Thermodynamics of Atmospheres and Oceans
Worksheet 13: Thermodynamic feedbacks in the climate system
Water vapor feedback: (section 13.3)
1. Recall the definition of water vapor path Wv (4.40). Derive an expression for dWv/dT0
in terms of dwv/dT0 where wv is water vapor mixing ratio and T0 is surface temperature.

1
Wv   v dz 
g
0
p

1
p av dp  g p wv dp
p0
dWv
d 1 0
1

wv dp 

dT
dT g p
g
p0
p0
dwv
 dT
0
dp
0
2. Recall the definition of water vapor mixing ratio (4.36). Starting from the result in #1,
derive an expression for dwv/dT0 in terms of es and H where H is the relative humidity
dwv  
H ( p)
e 
  es ( p )
 H ( p) s 
dT0 p 
T0
T0 
dwv  T ( p )  
H ( p )
e ( p) 
 es ( p)


 H ( p) s
dT0 p  T0 
T ( p )
T ( p) 
3. Starting from #2, write an expression for dwv/dT0 that eliminates the variables es (using
the Clausius-Clapeyron equation and the definition of mixing ratio) but retains H . Note,
you will need to use the definition of lapse rate to convert the dT0 derivatives into ∂T(p)
derivatives
dp Llv p

dT RvT 2
dwv
T ( p) d  H ( p) H ( p) Llv 


 ws

dT0
 dT0  T ( p) RvT 2 ( p) 
4. Using the result from #3, write the complete expression for dWv/dT0 (refer to 13.19)
F rad dWv
f  G0
so that from 13.19,
Wv dT0
p
dWv 1 0  T d  H HLlv 

dp
   ws

dT0 g 0   dT0  T RvT 2 
5. Simplify the result from #4 by assuming ∂H/∂T0 = 0, and write the expression in the form
(1/Wv) dWv/dT0.
1 Wv Llv
2

Wv T0 Rv 2 
R 
T0  2  v 
g 

6. In the tropics, the lapse rate is close to the moist adiabat. What sign would d/dT0
have in the tropics?
negative
7. In the polar regions, the mid- and upper-troposphere has been simulated in climate
models to warm less rapidly than the surface so that the lapse rate increases. What sign
would d/dT0 have in the tropics?
positive
Cloud feedback (section 13.4)
The cloud-radiative effect (sometimes referred to as “cloud forcing”) is defined as
F rad
 dA  CF net = F rad A – F rad 0
c
c
A c
where Ac is cloud fractional coverage. CF can be defined in the context of either the
TOA (top of atmosphere) or surface radiation fluxes.
1. Table 13.1 provides estimates of CF for the TOA. Estimate the sign of CF for the
surface radiation fluxes, in the table below for the polar regions and the tropics:
Region
LW
CF
CFSW
CFnet
________________________________
Tropics
+
Polar (summer)
+
Polar (winter)
+
0
+
In general clouds act to warm the arctic although they do show a brief period of cooling g
during the arctic summer.
2. The cloud distribution feedback involves changes in cloud fraction and cloud height
(temperature). Consider an altostratus cloud layer in midlatitudes. Lifting the cloud
(both the base and top, other cloud properties remain the same) would influence the
radiation fluxes:
a) TOA SW radiation fluxes would (increase, decrease, remain the same)
b) surface SW radiation fluxes would (increase, decrease, remain the same)
c) TOA LW radiation fluxes would (increase, decrease, remain the same)
d) surface LW radiation fluxes would (increase, decrease, remain the same)
Lifting the cloud will decrease its emission temp and as a result decrease the outgoing
longwave radiation (OLR). Since the emission temperature will decrease as the cloud is
lifted, the LW surface flux emitted by the cloud will also decrease. For this reason, low
clouds emit more infrared radiation than high clouds of comparable thickness.
3. The cloud optical depth feedback involves the term dc/dT0. Consider the shortwave
optical depth of liquid clouds. Using (8.12), write an expression for dc/dT0 that includes
Wl and re.
d c 3  1 dWl Wl drel 

 

dT0 2  rel  dT0 l dT0 
4. Focusing on the term dWl/dT0, write an expression in terms of wl (don’t forget to
include the possibility of cloud base and top height changing with T0).
dWl 1

dT0 g
pb
dwl
 dT
pt
0
dp 
1  dpt
dp 
 wlt
 wlb b 
g  dT0
dT0 
5. If wl is constant in the cloud
a) write an expression for wl in terms of Wl.
wl  gWl /( pb  pt )
b) write an expression for dWl/dT0 using the expression in a).
dWl 1

dT0 g
pb
dwl
pt
0
 dT
 d ln( pb  pt ) 

dp  Wl 
dT0


6. The following factors would influence dWl/dT0 as T0 increases
a) increasing air temperature
b) increasing rainfall
c) nucleation of ice particles (at same threshold temperature)
d) clouds become more convective in character
e) clouds become more stratiform in character
7. In the Arctic, clouds are frequently mixed phase. Assuming that the lapse rate
remains unchanged………
…
a) cloud temperatures would (increase, decrease, remain the same)
b) cloud phase would become (more crystalline, more liquid, remain the same)
c) cloud optical depth would (increase, decrease, remain the same). Explain
If cloud phase becomes more liquid, this increase in liquid water path will increase the
optical depth because emissivity and reflectivity is greater for liquid clouds.
d) surface downwelling longwave flux would (increase, decrease, remain the same)
e) surface downwelling shortwave flux would (increase, decrease, remain the same)