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Atmospheric and Climate Sciences, 2012, 2, 474-478
http://dx.doi.org/10.4236/acs.2012.24041 Published Online October 2012 (http://www.SciRP.org/journal/acs)
Temperature-Profile/Lapse-Rate Feedback:
A Misunderstood Feedback of the Climate System
Michael E. Schlesinger1, C. Bruce Entwistle2, Bin Li3
1Climate Research Group, Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Urbana, USA
2Aviation Weather Center, National Weather Service, Kansas City, USA
3I. M. Systems Group, Inc. at EMC/NCEP/NOAA, College Park, USA
Email: [email protected]
Received July 19, 2012; revised August 20, 2012; accepted August 31, 2012
ABSTRACT
This study shows that the heretofore assumed condition for no temperature-profile (TP)/lapse-rate feedback,

s
Tz T for all altitudes z, or


dd0Tz z
 
, in fact yields a negative feedback. The correct condition for no
TP feedback is
s
s
Tz TzTT

for all z, where Ts is the surface temperature. This condition translates into a
uniform increase (decrease) in lapse rate with altitude for an increase (decrease) in Ts. The temperature changes caused
by a change in solar irradiance and/or planetary albedo satisfy the condition for no TP feedback. The temperature
changes caused by a change in greenhouse gas concentration do not satisfy the condition for no TP feedback and, in-
stead, yield a positive feedback.
Keywords: Climate Feedback; Feedback Analysis; Lapse-Rate Feedback
1. Introduction
Feedback due to changes in the vertical temperature pro-
file has been called lapse-rate feedback [1]. It has been
assumed that the condition for this feedback to be zero is
that
s
Tz T
for all altitudes z, where
Tz is
the temperature at z and Ts is the surface temperature [2].
This condition is equivalent to

Tz
dd0z
, that is,
no change in lapse rate, hence the name lapse-rate feed-
back when


dd0Tz z. Here we first use a
one-layer atmospheric model, and then a multilayer at-
mospheric model, to show what we found 20 years ago
[3], namely, that the correct answer for zero feedback is

loglog
s
TzT

or, equivalently,

s
s Tz TzTT
for all z. When this condition is not satisfied, there is
feedback. In particular, if
s
Tz T for all altitudes
z, the feedback is negative. Since there is feedback when
the lapse rate does not change, it is recommended that the
name lapse-rate feedback be supplanted by the appella-
tion temperature-profile (TP) feedback.
2. Feedback Analysis
This section is based on the feedback analysis of
Schlesinger [4-6]. The net downward radiation at the top
of the atmosphere (TOA) per unit area, N, is given by
4
1
4
p
ps
NST




(1)
where S is the solar irradiance at TOA,
p
is the plane-
tary albedo, σ is the Stefan-Boltzmann constant, and
p
is the planetary emissivity. The change in N due to ex-
ternal radiative forcing F, say due to a change in S or the
anthropogenic increase in the concentrations of green-
house gases, can be written as
d
d
j
s
j
sjs
I
NN
NF T
TIT


 



(2)
where the second term on the right-hand side is the
change in N due to the change in Ts alone, and the third
term is the change in N due to the change in internal
quantities
j
I
—such as the temperature profile; water
vapor amount; cloud amount, height and optical depth—
through their dependence on Ts. From Equation (1) we
can also write
34
44
pp
p
sss
ss
S
NFTT T
TT
 
 

 





(3)
0NFor the new equilibrium, , hence by Equation
(2) we have
1
o
sf
G
TGF F
f

 


(4)
C
opyright © 2012 SciRes. ACS
M. E. SCHLESINGER ET AL. 475
where
1
o
f
G
G (5)
f
is the gain of the climate system with feedback,
d
d
j
o
j
j
s
I
N
fG
I
T
G
0N
0
pp
(6)
and o is the gain of the climate system with zero
feedback (f = 0). From Equation (3) with and
 
we obtain,

1
3
1
4
o
sps
N
GTT


 

 1
s
p
T
S
(7)
the latter from Equation (1) with N = 0, that is, the equi-
librium before the radiative forcing F is applied. For pre-
sent-day conditions, prescribed in Table 1 and calculated
in Table 2, 2
30 KWm
2
3.71 Wm
0N
0.
o
G

22xox
o
TGF 
. Thus if the climate
system had zero feedback, the temperature change due to
a doubling of the CO2 concentration would be
for [7].
1.11 C2x
F
From Equation (6) and Equations (2) and (3) with
it can be seen that
4
pp
os
ss
4
S
f
GT
TT





0
 (8)
For the feedback we consider here, p

. Thus, if
p
increases (decreases) with increasing (decreasing) Ts,
0T
ps
, the change in emissivity works to decre-
ase (increase)
s
T, hence as shown by Equation (8), f <
0. Conversely, if
p
decreases (increases) with in-
creasing (decreasing) Ts, 0T
ps
, the change in
emissivity works to increase (decrease)
s
T

, hence as
shown by Equation (8), f > 0. Below we show that the
heretofore assumed condition for zero feedback,
s
Tz T
for all altitudes z, actually yields a negative feedback, f < 0.
Table 1. Prescribed quantities.
Quantity Value
Stefan-Boltzmann constant,
3. Condition for No Temperature-Profile
Feedback
Consider the one-layer atmosphere shown in Figure 1
with temperature, a, infrared (IR) absorptance, a , and
solar absorptance b. Energy balance at TOA and the sur-
face gives
T
444 4
1
5.67 × 10–8 Wm–2·K–4
Solar irradiance, S 1367 Wm–2
Planetary albedo,
p
R
0.3
Longwave absorptance, a 0.8
Solar absorptance, b 0.1
Change in due to a CO2 doubling –3.71 Wm–2 [7]
s
aeps
aT aTRTT



44 4
1
(9)
s
ae
TaT bT

 (10)
Table 2. Calculated Quantities.
Quantity Equation Value
Planetary emissivity, p
(1) with N = 0 0.63
Equivalent blackbody temperature,
e
T(11) 254.86 K
Surface temperature,
s
T
a
(12) 285.89 K
Atmosphere temperature, Ta (13) 245.01 K
Gain of the climate system without
feedback, Go (7) 0.299
Change in longwave absorptance due
to a CO2 doubling,
∆ of (9)
44
R
a
TT
s
a



a
0.021
Longwave absorptance after the CO2
doubling,
aa a

p
0.821
Change in
due to a CO2 dou
b
ling,
pe
(23) –0.010
s
s
TT (26) 0.0045
aa
TT (27) 0.0037
s
T

TTT
s
ss
a
T
1.29 K

TTT
aaa
p
0.91 K
Change in
due to change in
temperatures,
pi
(15) –0.0014
Feedback due to change in
temperatures, f (8) 0.122
p
1S
4
4
s
1aT
4
a
aT
4
a
aT

p
1
1S
4
b
a
T
4
s
T
s
T
Figure 1. Schematic diagram of the solar and infrared
fluxes in a one-layer atmosphere.
Copyright © 2012 SciRes. ACS
M. E. SCHLESINGER ET AL.
476
where is the outgoing longwave radiation at TOA,
and
R


1/4
1
4
pS




e
T (11)
is the equivalent blackbody temperature of the planet.
Solving for Ts and Ta yields
1/4
2
2
s
e
b
TT
a





(12)
1/4
1
2
ae
a
TT
aa





ab
(13)
From Equation (9)

44
aa
ss
TT
a
TT



1a
1
paa

 
 (14)
where
 is the IR transmittance of the atmos-
phere. For fixed a,

4
4
pia



aas
sas
TTT
TTT
 

  (15)
where subscript i denotes “internal” in contrast to “ex-
ternal”, for example, by changing the concentration of
CO2. From Equation (8) with fixed
p
, the necessary
and sufficient condition for no TP feedback is 0
p
,
which by Equation (15) requires
as
as
TT
TT

log log
as
TT or, equivalently, .(16)
This result is readily generalizable to an atmosphere
with an arbitrary number of layers K by writing
4
1
K
k
k
ks
T
aT



ps

 (17)
where Tk is the temperature of layer k, 1/21/ 2kk k
a


,
and
j
is the transmittance from level j to TOA. For
fixed
j
,

4
1
4
K
kks
pk
iksks
TTT
aTTT
 

 


 

0
pi

(18)
Thus the sufficient condition for no TP feedback,
, is
for all or
log= log for al
ks
ks
ks
TT k
TT
TT


, equivalently,
l
k
(19)
For all practical purposes this is also the necessary
condition for no TP feedback.
From Equation (19), ksks
TTTT
T
for all k. Thus,
for no TP feedback the change in temperature with alti-
tude, k
, parallels the undisturbed temperature profile,
Tk. Figure 2 shows ksks
TTTT
 for the US Stan-
dard Atmosphere. It is seen that for no TP feedback,
ksks
TTTT

TT
decreases with increasing altitude in
the troposphere and increases with altitude in the strato-
sphere. A similar decrease then increase is needed in the
mesosphere and thermosphere, respectively. However, as
can be seen from Equation (18), these regions are of less
importance than the troposphere and stratosphere be-
cause of their smaller absorptance, ak. Figure 2 shows
that the temperature changes required for no TP feedback
is less than the heretofore assumed uniform temperature
change by as much as 25% at the tropopause and lower
stratosphere.
Now suppose ks
 for all k, as heretofore as-
sumed for no feedback. Then Equation (18) yields
4
s
1
T
4
K
pikk
k
k
ssks
TT
a
TTTT
 


 
(20)
Because
s
k
TT for at least the part of the atmos-
phere where

4
aTT

kks is largest, namely, the troposp-
here and lower stratosphere, 0
s
T

0f
p
i, hence by
Equation (8) the feedback is negative, , for a uniform
temperature change and thus no change in lapse rate.
How must the lapse rate change for there to be no TP
feedback? From the definition of lapse rate,
 
11kkk k
TTz z


where z is altitude and k increases from TOA to the sur-
face, it is straightforward to show from Equation (19)
that for no TP feedback,
1
s
s
T
T
(21)


Figure 2. The vertical profile of Tz T
s
 for zero TP
feedback for the US Standard Atmosphere.
Copyright © 2012 SciRes. ACS
M. E. SCHLESINGER ET AL. 477
where
 
11kk kk
TTT T

 
1kk
zz
0T
.
Thus for no TP feedback the lapse rate must increase
uniformly with altitude for surface warming, s
0T
,
and must decrease uniformly with altitude for surface
cooling, s. These changes in lapse rate are not
large—for a 3˚C global temperature change they are
about 1% of the undisturbed lapse rate.
4. Application to Solar Forcing
We now return to the one-layer atmosphere of Figure 1.
From Equations (11)-(13) for fixed a and b it is straight-
forward to show that

1/4
1
4
pS
1
ase
asee
TTT
TTTT







(22)
Thus the response of the atmospheric temperature to a
change in Ta, either through a change in the solar irradi-
ance or planetary albedo or both, satisfies the require-
ment for no TP feedback.
Figure 3 shows the profile of

kk ss
TTTT
in response to a 2% increase in the solar irradiance cal-
culated by our 26-layer stratosphere/troposphere radia-
tive-convective model [3] with the convective adjustment
turned off and no temperature dependence of the infrared
transmittances. It is seen that

1
ss
TT
kk
TT
for all 26 layers, hence TP feedback is zero for solar
forcing.
Figure 3. Vertical profile of (∆Tk/Tk)/(∆Ts/Ts) versus pres-
sure simulated for a 2% increase in solar irradiance and a
CO2 doubling by a 26-layer stratosphere/troposphere ra-
diative-convective model [3] with the convective adjustment
turned off and no temperature dependence of the longwave
transmittances.
5. Application to Infrared Forcing
In this section we show that condition (16) for no TP
feedback is not satisfied for radiative forcing in the infra-
red, such as from changing the concentration of green-
house gases, which changes a to . By Equa-
tion (14) this will change the planetary emissivity by
aa a


4
1a
pes
Ta
T

(23)
 


a
where the subscript e denotes an “external” change.
The new equilibrium is given by Equations (9) and (10)
with a replaced by
,
s
T by
s
s, and a
T by
aa
TT
TT
. This yields after using the binomial expansion
and linearizing,

4
114
x
xxx 

,

33 44
1
4
ssaasa
a
aT TaT TTT

 
(24)
334
4
s
saa a
a
TT aTTT
  (25)
Solving these Equations yields

42
s
s
Ta
Ta
(26)

4
1
42
sa
a
a
TT
Ta
Ta a

(27)
From Equations (26) and (27), together with Equations
(12) and (13), we obtain



2
12
1
aa
ss
ab
TT a
TTaa ba


aa
(28)
Setting Equation (28) equal to unity, assuming
and solving yields two solutions, b = 0 and a = 2, the
former having an unphysical solar absorptance and the
latter an unphysical infrared absorptance. Consequently,
for the case of infrared forcing, the condition required for
zero TP feedback given by Equation (16) cannot be satis-
fied.
The values of aa
TT
,
s
s, pi
TT

and f calc-
ulated for a CO2 doubling for the prescribed values
shown in Table 1 are presented in Table 2. It can be seen
that the change in the temperature profile yields a nega-
ive
pi
twhich is the same sign as the change in
pe
due to the doubling of the CO2 concentration.
As a result the feedback is positive and rather large.
This positive TP feedback increases the surface tem-
perature change by 16% from its zero feedback value of
1.11 K to 1.29 K. It occurs even in the absence of a
stratosphere, which the one-layer atmospheric model
does not possess. The presence of a stratosphere would
Copyright © 2012 SciRes. ACS
M. E. SCHLESINGER ET AL.
Copyright © 2012 SciRes. ACS
478
all the more result in TP feedback because the sign of the
stratospheric temperature changes induced by a change in
greenhouse-gas concentration is opposite to the sign of
the tropospheric temperature changes, thereby not satis-
fying condition (19) for no TP feedback.
Figure 3 shows the profile of

kk ss
in
response to a doubling of the CO2 concentration calcu-
lated by our 26-layer stratosphere/troposphere radiative-
convective model [3] with the convective adjustment
turned off and no temperature dependence of the long-
wave transmittances. It is seen that
TT TT

1
ss
TT

kk
TT
for all layers, hence the TP feedback is not zero. As
shown in Table 2, f = 0.122
6. Conclusions
This study has shown the following: 1) the heretofore
assumed condition for no temperature-profile (TP)/lapse-
rate feedback,
s
Tz T

for all altitudes z, which
gives no change in lapse rate,

dd0Tz z
, in fact
yields a negative feedback; 2) the correct condition for
no TP feedback is

s
s for all z; 3)
this condition translates into a uniform increase (decrease)
in lapse rate with altitude for an increase (decrease) in
surface temperature; 4) the temperature changes caused
by a change in solar irradiance and/or planetary albedo
satisfy the condition for no TP feedback; and 5) the tem-
perature changes caused by a change in greenhouse gas
concentration do not satisfy the condition for no TP
feedback and, instead, yield a positive feedback.
TzTz TT
7. Acknowledgements
This material is based upon work supported by the Na-
tional Science Foundation under Award No. ATM-
0084270. Any opinions, findings, and conclusions or
recommendations expressed in this publication are those
of the authors and do not necessarily reflect the views of
the National Science Foundation.
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