2.3 Global radiative equilibrium model

Figure 7: Global radiative equilibrium model.

At radiative equilibrium, the solar flux that is absorbed by the Earth, $F_{in}$, equals the infra-red radiation emitted by the Earth, $F_{out}$ (figure 7):

$$F_{in}=F_{out}$$

Fluxes $F_{in}$ and $F_{out}$ are expressed in $W/m^{2}$.

2.3.1 Absorbed solar flux

$F_{in}$ depends on the planetary albedo:

$$F_{in}=(1-A)cdot F_{0}^{^{in}}$$

$A$ is the Earth albedo, which depends on the ice sheet extent. It is computed as detailed in section 2.4.

$F_{0}^{^{in}}$ is the global-mean, annual-mean incoming solar flux at the top of the atmosphere. Since at any time, the Sun lights up only a quarter of the Earth, we have $F_{0}^{in}=frac{S_{0}}{4}$, where $S_{0}=1370 W/m^{2}$ is the solar constant.

2.3.2 Infrared radiation emitted by the Earth

$F_{out}$ depends on temperature according to Stefan-Boltzmann's law, and is modulated by the greenhouse effect:

$$F_{out}=(1-G)cdotsigmacdot T^{^{4}}$$

where:

$G$ is the greenhouse effect: it is the fraction of infrared radiation emitted by the Earth that is retained by the greenhouse effect and fails to escape to space;

$sigma$ is Stefan-Boltzmann's constant.

This relationship is illustrated for different $CO_{2}$ concentration in figure 8.

2.3.3 Equilibrium temperature

We calculate $T_{eq}(t)$ at each time step $t$, assuming radiative balance:

$$T_{eq}(t)=left(frac{left(1-A(t)right)cdot F_{0}^{^{in}}}{left(1-G(t)right)cdotsigma}right)^{1/4}$$

Graphically, $T_{eq}$ corresponds to the intersection point $T$ between $F_{in}(T)$ and $F_{out}(T)$ curves (figure 8).

Figure 8: Absorbed solar radiation ($F_{in}$) and infra-red radiation emitted by the Earth ($F_{out}$), as a function of temperature. Radiative equilibrium is reached for intersection points between $F_{in}(T)$ and $F_{out}(T)$ curves.

The temperature $T(t)$ simulated by SimClimat follows the equilibrium temperature $T_{eq}$, but with some delay to represent the effect of the thermal inertia of the oceans (section 6.1).