The greenhouse effect 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.
represents the fraction
of infra-red energy emitted by the Earth that escapes to space.
We note the reference greenhouse effect, chosen at the pre-industrial
time.
We assume that variations in the greenhouse effect are related
to changes in the atmospheric concentration in water vapor and in
. We neglect the effect of changes in the concentration of
other greenhouse gases such as
or
, or we consider
them in terms of “
-equivalent”.
We have:
where
is the greenhouse effect anomaly with
respect to the reference related to the water vapor concentration
anomaly and
is that related to the
concentration anomaly.
is calculated as a function of
concentration:
. In the usual CO2 concentration
range (between 100 and 10,000 ppm), we assume a logarithmic relationship
between
and
([Myhre et al., 1998,Pierrehumbert et al., 2006]):
Around this range, a linear approximation extends the logarithmic relationship.
The effect of the concentration on he infra-red energy emitted
by the Earth escaping to the space (
) is illustrated in
figure 8.
is calculated as a function of the global-mean
amount of water vapor integrated in the atmospheric column,
:
where is the ratio between the amount of water vapor
at time
and its reference quantity:
and limits the greenhouse effect when
becomes very
strong, avoiding too strong a runaway greenhouse effect when the temperature
becomes very strong: :
.
To satisfy the observational constraints (section 2.1),
we take and
.
In order to simulate the positive feedback of water vapor on the climate,
the ratio is expressed as a function of the temperature
assuming that the relative humidity remains constant. Then
equals the ratio of partial saturation pressures
.
The saturation vapor pressure is calculated by the Rankine formula:
The temperature is in K and
.