# Modelling of climate indicators - IMACLIM

Model Documentation - IMACLIM | |
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Corresponding documentation | |

Model information | |

Institution | Centre international de recherche sur l'environnement et le développement (CIRED), France, http://www.centre-cired.fr., Societe de Mathematiques Appliquees et de Sciences Humaines (SMASH), France, http://www.smash.fr. |

Solution concept | General equilibrium (closed economy) |

Solution method | SimulationImaclim-R is implemented in Scilab, and uses the fonction fsolve from a shared C++ library to solve the static equilibrium system of non-linear equations. |

Anticipation | Recursive dynamics: each year the equilibrium is solved (system of non-linear equations), in between two years parameters to the equilibrium evolve according to specified functions. |

## Modelling of Climate indicators

The impact of emissions scenarios on climate indicators is computed using a simplified 3-box carbon cycle model and a simplified 2-box climate model (Ambrosi, 2003).

### Radiative Forcing from Other Gases

The radiative forcing from other gases are taken as exogenous assumptions.

### Carbon Cycle Model and Climate Model

The carbon cycle is a three-box model, after Nordhaus and Boyer (2010)1. The model is a linear three-reservoir model (atmosphere, biosphere + ocean mixed layer, and deep ocean). Each reservoir is assumed to be homogenous (well-mixed in the short run) and is characterized by a residence time inside the box and corresponding mixing rates with the two other reservoirs (for longer timescales). Carbon flows between reservoirs depend on constant transfer coefficients. GHGs emissions (*CO _{2}* solely) accumulate in the atmosphere and are slowly removed by biospheric and oceanic sinks.

The stocks of carbon (in the form of *CO _{2}*) in the atmosphere, in the biomass and upper ocean, and in the deep ocean are, respectively,

*A*,

*B*, and

*O*. The variable

*E*is the

*CO*emissions. The evolution of

_{2}*A*,

*B*, and

*O*is given by

The initial values of *A*, *B*, and *O*, and the parameters *a12*, *a21*, *a23*, and *a32* determine the fluxes between reservoirs. The main criticism which may be addressed to this Carbon-cycle model is that the transfer coefficients are constant. In particular, they do not depend on the carbon content of the reservoir (e.g. deforestation hindering biospheric sinks) nor are they influenced by ongoing climatic change (e.g. positive feedbacks between climate change and the carbon cycle).

Nordhaus' original calibration has been adapted to reproduce both; data until 2010 and; results from the IMAGE model for a given trajectory of *CO _{2}* emissions. This gives the following results (for a yearly time step):

*a12*= 0.02793,

*a21*=0.03427,

*a23*=0.007863,

*a32*=0.0003552, with the initial conditions:

*A*2010=830

*GtC*(i.e. 391ppm),

*B*2010=845

*GtC*and

*O*2010=19254

*GtC*. The additional forcing caused by

*CO*and

_{2}*non-CO*gases is given by

_{2}where *A _{PI}* is the pre-industrial

*CO*concentration (280 ppm),

_{2}*F*is the additional radiative forcing for a doubling of the

_{2x}*CO*concentration (3.71 W.m^-2^), and

_{2}*F*is the additional radiative forcing of

_{non-CO2}*non-CO*gases.

_{2}The temperature model is a two-box model, after Schneider and Thompson (1981)2 and Ambrosi et al. (2003)3, with the atmosphere temperature *T _{A}* and the ocean temperature

*T*as follows:

_{O}where *T2x* is the equilibrium temperature increase at the doubling of the *CO _{2}* concentration, that is, it represents climate sensitivity. All parameters have been calibrated to reproduce results from CMIP5 from CNRM-CERFACS global climate model, CNRM-CM5, over the 21st century for RCP3-PD and RCP4.5 radiative forcing trajectories (using a least squares method). This calibration leads to the following parameter values for heat transfer rates (for a yearly time step):

*σ1*= 0.054

*C.W*,

^{-1}-1.m^{2}*σ2*= 0.664

*C.W*and

^{-1}-1.m^{2}*σ3*= 0.0308, and a climate sensitivity of 2.6°C.

## References

- ^ | William D Nordhaus, Joseph Boyer (2003).
*Warming the world: economic models of global warming*. MIT press. - ^ | Stephen H Schneider, Starley L Thompson (1981). Atmospheric CO2 and climate: importance of the transient response.
*Journal of Geophysical Research: Oceans, 86 (C4)*, 3135--3147. - ^ | Philippe Ambrosi, Jean-Charles Hourcade, Stéphane Hallegatte, Franck Lecocq, Patrice Dumas, Minh Ha Duong (2009). Optimal control models and elicitation of attitudes towards climate damages. In
*Uncertainty and environmental decision making*(pp. 177-209). Springer.