5.2 The greenhouse effect

5.2.1 The two components of the greenhouse effect

The greenhouse effect $G$ is defined here as the fraction of infrared radiation emitted by the Earth that is retained by the greenhouse effect and fails to escape to space. $1-G$ represents the fraction of infra-red energy emitted by the Earth that escapes to space.

We note $G_{0}$ the reference greenhouse effect, chosen at the pre-industrial time.

We assume that variations in the greenhouse effect $G$ are related to changes in the atmospheric concentration in water vapor and in $CO_{2}$. We neglect the effect of changes in the concentration of other greenhouse gases such as $CH_{4}$ or $N_{2}O$, or we consider them in terms of ``$CO_{2}$-equivalent''.

We have:

$$G=G_{0}+G_{H_{2}O}^{serre}+G_{CO_{2}}^{serre}$$

where $G_{H_{2}O}^{serre}$ is the greenhouse effect anomaly with respect to the reference related to the water vapor concentration anomaly and $G_{Co_{2}}^{serre}$ is that related to the $CO_{2}$ concentration anomaly.

5.2.2 The greenhouse effect related to $CO_{2}$ as a function of $CO_{2}$ concentration

$G_{Co_{2}}^{serre}$ is calculated as a function of $CO_{2}$ concentration: $CO_{2}(t)$. In the usual CO2 concentration range (between 100 and 10,000 ppm), we assume a logarithmic relationship between $G_{Co_{2}}^{serre}$ and $CO_{2}(t)$ ([Myhre et al., 1998,Pierrehumbert et al., 2006]):

$$G_{Co_{2}}^{serre}=a_{CO_{2}}cdot ln(frac{CO_{2}(t)}{CO_{2}^{ref}})$$

The $a_{CO_{2}}$ factor is adjusted to obtain realistic climate projections, and is currently set at 2.2$10^{-2}$. Around this range, a linear approximation extends the logarithmic relationship.

The effect of the $CO_{2}$ concentration on he infra-red energy emitted by the Earth escaping to the space ($F_{out}$) is illustrated in figure 9.

5.2.3 The greenhouse effect related to water vapor as a function of the water vapor concentration

$G_{H_{2}O}^{serre}$ is calculated as a function of the global-mean amount of water vapor integrated in the atmospheric column, $H_{2}O(t)$:

$$G_{H_{2}O}^{serre}=-Qcdot G_{0}cdotleft(1-left(R_{H_{2}O}(t)right)^{p}right)cdot L$$

where $R_{H_{2}O}(t)$ is the ratio between the amount of water vapor at time $t$ and its reference quantity:

$$R_{H_{2}O}(t)=frac{H_{2}O(t)}{H_{2}O^{ref}}$$

and $L$ limits the greenhouse effect when $R_{H_{2}O}$ becomes very strong, avoiding too strong a runaway greenhouse effect when the temperature becomes very strong: : $L=0.3cdot e^{-sqrt{R_{H_{2}O}(t)-1}}+0.7$.

To satisfy the observational constraints (section 2.1), we take $Q=0.6$ and $p=0.23$.

5.2.4 The water vapor concentration as a function of temperature

In order to simulate the positive feedback of water vapor on the climate, the ratio $R_{H_{2}O}(t)$ is expressed as a function of the temperature $T(t)$ assuming that the relative humidity remains constant. Then $R_{H_{2}O}(t)$ equals the ratio of partial saturation pressures $p_{sat}$.

$$R_{H_{2}O}(t)=frac{p_{sat}(T)}{p_{sat}(T_{ref})}$$

The saturation vapor pressure is calculated by the Rankine formula:

$$p_{sat}(T)=exp(13.7-5120./T)$$

The temperature is in K and $T_{ref}=14.4^{circ}C$.