend on prox-ies (tree rings, ice cores, etc.). The instrumental data (Figs. 1–4)are more accurate and will be used whenever possible. In cer-tain cases, however, the correct identification of some trans-fer functions will require data older than 1750 (when anthro-pogenic emission is negligible) and reconstructed data will betaken into account. The instrumental data are taken from theweb site of the Goddard Institute for Space Studies availableon-line at http://www.giss.nasa.gov (last visited in November2006), while the reconstructed data are from Jones et al. (1998)(temperature),2 Robertson et al. (2001) (carbon dioxide concen-tration) and Lean et al. (1995) (solar forcing) and downloadedfrom the web site of the National Climate Data Center ofthe U.S. Department of Commerce, http://www.ncdc.noaa.gov(last visited in November 2006). Anthropogenic emission datafrom 1751 were downloaded from the Carbon Dioxide Infor-mation Analysis Center, http://cdiac.esd.ornl.gov (last visitedin November 2006). Although for a complete analysis nitrousoxide and methane atmospheric concentrations should be in-cluded (see Liu and Rodriguez, 2005; Hansen et al., 2000), inthis paper they are not considered because the trends of thesegases are similar to that of the carbon dioxide (small incre-ments until 1945–1950 and higher increments afterwards). Ifthe trends were exactly the same, the resulting difference inthe model parameters would be only a multiplicative constantand would not affect the qualitative results discussed in thispaper. In the present case, the trends are not exactly the sameand small differences should be taken into account; however,the loss of accuracy in neglecting these gasses is compensated

difference that the concentration of the CFCs did not increase

2 There are many different reconstructed temperature dataavailable in the literature and Jones et al. (1998) is not necessar-ily the most accurate. The use of a different set, however, wouldhad not affected the results after 1880 very much since, in thisrange, T′′ > T′ (see Fig. 9).

http://www.giss.nasa.gov

http://www.ncdc.noaa.gov

http://cdiac.esd.ornl.gov

e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 243–256 245

ic

4

Iifetprtsocrw

Ft

Fig. 3 – Carbon dioxide emission due to human activity E[GtC] vs. time [years].

Fig. 1 – Global mean temperature T [◦C] vs. time [years].

n the last 10 years and that aerosols give a negative forcingontribution.

. Feedback effects

n many fields of science, process dynamics can often be stud-ed by means of model-based methods. In control engineering,or instance, industrial plants are often represented by blocks;ach block is studied as a single black-box model but the wayhey are connected provides precise information about therocess as a whole. Initially, the blocks were associated to theeal devices of industrial plants but, very soon, it was noticedhat they could advantageously be related to abstract repre-entations of physical phenomena and, connected with each
ther, used to describe very complex behaviors. In our spe-ific case, the block representation can be used to describe theole of the different forcing variables in the process of globalarming. Solar irradiance is considered to be an important
ig. 2 – Mean atmospheric CO2 concentration C [ppm] vs.ime [years].

Fig. 4 – Total spectral irradiance (annual mean) S [W m−2]vs. time [years].

factor influencing atmospheric temperature (e.g. Loehle, 2004;Hansen et al., 2000) and is included as first input variable in themodel. The connection between CO2 concentration and tem-perature is more complex (e.g. Lashof, 1989; Lashof et al., 1997;Woodwell et al., 1998; Kirchner, 2002; Levy et al., 2004). On theone hand, carbon dioxide is a greenhouse gas and affects di-rectly the thermal balance of the planet. On the other hand, asresponse to subsequent warming, various feedback processescome into operation (water vapor feedback, ice albedo feed-back, cloud-climate feedback, etc.). Some of them, in turn, af-fect the cycling and amount of the sinks and sources of car-bon dioxide (changes in soil respiration rates, photosynthe-
sis, sequestration in biomass, marine biology-CO2 pump, etc.).In works based on pure statistical analysis (Kaufmann andStern, 1997; Thomson, 1997; Liu and Rodriguez, 2005; Krivovaand Solanki, 2004), the specific issue of the feedback is, usu-

246 e c o l o g i c a l m o d e l l i n g 2 0 3 ( 2 0 0 7 ) 243–256

Table 1 – Fitting parameters of the transfer functions G1, G2, G3, G4

G1 G2 G3 G4

a1 −0.602 ± 0.0677 −1.992 ± 0.00484 −0.489 ± 0.0277 −0.998 ± 0.00157a2 0.992 ± 0.00482

37 0.125 0.02255 0.0107 0.03215 0.0107 0.0325

Fig. 5 – Block representation of the feedback model with S(solar forcing), E (carbon dioxide emission due to humanactivity), T (temperature) and C (carbon dioxide

with G2(q) = B2(q)/A2(q). In all the considered cases, the distur-bance has always the form e(t)/A(q). The best fitting is found

Fig. 6 – The feedback model (Fig. 5) divided in twosubsystems.

b0 0.0855 0.0002Lf 0.00823 2.15e−FPE 0.00845 2.17e−

ally, not directly considered. In these studies, the tempera-ture (T) is considered direct function of certain input vari-ables such as C (greenhouse gasses concentration) and S (so-lar forcing) but feedback connection between C and T is notconsidered.

In this paper, no implicit assumptions are made about therelative strength of positive and negative feedback and bothare included in the model (Fig. 5). The input variables of themodel are the solar forcing (S) and the carbon dioxide cumula-tive emission due to human activity (E). Temperature (T) andcarbon dioxide concentration (C) are internal variables play-ing the role of input and output variables simultaneously. Thefinal temperature T is the sum of T′ and T′′. T′ is the tempera-ture fraction originated by solar forcing. T′′ includes the effectof CO2 concentration on temperature. According to this model,solar forcing has a direct effect (through the transfer functionG1) on temperature (T′), which, in turn, affects (through G2)the CO2 concentration (C′). The CO2 emission due to humanactivity E (through G4) contributes to the level of carbon diox-ide (C′′) and the total CO2 concentration (C = C′ + C′′) closesthe feedback loop affecting the temperature through G3. Thetransfer functions G1, G2, G3 and G4, even though they areblack-box models, if considered alone, gain specific physicalmeaning (feedback between CO2 concentration and tempera-ture) when integrated in the feedback structure of Fig. 5. In thefollowing sections the way to determine the transfer functionsfrom the available data is discussed. Dimensionless variablesT∗ = (T − Tm)/�T, C∗ = (C − Cm)/�C, S∗ = (S − Sm)/�S and E∗ =(E − Em)/�E with Tm = 13.83 ◦C, �T = 1.56 ◦C, Cm = 280.6 ppm,�C = 50.9 ppm, Sm = 1365 W m2, �S = 4.1 W m2, Em = 0 GtCand �E = 156.2 GtC are used. The software MATLAB SystemIdentification Toolbox is used for the computations reportedin the next sections. In all cases, ARX models with differentdegrees of complexity were analyzed and the one with thebest fitting chosen. Only stable models were considered and,in case of more than one model with about the same fitting,the simplest was preferred. In all cases, low residuals autocor-relation was checked.

5. Identification of the transfer functions

The usual procedure to identify G1,2,3,4 in a classic controlcase would be to open the feedback loop and run a newset of experiments. In this case, of course, this is impossi-ble and a different approach must be used. The system can
be cut along T∗ and C∗ and rearranged as shown in Fig. 6.In this way, the system is divided in two subsystems, whichcan be evaluated separately since both C∗ and T∗ series areavailable.
concentration).

5.1. The transfer functions G2 and G4

The transfer function G2 has as input the temperature T∗ andas output the carbon dioxide concentration C′.

A2(q)C′(t) = T∗(t)B2(q) (5)

Fig. 7 – Comparison between carbon dioxide concentrationhistorical data and feedback model simulation.


Fig. 8 – Comparison between mean temperature historicald

w

G

Tfitt1icsitv

Ff

Fig. 10 – Anthropogenic carbon dioxide emission in thefour scenarios.

ata and feedback model simulation.

ith the model

2(q) = b0

1 + a1q−1 + a2q−2(6)

he parameters a1,2 and b0 found in order to optimize thetting of the data are reported in Table 1. The loss func-ion Lf is the determinant of the estimated covariance ma-rix and FPE is the Akaike final prediction error (see Ljung,997). According to the feedback hypothesis the model takesnto account only the ‘natural’ component C′ of the CO2

oncentration. The anthropogenic carbon dioxide emission
tarted only after 1750; consequently the 1610–1750 histor-cal data (where C∗ = C′) were used for identification of theransfer function, while the range 1751–2004 was used foralidation.
ig. 9 – Comparison between temperature T′ (due to solarorcing) and temperature T′′ (due to CO2 forcing).

Fig. 11 – Carbon dioxide concentration from Scenario 1.

The transfer function G4 connects the CO2 emission E∗ withC′′.

C′′(t) = E∗(t)G4(q) (7)

The best fitting is found with the model

G4(q) = b0

1 + a1q−1(8)

Data in the instrumental range are used for identification anddata in the reconstructed range (after 1750) for validation.Model parameters are shown in Table 1.

In theory, the atmospheric concentration can be functionof the cumulative emission, which is the integral over time of
E∗ but, in Eq. (7), only E∗ is used as input. In the present case,however, this makes no difference because an ARX model witha pole located at (1, 0) in the complex plane gives the exactintegral of the input multiplied by b0. Consequently, the model


Fig. 12 – Average temperature deviation (−1870) fromScenario 1.


Fig. 15 – Comparison among different models for the


can automatically take into account possible integrals in theoutput.3

5.2. The transfer functions G1 and G3

Solar forcing and total atmospheric CO2 concentration are, re-spectively, the input of G1 and G3.

A1(q)T∗(t) = S∗(t)B1(q) (9)

and

A3(q)T∗(t) = C∗(t)B3(q) (10)

where G1 = B1(q)/A1(q) and G3 = B3(q)/A3(q).

3 In order to take into account carbon dioxide sinks, C′′(t) mustbe a fraction of the integral.

“business as usual scenario”.

The best fitting is found with the same model structure inboth cases4

G1,3(q) = b0

1 + a1q−1(11)

Basically, it would be better to use the instrumental data foridentification and the reconstructed data for validation. In theinstrumental range, however, the CO2 forcing is predominant(see Hansen et al., 2000; Crowley, 2000). For this reason, lowerstatistical errors are found using the reconstructed data toidentify G1 and the instrumental data to identify G3. In Table
1, the model parameters are reported.
4 The solution of a linear difference equation is not necessarylinear. The response of G1,3 to a step input, for instance, is loga-rithmic.


6

IIaiwTai1dowoa

FS

. The feedback model

n the previous sections, single transfer functions were found.n this section, the transfer functions are connected togetherccording to the feedback scheme shown in Fig. 5. The onlynput variables are E∗ and S∗, while C∗ and T∗ are computed

ith the overall model and compared with the historical data.he software Simulink was used for these simulations. Resultsre shown in Figs. 7 and 8. Both temperature and carbon diox-de concentration are well simulated except for the interval870–1950 where carbon dioxide concentration is slightly un-erestimated. Interestingly, the same discrepancy appears inther models (see Lenton, 2000; Ichii et al., 2003). It is worth-
hile noting that the transfer functions G1,2,3,4 are a seriesf black-box models computed from historical data with nossumption about the feedback structure. There would be no
ig. 17 – Average temperature deviation (−1870) fromcenario 3.


reasons for them to work together if the feedback hypothesiswas not correct. The fact that simulated and historical data arein good agreement, consequently, supports the ideas behindthe model structure proposed in Fig. 5. In Fig. 9, the compar-ison between T′ and T′′ is shown. According to the figure, therecent temperature increment is due mostly to T′′. This con-clusion is in agreement with calculations that show the rel-ative forcing of carbon dioxide and solar radiation with time(e.g. Hansen et al., 2000; Crowley, 2000).

7. Scenarios

The model is used to simulate four future scenarios. The
first three scenarios are similar to the IS92e, IS92a and IS92cscenarios, simply rescaled in order to take into account thereal data from 1992 to 2004. The rescale factor is almostnegligible in Scenario 1 and Scenario 2 (1.04 and 1.07 with


iatio
Fig. 20 – CO2 concentration and temperature dev
respect to IS92e and IS92a) but it is more consistent (1.31with respect to IS92c) in the case of Scenario 3. The fourthscenario is chosen in order to show some characteristics ofthe model proposed in this paper. The future values of solarforcing are the same for all the four scenarios and are com-puted according to a two-cycle solar model based on a 11-years and a 88-years cycle (Loehle, 2004). The anthropogeniccarbon dioxide emission (E), on the other hand, changes inthe four scenarios (see Fig. 10). Scenario 1 is similar to theIS92e scenario and has the higher E increment. Scenario 2 isderived from the IS92a “business as usual scenario”, which isoften used to compare the results of different models. Sce-nario 3 is similar to the IS92c scenarios; the emission af-
ter an initial increment gradually decreases. Scenario 4 takesinto account a sudden reduction of emission to zero; it isnot a likely scenario but it is an interesting test for themodel.
n from Scenario 4; three centuries calculations.

In the following figures, both the average behavior and, forthe temperature, one of the possible profiles, computed con-sidering the stochastic noise, are reported.

During the identification phase, the model was trained toreproduce the main characteristics of the historical data. Inthis way, however, only phenomena with time-scale largerthan the time-step (1 year) but smaller than the identifica-tion range (150 years) can be simulated by the model. A sim-ilar consideration can be done for T and C. The model was,trained for a range of temperature and carbon dioxide con-centration, respectively, of �T ∼= 2

◦C and �C ∼= 100 ppm. The

results are statistically more reliable when T and C, calculatedfrom year 2005, do not increase excessively beyond T2005 + �T

and C2005 + �C. For the first three scenarios, the more reliableresults are calculated, respectively, for year < 2060 (Scenario1), year < 2080 (Scenario 2), and year < 2100 (Scenario 3). InScenario 4, this is not an issue since T and C do not increase.


ction

Tpil

7

Bprmsp

a(p

Fig. 21 – Poles of the transfer fun

hese preliminary considerations notwithstanding, in this pa-er simulations are carried out for 100 years in all the scenar-

os in order to compare the results with others available in theiterature.5

.1. Scenario 1 (Figs. 11 and 12)

oth temperature and CO2 concentration increase almostarabolically to higher values. The model gives results compa-
able with other models. In Enting et al. (1994) results from 18
odels are compared: the CO2 projection to 2100 for the IS92ecenario is between 856.9 and 913.1 ppm. With the model pro-osed in these paper, it is 895.5 ppm.

5 The fact that new, unknown phenomena can arise at high Tnd C and that these phenomena cannot be included in the modelprecisely because unknown), it is a limit not only of the modelroposed here, but, in general, of all the global warming models.

s G1,2,3,4 in the complex plane.

7.2. Scenario 2 (Figs. 13 and 14)

This scenario is known as “business as usual scenario” andis often used as benchmark in order to compare results fromdifferent models. In Fig. 15, the temperature increment andthe carbon dioxide concentration at year 2100 for differentmodels are compared. All the models in Fig. 15 take explicitlyinto account the feedback effect between T and C. The modelsmarked with ‘∗’ did not use the IS92a scenario but the slightlydifferent SRES-A2 scenario. Comparison can be done also withEnting et al. (1994), which gives carbon dioxide concentrationin 2100 in the range 628.8–734.0 ppm; the value found with mymodel is 767.7 ppm.


In Scenario 3 the emission increases till 2050 and diminishesafterwards. The carbon dioxide concentration in the atmo-sphere and the average temperature, however, do not decreasein the considered time. This behavior is in agreement with


iffer
Fig. 22 – Carbon dioxide concentration at 2100 for d
other models, although the final CO2 value I obtained is higher.The range of variation in the models taken into account inEnting et al. (1994) is 420.6–474.2 ppm, while the present modelgives a final concentration of 623.8 ppm. If the fact that Sce-nario 3 is rescaled with a factor 1.31 with respect to IS92c istaken into account, however, the final concentration is only476.1 ppm.


The emission in this scenario is drastically reduced but bothtemperature and concentration need a certain time (∼5 years)to show any reduction. After that time, both these variables be-gin to decrease and after a transition period a constant plateauis reached. The plateau, however, is at a higher value thanthe preindustrial era (366.2 ppm). This behavior is consistentwith result from Lenton (2000) and Ichii et al. (2003), althoughnumerical comparison with these works is not possible sincethe scenario they used is different (a century of “business asusual scenario”, plus a century of linear decay to zero, pluseight centuries of zero emission) and takes into account toolong a period for the model presented here. Although phe-nomena with characteristic times longer than 150 years arenot captured by the model, calculations were extended to 300years for this particular scenario in order to facilitate compar-ison with Lenton (2000) and Ichii et al. (2003). In these studies,the concentration as a consequence of the emission reductionreaches asymptotically a new value. Calculations carried out
with the model proposed in this paper initially have a simi-lar behavior (Fig. 18) but the longer simulation (Fig. 20) showsthat the plateau is only the first of a series of steps throughwhich the concentration decreases. This is the consequence
ent values of the model’s parameters (Scenario 2).

of a complex pole in G2 as it is briefly discussed in the nextsection.

The first consideration the four scenarios suggest is thatthe CO2 concentration has reached such a high level as tobecome the main driving force of global warming. In all thescenarios, the mean temperatures have an oscillating compo-nent due to solar forcing but the overall dynamics seem to bemore affected by the E-dynamics than the S-dynamics. Thesecond important consideration regards the fact that the sys-tem presents a kind of ‘inertia’ when the value of the emis-sion is reduced. Scenario 3 is particularly clear on this point.Although the CO2 emission is reduced the atmospheric con-centration keeps increasing. A final consideration regards theconfidence intervals shown in Figs. 11, 13, 16 and 18 (carbondioxide concentration results). The confidence intervals arewider when the concentration decreases while they are morenarrow when the concentration increases. This is logical if wethink that the model is trained from the historical data. Inthese data, the CO2 concentration always increases. For thisreason, the model is more precise for CO2 increments thandecrements. A consequence of this is that the confidence in-tervals are wider in Fig. 18 than Fig. 11.

8. Sensitivity and stability analysis

The parameters a and b shown in Table 1 characterize themodel and have been determined according to the histori-
cal data. As mentioned before, however, the model is linearand possible non-linearities are linearized in the consideredranges of temperature and CO2 concentration. If the systemis not linear and T∗ or C∗ increases far beyond the previous


ent values of the model’s parameter aG41 (Scenarios 2 and 4).

itwtTtttscalutaip

c(teoi(s

their influence on the output is linear and, consequently, thesensitivity of the system to their variation is completely de-scribed by the derivatives dC∗/db and dT∗/db (Table 2). Fig. 22shows that a

G11 and a

G31 are not particularly critical parame-

ters. They are located in a relatively flat part of the curve andthey are far from the instability zone. According to Table 2,the higher sensitivity comes from a

G41 , which is also located

near the unit circle (see Fig. 21). In this case, however, there isnot any risk of instability since the case a

G41 = 1 is physically

impossible. It would imply, in fact, spontaneous generation ofmass in the atmosphere. In Fig. 23, the concentration profilesfor three different values of a

G41 using Scenario 2 and Scenario

4 are shown.

Table 2 – Sensitivity of the results (Scenario 2 at 2100) todifferent values of the model’s parameters

G1 G3 G4 G2

a1

C∗dC∗

da10.0082 0.080 41.92

�

C∗dC∗

d�1.00

a1 dT∗0.025 1.06 40.73

� dT∗0.22

Fig. 23 – Carbon dioxide concentration profiles for differ

ntervals, the optimal values of the parameters can vary fromhe one reported in Table 1. Since it is not possible to foreseehich parameter will change and to what extent, the sensi-

ivity of the results to different a and b must be investigated.he simplest way to analyze the sensitivity would be to vary

he parameters and compare the results. This is not, however,he common procedure in control theory since the analysis ofhe poles (the zeros of the denominator of G1,2,3,4) gives moreignificative insights into the system’s behavior. The poles’ lo-ation in the complex plane, in particular, gives informationbout the stability of the system. For discrete-time systemsike the one investigated, in fact, the poles must be within thenit circle for the system to be stable. Outside the unit circlehe system is called unstable, which means that, as effect ofbounded input, the output grows indefinitely towards infin-

ty, minus infinity or oscillates between the two. In Fig. 21, theoles of the four transfer functions are shown.

The transfer functions G1,3,4 have only a single pole; in thisase the pole coincides with the parameter −a1. Scenario 2IS92a) is chosen as reference; concentrations and tempera-ures in 2100 are calculated for different values of the param-ter a1 (in case of G1,3,4) and � and � (in case of G2). The meaning
f � and � is explained later. Results are shown in Fig. 22 and
n Table 2 the local derivatives dC∗/da and dT∗/da are reportedthe factors a/C∗ and a/T∗ are used to compare the values at theame scale). The parameters b0 are multiplicative constants;

T∗ da1 T∗ d�b0

C∗dC∗

db00.005 0.081 0.795

�

C∗dC∗

d�−0.017

b0

T∗dT∗

db00.023 1.079 0.778

�

T∗dT∗

d�−0.016

254 e c o l o g i c a l m o d e l l i n g

Fig. 24 – Carbon dioxide concentration profiles for different

dence of the interval tangent to the unit circle (Fig. 25) is 0.86.This means that there is a 14% chance that the pole is alreadyin the unstable region and that the temperatures and the con-centrations that we experience today are just the initial tran-

values of the model’s parameters � and � (Scenarios 2 and4).

The transfer function G2 is more complicated because it isa second order transfer function. It has two complex conju-gated poles p1 and p2 (see Fig. 21), which can be representedin exponential complex notation as

p1,2 = � e±i� (12)

where

� = √a2 ∼= 0.996 (13)

is the modulus and

� = tan−1

(−a1√

4a2 − a21

)∼= 0.02 rad (14)

is the argument of the complex pole.
The imaginary part is responsible for an oscillating re-
sponse to the input. In these circumstances, however, � is verysmall and it does not affect the final result very much (see thefirst Fig. 24). More critical is �, which is very close to the unit

2 0 3 ( 2 0 0 7 ) 243–256

circle. This time, there are no physical reasons that prevent �

from reaching values higher than one. The CO2 forcing wouldincrease the temperature which, in turn, would increase theCO2 concentration further. If � > 1, this cycle is unstable and itshows a typical run-away behavior. In the second and third Fig.24, the concentration profile for different values of � are shown(Scenarios 2 and 4). Basically, the results are not particularlysensitive to � (according to Table 2 the results are much moresensible to a

G41 ) if the system is stable. But as soon as � > 1,

even in the case of Scenario 4 when the emission is drasticallyreduced to zero, the concentration grows indefinitely.

From the mathematical point of view T and C can increasead libitum but, of course, this is not physically possible. Themost probable outcome is that, after a dramatic transient, thesystem shifts to a new stable state at higher temperature andcarbon dioxide concentration.6 This situation is known to Pa-leoclimate scientists as ‘Abrupt Climate Change’. It happened,for instance, about 11,500 years ago at the end of the periodcalled Younger Dryas (e.g. Cuffey and Clow, 1997; Alley, 2000).Probably triggered by a surplus of fresh water into the NorthAtlantic (Clark et al., 2001) it was particularly abrupt (in Green-land, for instance, temperatures rose 10 ◦C in a decade).

A more technical approach concerning feedback loops sta-bility is based on the so called ‘root locus diagram’, which an-alyzes the sensitivity of the results respect the ‘feedback gain’b

G20 b

G30 . This approach is reported in Appendix A.

9. Conclusions

In this paper, a feedback model is used to describe the influ-ence of the carbon dioxide emission due to human activityon the global temperature and the atmospheric CO2 concen-tration. The results presented show that the anthropogeniccarbon dioxide has become the main driving force in globalwarming and even in the case of reduction of the emissions,the temperature will keep increasing for a certain time. Themain characteristic of the present model, however, regards theoption to test the stability of the system. Other global warm-ing models available in the literature are stable by definition,which means that they respond to a bounded input with abounded output. Consequently, the temperatures calculatedcan change according to the model and the scenario, but thepossibility of an unstable, run-away behavior is excluded apriori. Stability analysis, on the other hand, can be easily per-formed on the model proposed in this paper. This type of anal-ysis shows that the most important factor concerning the sta-bility of the CO2-temperature feedback system concerns thepoles of the transfer function G2 (see Fig. 21). They are, in fact,very close to the unit circle, which means very close to theinstability region. Actually, the 99% confidence intervals forthese poles are already outside the stable region. The confi-

6 Since the model is linear, it can describe the run-away from thecurrent state but not the settling around a new equilibrium.


sBcgriibwtsstbctw

A

Tat

A

IG

mcfi

T

(st

positive is the feedback between climate change and the

Fig. 25 – Different confidence intervals for the poles of G2.

ient of the typical run-away behavior of an unstable system.esides this, the effect of the variables that have not been in-luded as inputs (i.e. aerosols, ice albedo, other greenhouseases, etc.) must be considered. Their effect in the past is al-eady in the parameters since the model is trained from histor-cal data. When the model is used to calculate future scenar-os, however, it automatically assumes that these phenomenaehave with the same dynamic they showed in the past. Buthat happens if some of them change drastically the behavior

hey had in the historical series, if the aerosols concentrationuddenly decreases or the glaciers melting accelerates, for in-tance? In this case, the poles will be pushed further towardshe unstable region and the probability to cross the boundaryetween the two regions will increase. Once this boundary isrossed, the feedback effect, which usually regulates the CO2-emperature mechanism, will enter in an auto-excited modeith unpredictable consequences on climate change.

cknowledgment

he authors gratefully acknowledge the very useful commentsnd suggestions to the paper offered by Dr. Osvaldo Rojas ofhe University of New South Wales.

ppendix A. Root locus diagram

n the paper, the stability of the single transfer functions

1,2,3,4 and the sensitivity of the results to a change of theodel parameters were studied. A feedback loop, however,

an have an unstable behavior even if all the single transferunctions are stable. The global transfer function of the systemn Fig. 5, in fact, is

∗Gf = S∗G1

1 − G2G3+ E∗G3G4

1 − G2G3(15)

The equation 1 − G2G3 = 0 is called characteristic equationmore frequently, it is written as 1 + G2G3 = 0 since a differentign convention is used). The roots of the characteristic equa-ion, which give the poles of Gf , vary with the ‘feedback gain’

Fig. 26. – Root locus diagram of 1 − G3G2.

that, in this particular case, is given by the product bG20 b

G30 . The

root locus diagram (Fig. 26) shows how the root of

G� = 1 − �G2G3 = 0 (16)

vary with � ∈ [0 − ∞]. When � = 0, the roots of G� coincide withthe poles of G3 (pG3 ) and G2 (pG2

1,2). If � increases, two poles (pG3

and pG21 ) moves towards the origin and the third (pG2

2 ) moves

towards the unit circle. When � = 3.46, pG22 crosses the unit cir-

cle. This means that the gain bG20 b

G30 must increase 3.46 times

in order to make the system unstable and that the location ofthe poles p

G21,2 is more critical, from the stability prospective,

than the feedback gain.

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