Lecture 12- Surface temperature change caused by 2XCO2
Key concept: Stefan-Boltzman feedback; water vapour feedback
Preliminary estimate
When CO2 concentrations increase by a factor of two, the optical thickness of the atmosphere increases
(cf. 10.4) and the emission level moves upward (cf. 11.7) by an amount Z (CO2 )  150m (Fig. 1).
Assuming fixed emission temperature and lapse-rate, we can readily predict the associated (assumed uniform)
change in atmospheric temperature T  Tsref from eq. (9.3)
Ts( ref )  Z (CO2 )  1K
(12.4)
Z
Fig. 1: Illustration of the
ZE+ΔZ
2 X CO2
ZE
1 X CO2
TE
changes in temperature
(assumed constant with height)
caused by an increase in CO2
concentrations assuming fixed
lapse-rate and emission
temperature.
T
TS TS+ΔT
Physical interpretation
The prediction (12.4) illustrates a balance between (i) a reduced upward radiation at the top-of-the atmosphere
associated with the larger CO2 concentrations (denoted by I (1) ) since the atmosphere is more opaque and
absorbs more upward radiation with larger CO2 concentrations and (ii) an increased upward radiation at the
top-of-the atmosphere associated with a warmer atmospheric column (denoted by I ( 2) ),
I (1)   [T ( Z E  Z (CO2 ))] 4   [T ( Z E )] 4  4 [T ( Z E )]3
I ( 2)  TE4   [TE  T ] 4  4TE3 T   SB T  0
dT
Z (CO2 )  4TE3 Z (CO2 )  4Wm  2 (12.1)
dz
(12.2)
in which
 SB  4TE3  4Wm 2 K 1
(12.3)
is the (negative) Stefan-Boltzman feedback (a warmer atmosphere emits more radiation and thereby cools).
This feedback is the main effect which prevents the Earth from warming too much when greenhouse gas
concentrations increase. Note that the prediction (12.4) is readily recovered from the balance I (1)  I ( 2)  0 .
Water vapour feedback
The above discussion neglects the crucial fact that, as the surface gets warmer there is more surface
evaporation, and thus more water vapour in the atmosphere. Since water vapour is the main greenhouse gas,
this further raises the emission level: in other words, water vapour acts as a positive feedback, amplifying our
initial estimate of surface temperature change. Fig. 1 is still valid but we need to interpret the change in
emission level Z as resulting jointly from changes in CO2 concentrations and water vapour.
To take this effect into account and improve our estimate of T , we can still use the balance I (1)  I ( 2)  0
but we must acknowledge that the presence of a varying water vapour distribution complicates the estimate
of I ( 2) : numerical results from Manabe & Wetherald show a dependence of radiation upon surface
temperature different from that predicted from Stefan-Boltzman (linear with respect to T rather than depending
upon its fourth power). We empirically write
I ( 2 )   MW T  0
(12.6)
where
 MW  2Wm 2 K 1
(12.7)
is the (negative) Manabe-Wetherald feedback: a warmer (and moister) atmosphere emits more radiation to
space but not as much as Stefan-Boltzman (eq. 12.2) would predict. We now obtain, using I (1)  I ( 2)  0
with (12.6) that
T  Ts  Ts( ref )
SB
 2K
MW
(12.8)
As a result of a smaller sensitivity of the top-of-the atmosphere radiation to Ts introduced by the water vapour
feedback ( | MW |  | SB | ), the atmosphere and the surface must be at a higher temperature to balance the
radiative forcing of 2XCO2 ( I (1)  4Wm 2 K 1 ). A measure of the positive feedback of water vapour is then
simply
H O  MW  SB  2  (4)  2Wm 2 K 1
2
(12.9)
The water vapour feedback is the largest positive feedback in the climate system, being on the same order of
magnitude than, but in opposite direction to, the Stefan-Boltzman feedback.