# Radiative Forcing

- model description
- carbon cycle
- non-CO2 concentrations
**radiative forcing routines**- climate model
- upwelling-diffusion-entrainment implementation

## Contents

## Radiative forcing

The following section highlights the key parameterizations used for estimating the radiative forcing due to human-induced changes in greenhouse gas concentrations, tropospheric ozone and aerosols. The radiative forcing applied in MAGICC is in general the forcing at tropopause level after stratospheric temperature adjustment. Efficacies of the forcings, as discussed by Hansen et al. (2005) and Meehl et al. (2007) can be applied.

### Carbon dioxide

Taking into account the "saturation" effect of CO_{2} forcing, i.e., the decreasing forcing efficiency for a unit increases of CO_{2} concentrations with higher background concentrations, the first IPCC Assessment (Shine et al., 1990) presented the simplified expression of the form:

**A35**)

where <math>\Delta Q_{\rm CO_2}</math> is the adjusted radiative forcing by CO_{2} (Wm^{-2}) for a CO_{2} concentration <math>C</math> (ppm) above the pre-industrial concentration <math>C</math>_{0} (278 ppm). This expression proved to be a good approximation, although the scaling parameter <math>\alpha_{\rm CO_2}</math> has since been updated to a best-estimate of 5.35 Wm^{-2} (=<math>\frac{3.71}{ln(2)} </math>Wm^{-2})Myhre et al. (1998), used as default in MAGICC. When applying AOGCM-specific CO_{2} forcing, <math>\alpha_{\rm CO_2}</math> is set to:

**A36**)

### Methane and nitrous oxide

Methane and nitrous oxide have overlapping absorption bands so that higher concentrations of one gas will reduce the effective absorption by the other and vice versa. This is reflected in the standard simplified expression for methane and nitrous oxide forcing, <math>\Delta Q_{\rm CH4}</math> and <math>\Delta Q_{\rm N2O}</math>, respectively (see Ramaswamy et al., 2001, Myhre et al. 1998):

<math>\label{eq_methane_forcing}\Delta Q_{\rm CH_4} = \alpha_{\rm CH_4}(\sqrt{\rm C_{\rm CH_4}}-\sqrt{\rm C_{\rm CH_4}^0}-f(\rm C_{\rm CH_4},{\rm C}_{\rm N_2O}^0)-f({\rm C}_{\rm CH_4}^0,{\rm C}_{\rm N_2O}^0)</math>**A37**)

**A38**)

where the overlap is captured by the function

<math>\label{eq_methane_forcing_overlap}f(\rm M,N)=0.47{\rm ln}\,(1+0.6356(\frac{\rm MN}{10^6})^{0.75}+0.007\frac{\rm M}{10^3}(\frac{\rm MN}{10^6})^{1.52})</math>**A39**)

with M and N being CH_{4} and N_{2}O concentrations in ppb. For methane, an additional forcing factor due to methane-induced enhancement of stratospheric water vapor content is included. This enhancement is assumed to be proportional to (default <math>\beta</math>=15 %) the ``pure´´ methane radiative forcing, i.e., without subtraction of N_{2}O absorption band overlaps:

**A40**)

### Tropospheric ozone

From the tropospheric ozone precursor emissions and following the updated parameterizations of OxComp as given in footnote a of Table 4.11 in Ehhalt et al. (2001), the change in hemispheric tropospheric ozone concentrations (in DU) is parameterized as:

<math>\label{eq_tropospheric_ozone} \Delta (\rm trop O_3)={\rm S}_{\rm CH_4}^{O_3}\Delta ln(\rm CH_4)+ {\rm S}_{\rm NOx}^{\rm O_3} E_{\rm NOx} + {\rm S}_{\rm CO}^{\rm O_3} E_{\rm CO} + {\rm S}_{\rm VOC}^{\rm O_3} E_{\rm VOC}</math>**A41**)

where <math>S_{\rm x}^{\rm O_3}</math> are the respective sensitivity coefficients of tropospheric ozone to methane concentrations and precursor emissions. The radiative forcing is then approximated by a linear abundance to forcing relationship so that <math>\Delta Q_{\rm trop O_3}=\alpha_{\rm trop O3} \Delta (\rm trop O_3)</math> with <math>\alpha_{\rm trop O3}</math> being the radiative efficiency factor (default 0.042).

### Halogenated gases

The global-mean radiative forcing <math>\Delta Q_{t,i}</math> of halogenated gases is simply derived from their atmospheric concentrations C (see Non-CO2 concentrations) and radiative efficiencies <math>\varrho_i</math> Ehhalt et al. (2001) table 4.11.

<math>\Delta {Q_{t,i}} = \varrho_i (C_{t,i}-C_{0,i})\label{eq_halogas_RF}</math>**A42**)

The land-ocean forcing contrast in each hemisphere for halogenated gases is assumed to follow the one Hansen et al. (2005) estimated for CFC-11. The hemispheric forcing contrast is dependent on the lifetime of the gas. For short-lived gases (<math>{<}</math>1\,yr) the hemispheric forcing contrast is assumed to equal the time-variable hemispheric emission ratio. For longer lived gases (default <math>{>}</math>8\,yrs), the hemispheric forcing contrast is assumed to equal the one from CFC-11 with linear scaling in between these two approaches for gases with a medium lifetime.

### Stratospheric ozone

Depletion of the stratospheric ozone layer causes a negative global-mean radiative forcing <math>\Delta Q_{t}</math>. The depletion and hence radiative forcing is assumed to be dependent on the equivalent effective stratospheric chlorine (EESC) concentrations as follows:

<math>\Delta Q_{t} = \eta_1 (\eta_2 \times \Delta {\rm EESC}_{t})^{\eta_3} \label{eq_stratospheric_ozone} </math>**A43**)

where <math>\eta_1</math> is a sensitivity scaling factor (default <math>-</math>4.49e-4 Wm^{-2}), <math>\Delta {\rm EESC}_{t}</math> the EESC concentrations above 1980 levels (in ppb), the factor <math>\eta_2</math> equals <math>\frac{1}{100}</math> (ppb<math>^{-1}</math>) and <math>\eta_3</math> is the sensitivity exponent (default 1.7).

EESC concentrations are derived from the modeled concentrations of 16 ozone depleting substances controlled under the Montreal Protocol, their respective chlorine and bromine atoms, fractional release factors and a bromine versus chlorine ozone depletion efficiency (default 45) (Daniel et al., 1999).

### Tropospheric aerosols

\label{section_TropAerosolParameterization}

The direct effect of aerosols is approximated by simple linear forcing-abundance relationships for sulfate, nitrate, black carbon and organic carbon. Time-variable hemispheric abundances of these short-lived aerosols are in turn approximated by their hemispheric emissions, justifiable because of their very short lifetimes. The ratio of direct forcing over land and ocean areas in each hemisphere is taken from Hansen et al. (2005) (available at http://data.giss.nasa.gov/efficacy/). Specifying the direct radiative forcing patterns for one particular year, and knowing the hemispheric emissions in that year, allows us to define the future forcing as a function of future emissions.

The indirect radiative forcing, formerly modeled as dependent on SO<math>_{\rm x}</math> abundances only Wigley, 1991a, is now estimated by taking into account time-series of sulfate, nitrate, black carbon and organic carbon optical thickness:

<math>\label{eq_forcing_indirectAer} \Delta Q_{\rm Alb,i} = r\times {\rm P}_{\rm Alb,i}\times \log(\frac{\sum_g w_g {\rm N}_{g,i}}{\sum_g w_g {\rm N}_{g,i}^0})</math>**A44**)

where <math>\Delta Q_{\rm Alb,i}</math> is the first indirect aerosol forcing in the four atmospheric boxes <math>i</math>, representing land and ocean areas in each hemisphere; P<math>_{\rm Alb}</math> is the four-element pattern of aerosol indirect effects related to albedo (HTwomey, 1977) in a reference year. The second indirect effect on cloud cover changes (Albrecht, 1989) is modeled equivalently -- using a reference year pattern P<math>_{\rm Cvr,i}</math>. The respective default patterns are derived from data displayed in Fig. 13 of Hansen et al. (2005). The scaling factor <math>r</math> allows one to specify a global-mean first or second indirect forcing for a specific reference year. The time-variable number concentrations of soluble aerosols N<math>_{g,i}</math> relative to their pre-industrial level in each hemisphere N<math>_{g,i}^0</math> are normed to unity in that reference year. This is done separately for sulfates, nitrates, black carbon and organic carbon. For the latter, the differential solubility from industrial (fossil fuel) and biomass burning sources is taken into account (default solubility ratio 0.6/0.8) (Hansen et al., 2005). The default contribution shares w<math>_g</math> of the individual aerosol types g to the indirect aerosol effect were assigned to reflect the preliminary results by Hansen et al. (2005), namely 36 % for sulfates, 36 % for organic carbon, 23 % for nitrates and 5 % for black carbon. Note, however, that these estimates of the importance of non-SOx aerosol contributions are very uncertain, not least because the solubility, e.g. for organic carbon and nitrates have large uncertainties. The number concentrations <math>N_{g,i}</math> are here approximated by historical optical thickness estimates (as provided on http://data.giss.nasa.gov/efficacy/ see as well Supplement) and extrapolated into the future by scaling with hemispheric emissions. The general logarithmic relation between number concentrations and forcing is based on the findings by Wigley and Raper (1992); Wigley (1991a); Gultepe and Isaac(1999) and used in Hansen et al. (2005)}.