The benefits of climate change mitigation in integrated assessment models: The role of the carbon cycle and climate component
Supplementary Material
Andries F. Hof, Chris W. Hope, Jason Lowe, Michael D. Mastrandrea, Malte Meinshausen, Detlef P. van Vuuren
This Supplementary Material contains the important equations of the stand-alone versions of the carbon cycle and climate models used in the article, as well as the damage functions used. The equations are written in the MyM language (http://www.my-m.eu/).
Important functions of the carbon cycle
DICE-2007 and DICE-2009
Both DICE model versions start in 2005. The model runs in steps of 10 years. The functions of DICE-2007 were derived from Nordhaus (2007) and of the recalibrated DICE-2009 model version from the Excel file available at http://nordhaus.econ.yale.edu/ DICE2007.htm.
The equations used are:
The variable MMAT stands for atmospheric concentration; MML for lower ocean concentration and MMU for shallow ocean concentration, all expressed in GtC. The variable MMATAVEQ denotes the average concentration level of the 10-year period; EE denotes annual CO2 emissions expressed in GtC per year.
The parameters b11, b12, b21, b22, b23, b32 and b33 are used for the carbon cycle transition matrix. The values of these parameters are as follows:
DICE-2007 / DICE-2009b11 / 0.810712 / 0.88
b12 / 0.189288 / 0.12
b21 / 0.097213 / 0.047
b22 / 0.852787 / 0.948
b23 / 0.05 / 0.005
b32 / 0.003119 / 0.001
b33 / 0.996881 / 0.999
Base year level MMAT / 808.9 / 801.57
Base year level MMU / 1255 / 1600
Base year level MML / 18365 / 10010
MERGE 5.1
The original MERGE 5.1 model runs in steps of 10 years and starts in 2000. The carbon cycle model is based on the “box-model” of Maier-Reimer and Hasselmen (1987), which divides the total stock of anthropogenic CO2 in the atmosphere in boxes, each with different decay time. Therefore, it needs to assume an initial stock of CO2 for each box in 2000. If MERGE is run with our historical emissions data from 1860 onwards with an initial stock of zero CO2 in all boxes, the levels of CO2 in the boxes in 2000 differ from the initial levels as stated in the MERGE model code. To rule out differences between model results due to different assumed historical emission datasets, we have ran the stand-alone version of MERGE 5.1 from 1860 onwards, assuming zero CO2 levels within each box. The equations were obtained from the original GAMS model code (http://www.stanford.edu/group/MERGE/):
The parameters decay1 and decay2 denote the decay of CO2 concentrations for each box i, where decay2 denotes the decay of concentrations resulting from emissions during the last time step. CO2 denotes the stock of anthropogenic CO2 in each individual box i (expressed in GtC); co2stock defines the total stock of CO2 in the atmosphere. The parameter frac denotes the fraction of CO2 emissions decaying within a certain box and EE denotes annual CO2 emissions in GtC. Finally, NCO2 denotes the pre-industrial level of CO2 assumed never to decay and amounts to 594 billion tonne. The values of the parameters decay1, decay2 and frac are as follows:
decay1 / fracbox 1 / 1.0 / 0.142
box 2 / exp(-1/313.8) / 0.241
box 3 / exp(-1/79.8) / 0.323
box 4 / exp(-1/18.8) / 0.206
box 5 / exp(-1/1.7) / 0.088
FUND 2.8 and FUND 3.3
The carbon cycle models of FUND 2.8 and 3.3 are the same, with the exception that the decay parameters of FUND 3.3 are not deterministic, but based on normal probability distributions. The mean values of these distributions (used in our study) are identical to the deterministic values of FUND 2.8. Both model versions start to run in 1950 and run in time steps of 1 year. The model code of both versions can be downloaded from http://www.mi.uni-hamburg.de/FUND.5679.0.html. In some cases, the model code differed from the technical description of the model. In these cases, the model code was used. The basic equations are the same as in MERGE 5.1; the only differences are due to the fact that FUND runs in time steps of 1 year instead of 10 years:
In contrast to MERGE 5.1, both FUND model versions express concentrations in ppm. Therefore, the term 0.471 in the equation CO2 is needed to convert GtC emissions to ppm concentrations. The base year values of CO2 and the parameters decay and frac differ from MERGE, due to a different starting year and because the impulse-response function of FUND is based on a pulse of 2x CO2, while MERGE 5.1 is based on 1.25x CO2.
Base year level CO2 (ppm) / decay / fracbox 1 / 283.53 / 1.0 / 0.13
box 2 / 5.62 / exp(-1/363) / 0.20
box 3 / 6.29 / exp(-1/74) / 0.32
box 4 / 2.19 / exp(-1/17) / 0.25
box 5 / 0.15 / exp(-1/2) / 0.10
Important functions of the climate models
DICE-2007 and DICE-2009
The equations of the climate model in both DICE model versions are:
The variable forcing denotes the radiative forcing resulting from the increased CO2 concentrations; LAM is a climate model parameter, temp denotes average global temperature increase since 1900 and TOCEANEQ denotes the average temperature of deep oceans (in terms of degrees Celsius above 1900 levels). FCO22X denotes radiative forcing resulting from a doubling of CO2 concentrations and T2XCO2 the equilibrium temperature impact of such a doubling; C1, C3 and C4 are temperature transfer coefficients. The values of the parameters are:
DICE-2007 / DICE-2009C1 / 0.22 / 0.19
C3 / 0.30 / 0.31
C4 / 0.05 / 0.05
FCO22X / 3.8 / 3.8
T2XCO2 / 3.0 / 3.0
Base year level temp / 0.7307 / 0.7307
Base year level TOCEANEQ / 0.0068 / 0.0068
MERGE 5.1
The equations of the climate model in MERGE 5.1 are:
The variable forcing again denotes the radiative forcing resulting from the increased CO2 concentrations; ptdf denotes the potential temperature (or equilibrium temperature); lag1 denotes the 1-year transient temperature response time (equal to 0.038); lag2 the 10-year transient response time; and temp denotes average global temperature. In the original MERGE 5.1 model, temperature increase is measured relative to 2000, but our stand-alone version measures temperature increase relative to pre-industrial times (see also the section about the carbon cycle of MERGE 5.1). The parameter rfconv (equal to 0.618) is used to convert radiative forcing into equilibrium temperature change.
FUND 2.8 and FUND 3.3
The structure of the climate models of FUND 2.8 and FUND 3.3 are the same, but some of the parameters have been updated in FUND 3.3. The equations are:
(FUND 2.8)
(FUND 3.3)
(FUND 2.8)
(FUND 3.3)
The variable forcing again denotes radiative forcing of CO2, co2pre denotes the pre-industrial concentration of CO2, CS denotes the equilibrium climate sensitivity, delaytemp denotes the temperature response time and temp denotes the average global temperature increase relative to the pre-industrial level. In FUND 3.3, the parameters are based on probability distributions, but in our study, we only use the mean and mode values:
FUND 2.8 / FUND 3.3 (mean) / FUND 3.3 (mode)co2pre / 275 / 275 / 275
CS / 2.5 / 2.85 / 2.5
lifetemp / 50 / 75 / 75
Damage functions
Two damage functions are used in the study; the default damage function is identical to the damage function applied in the DICE-2007 and DICE-2009 models (Nordhaus 2007):
Damage denotes the damages as share of world GDP. The alternative damage function is based on the equity-weighted damage function of Tol, as depicted in Figure 19-4 in Smith et al. (2001). This function is approximated by:
References
Maier-Reimer E, Hasselmann K (1987) Transport and storage of carbon dioxide in the ocean-an inorganic ocean-circulation carbon cycle model. Clim Dyn 2:63-90
Nordhaus WD (2007) The challenge of global warming: Economic models and environmental policy. Yale University, New Haven
Smith J, Schnellnhubner H-J, Mirza MQM (2001) Vulnerability to climate change and reasons for concern: A synthesis. In: McCarthy JJ, Canziani OF, Leary NA, Dokken DJ, White KS (eds) Climate change 2001: Impacts, adaptation, and vulnerability. Cambridge University Press, Cambridge, UK
1