## 7.2 Climate module

WITCH included an internal climate module, which translates the regional emissions into global temperature through atmospheric concentrations. It has been building upon the DICE climate equations (Nordhaus and Sztorc 2013). Alternatively, and to make the climate outcome comparable with other models, it allows a soft link with the MAGICC6 climate model (Meinshausen, Raper, and Wigley 2011) for reporting a number of climate outcomes based on this widely used model. IN the following, we describe the internal WITCH climate module.

The greenhouse gases ghg that are accounted for in the model are: carbon dioxide CO2, methane CH4, oxide nitrous N2O, short-lived greenhouse gases slf, long-lived greenhouse gases llf. The regional emissions $$Q_E(\text{ghg},t,n)$$ are summed up into global emission $$WE(\text{ghg},t)$$ and converted into Gton equivalent carbon.

$WE(\text{ghg},t) = \sum_n \frac{Q_E(\text{ghg},t,n)}{\text{wemi2qemi}(\text{ghg})}$

Non-CO2 greenhouse gases are converted in CO2 equivalent using the global warming potentials over a time horizon of 100 years (AR4 IPCC report, 2007) and in carbon equivalent by multiplying by 12/44.

$\text{wemi2qemi}(\text{oghg}) = \text{gwp}(\text{oghg}) \times 12 / 44$

GHG emi_gwp
CO2 1
CH4 25
N2O 298
slf 1430
llf 22800

### 7.2.1 Carbon-cycle

The carbon-cycle model is a 3-layer model calibrated to MAGICC (Meinshausen, Raper, and Wigley 2011). The carbon dioxide emissions go into the atmosphere box a and alter the atmospheric carbon concentration, then the carbon is exchanged through the ocean-biosphere-atmosphere carbon fluxes. The ocean carbon sink is divided in two layers: the upper layer u (shallow oceans) and the lower layer l (deep oceans). In this representation, the upper layer also includes the biosphere, and $$M_{\text{box}}(t)$$ denotes the CO2 concentrations expressed in GtC.

The carbon cycle is described in the model as

\begin{align*} M_a(t+1) & = \mathbf{A}_{a,a} \times M_a(t) + \mathbf{A}_{u,a} \times M_u(t) + \Delta_{\text{t}} \times WE(\text{CO}_2,t)\\ M_u(t+1) & = \mathbf{A}_{a,u} \times M_a(t) + \mathbf{A}_{u,u} \times M_u(t) + \mathbf{A}_{l,u} \times M_l(t)\\ M_l(t+1) & = \mathbf{A}_{u,l} \times M_u(t) + \mathbf{A}_{u,l} \times M_l(t)\\ \end{align*}

$$\Delta_{\text{t}}=5$$ and is the number of years in one time step, and the transition matrix is

$\mathbf{A}= \begin{pmatrix} {0.88} & {0.04704} & \\ {0.12} & {0.94796} & {0.00075} \\ & {0.005 } & {0.99925} \\ \end{pmatrix}$

Initial CO2 stocks in 2005 are reported below.

Stock in 2005 emi_gwp
$$M_a(2005)$$ 808.9
$$M_l(2005)$$ 10000
$$M_u(2005)$$ 1000

### 7.2.2 Accumulation of non-CO2 ghg in the atmosphere

the non-CO2 greenhouse gases are accumulated in the atmosphere as follows:

\begin{align*} M(\text{oghg},\text{atm},t+1) =& d_1(\text{oghg})^{\Delta_t}\times M(\text{oghg},\text{atm},t) +\\ &d_2(\text{oghg},t)\times \frac{1}{2}\left(E(\text{oghg},t)+E(\text{oghg},t+1)\right)+\\ &(1-d_1(\text{oghg})^{\Delta_t})\times \overline{\text{stock}}(\text{oghg}) \end{align*}

where $$d_1(\text{oghg})$$ is the yearly retention factor, and $$d_2(\text{oghg},t)$$ is the one-period retention factor, which is however kept constant.

stock(oghg) is the stock of each greenhouse gas at equilibrium, which is not subject to decay, and expressed in GtC. It is calculated as a fraction of $$stock_0$$, the stock of non-CO2 greenhouse gases in 2005.

ghg cmdec1 cmdec2 emi_eqstock emi_stock0
CH$$_4$$ 0.917 4.2361 1.9352 4.838
N$$_2$$O 0.9917 4.9177 1.3411 1.524
slf 0.931 4.3560 0.000347
llf 0.9997 4.9969 0.000848

The radiative forcing of all greenhouse gases $$F(t)$$ are summed up in the following equation:

$F(t) = \sum_{\text{ghg}} RF(\text{ghg},t) + RF_{\text{aerosols}}(t)$

The exogenous radiative forcing from aerosols are based on the average response obtained from many runs and the MAGICC outputs.

The CO2 radiative forcing is calculated according to the IPCC Third Assessement Report expressions, i.e. proportional to the natural logarithm of the ratio of the current concentration to pre-industrial levels. $$M_{\text{pre}}$$ are the pre-industrial CO$$_2$$ concentration level in GtC.

\begin{align*} RF(\text{CO}_2,t) = \alpha \times \left(\ln(M(\text{CO}_2,\text{atm},t))-\ln(M_{\text{pre}})\right) \end{align*}
parameter value
$$\alpha$$ 5.35
$$M_{\text{pre}}$$ 592.14

The radiative forcing of the greenhouse gases CH4 and N2O is given below. In reality, these radiative forcing are interdependent and have a complex formulation. An approximation is obtained using the term inter in the equations.

\begin{align*} RF(\text{oghg},t) =& \text{inter} \times \text{fac} \times \left( \sqrt{\text{stm}\times M(\text{oghg},t)} - \sqrt{\text{stm}\times M_{\text{pre}}(\text{oghg})} \right),\\ &\text{oghg}\in \{\text{CH}_4,\text{N}_2\text{O}\} \end{align*}
Parameter CH$$_4$$ N$$_2$$O
inter 0.85 0.92
fac 0.036 0.12
stm 351 206.7
$$M_{\text{pre}}$$ 1.95 1.29

The radiative forcing of slf and llf are proportional to the concentration levels.

$RF(\text{oghg},t) = \text{fac}(\text{oghg}) \times M(\text{oghg},t), \text{oghg}\in \{\text{slf},\text{llf}\}$

parameter slf llf
fac 2.571 13.026

### 7.2.4 Global temperature increase from pre-industrial levels

The global temperature increase from pre-industrial level $$T(t)$$ is obtained from the energy balance 2-layer model.

$T(t+1)= T(t) + \sigma_1 \times \left(F(t) - \lambda \times T(t)- \sigma_2 \times \left(T(t)-T^{o}(t)\right)\right).$

$$\sigma_1$$ is a lag parameter. $$\sigma_2$$ is a atmosphere-ocean exchange coefficient. The climate feedback parameter $$\lambda=\frac{4.1}{s}$$, where $$s$$ is the climate sensitivity.

The ocean temperature $$T^{o}(t)$$ is given as $T^{o}(t+1) = T^{o}(t) + \sigma_{ho} \left(T(t) - T^{o}(t)\right),$ where $$\sigma_{ho}$$ is a coefficient of the heat capacity of the ocean.

Parameter Value
$$\sigma_1$$ 0.208
$$\sigma_2$$ 0.31
$$\sigma_{ho}$$ 0.05
$$\lambda$$ 1.36667
$$s$$ 3.0

### References

Nordhaus, William, and Paul Sztorc. 2013. User’s Manual for Dice-2013r.

Meinshausen, M., S. C. B. Raper, and T. M. L. Wigley. 2011. “Emulating Coupled Atmosphere-Ocean and Carbon Cycle Models with a Simpler Model, Magicc6: Part I – Model Description and Calibration.” Atmospheric Chemistry and Physics 11: 1417–56.