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Climate Sensitivity & Climate Feedback
Instructor: Prof. Johnny Luo
http://www.sci.ccny.cuny.edu/~luo
Te  4
4
(S0 /4)(1   )

(1367 /4)(1  0.3)
5.67  10 8
 255K  18 0 C

Te  4
4
(S0 /4)(1   )

(1367 /4)(1  0.3)
5.67  10 8
 255K  18 0 C

Considering the
Greenhouse Effect
Ts = 15 0C > -18 0C
EAS 488/B8800 Climate & Climate Change
Part I: Fundamentals of Climate Science
1.Introduction to the climate system
2.The Earth’s energy balance
3.Atmospheric radiation and climate
4.Surface energy balance
5.Atmosphere general circulation
6.Ocean general circulation
Energy budget (global
balance & local imbalance)
Fluid movement (due to
local energy imbalance)
Part II: Climate Change
1.Climate sensitivity & climate feedback
2.Natural & anthropogenic climate change
3.IPCC assessment of past & future climate
change
What will happen if
energy imbalance occurs
at a global level?
Outlines
1.
Basic concepts: climate forcing, response,
sensitivity and feedbacks
2.
Climate sensitivity w/o feedback
3.
Water vapor feedback
4.
Ice albedo feedback
5.
Cloud feedback
6.
Tropical SST regulatory mechanism
7.
Daisy world
Global energy balance: the starting point
This chapter deals with:
S0
(1   )  Te 4
4
1)
what may break this
balance?
2)
what will happen
when this balance is
violated?

First, we will look at a few
fundamental concepts:
1)
2)
3)
4)
climate forcing,
climate response,
climate sensitivity
climate feedback
Forcing & Response
Climate Forcing: change in external factors that breaks the
aforementioned energy balance (usually measured in changes in
energy flux density in W m-2 at TOA).
Climate Response: adjustment of the climate system in response to
the external forcings (usually measured as change in surface
temperature, Ts).
Climate Sensitivity: climate response (Ts) over climate forcing (Q).

dTs
dQ
Example:

Q

S0
(1   )  T 4
4
Forcing: When CO2 is doubled, OLR will
change from 240 W m-2 to 236 W m-2 (is
this a warming or cooling for the climate
system?).
Response: For planet A: Ts increases by
1 K; for planet B: Ts increases by 10 K.
Sensitivity:
λ(A) = 1K/(4 W m -2) = 0.25 K/(W m -2).
λ(B) = 10K/(4 W m -2) = 2.5 K/(W m -2).
Outlines
1.
Basic concepts: climate forcing, response,
sensitivity and feedbacks
2.
Climate sensitivity w/o feedback
3.
Water vapor feedback
4.
Ice albedo feedback
5.
Cloud feedback
6.
Tropical SST regulatory mechanism
7.
Daisy world
Sensitivity of the Earth’s climate
RTOA
S0
 (1   )  Te 4  0
4
equilibrium



Suppose a forcing dQ is imposed on RTOA. Let’s calculate the climate
sensitivity dTs/dQ.
RTOA  RTOA (Q,Ts (Q))
dRTOA RTOA RTOA dTs


0
dQ
Q
Ts dQ
=1
New equilibrium:
RTOA = 0

dTs
1

dQ
RTOA /Ts
Sensitivity parameter
Sensitivity of the Earth’s climate
RTOA
S0
 (1   )  Te 4  0
4
equilibrium


Suppose a forcing dQ is imposed on RTOA. Let’s calculate the climate
sensitivity dTs/dQ.
dQ: forcing; dTs: response
RTOA  RTOA (Q,Ts (Q))
Sensitivity of the Earth’s climate
RTOA
S0
 (1   )  Te 4  0
4
equilibrium



Suppose a forcing dQ is imposed on RTOA. Let’s calculate the climate
sensitivity dTs/dQ.
dQ: forcing; dTs: response
RTOA  RTOA (Q,Ts (Q))
dRTOA RTOA RTOA dTs


0
dQ
Q
Ts dQ
= 1 (b/c instantaneous
changes in RTOA & dQ
are the same)
New equilibrium
at the TOA
Sensitivity of the Earth’s climate
RTOA
S0
 (1   )  Te 4  0
4
equilibrium



Suppose a forcing dQ is imposed on RTOA. Let’s calculate the climate
sensitivity dTs/dQ.
dQ: forcing; dTs: response
RTOA  RTOA (Q,Ts (Q))
dRTOA RTOA RTOA dTs


0
dQ
Q
Ts dQ
= 1 (b/c instantaneous
changes in RTOA & dQ
are the same)
New equilibrium 
at the TOA
dTs
1


dQ
RTOA /Ts
Sensitivity parameter
Now we calculate:
RTOA /Ts
S0
(1  )  Te4
4
RTOA  (Te4 )

 4Te3
Ts
Ts
RTOA 

Assuming: 1) solar constant is unchanging, and
2) Te and Ts change at the same rate
Now we calculate:
RTOA /Ts
S0
(1  )  Te4
4
RTOA  (Te4 )

 4Te3
Ts
Ts
RTOA 

Assuming: 1) solar constant is unchanging, and
2) Te and Ts change at the same rate
Estimating the sensitivity parameter (Te = 255 K for current climate)
dTs
1
1
2 1



0.26
K(W
m
)
dQ
RTOA /Ts 4Te3
What this means is: for every 1 W m-2 of energy we add to or subtract from
the climate system, change of effective temperature (or surface temperature)
will be 0.26 K.

This is dictated by the Stefan-Boltzmann relation. Note that other factors
(e.g., albedo, water vapor) are held unchanged at this point.

dTs
 0.26 K(W m2 )1
dQ
This is the climate sensitivity that is built-in of the σTe4 relationship.
Think-Pair-Share Questions:

1)For this kind of climate system, i.e., λ=0.26 K (W m-2)-1, what dQ
is needed to warm up the Earth’s surface by 1K (i.e., dTs=1K) ?
2)How many W m-2 does the Solar Constant (S) have to increase to
achieve dTs=1 K? Assume the albedo is 0.3
RTOA

S0
 (1   )  Te 4
4
Think-Pair-Share Questions:
1)For this kind of climate system, i.e., λ=0.26 K (W m-2)-1, what dQ
is needed to warm up the Earth’s surface by 1K (i.e., dTs=1K) ?
2)How many W m-2 does the Solar Constant (S) have to increase to
achieve dTs=1 K? Assume the albedo is 0.3
1 W m-2 -> 0.26 K  about 4 W m-2 is needed for 1 K.
To achieve 4 W m-2 thru changing the
Solar Constant (S0)
S0
(1  )
4
S0
(1 0.3)  4 W m2  S0 22 W m2
4
S0  1370 W m2

Observations show that S0
varies in magnitude of 1 W m2 (historical data dated back to
1870 can also support this
estimate; however, over a
longer history such as millions
of years, there are larger
variations).

dTs
 0.26 K(W m2 )1
dQ
Conclusion: the σTe4 type of climate
system is a rather stable one because
of the fundamental way energy
balance is achieved.
So, S0(1-0.3)/4 = 0.175 W m2. With this climate forcing,
the response will be 0.175 ×
0.26 = 0.0455 K.
Outlines
1.
Basic concepts: climate forcing, response,
sensitivity and feedbacks
2.
Climate sensitivity w/o feedback
3.
Water vapor feedback
4.
Ice albedo feedback
5.
Cloud feedback
6.
Tropical SST regulatory mechanism
7.
Daisy world
Feedback
mechanism:
Sensitivity = Output/Input. With feedback, the sensitivity parameter
will be different.

dTs
 0.26 K(W m2 )1
dQ
T-P-S: How
 will water vapor affect the intrinsic climate
sensitivity parameter? In other words, given the same
forcing, how will water vapor changes the Ts response?
Feedback
mechanism:
Water vapor: a strong positive feedback in global warming scenario
dQ
Increasing
CO2
dTs
Temperature
H2O
IR absorption spectra
(0 means no absorption; 100
means total absorption)
Much of the infrared
absorption
(greenhouse effect)
comes from the
contribution of H2O
Clausius-Clapeyron relationship (C-C): saturation vapor pressure
increases with temperature
des
L dT
(
)
es
RvT T

For current terrestrial conditions,
for every 1 K increase in
temperature, es increases by ~ 6%.
Calculate OLR as a
function of surface
temperature (holding RH
constant so vapor
pressure increases with
Ts).
This will need a radiative
transfer model. For each
Ts, we calculate I (OLR),
so we have dTs/d(OLR)
OLR increases with
increasing Ts, but at a
SLOWER rate than what
the stefan-Boltzmann
relationship gives: σ(Ts30)4.
Red: assume clear sky
Green: average cloudiness
T* is the surface temperature (Ts). T* - 10, T* - 20,
…, T* - 50 are attempts to estimate the effective
temperature (Te) from the surface temperature.
For global average, T* = 288 K, Te = 255 K, so T* 30 is a good approximation for global average
curve.
Conclusion: because of
the water vapor feedback,
climate sensitivity is
HIGHER than a sigma-Tto-the-4th relationship.
dOLR 1
(
)
dTs
With water feedback  



dTs
 0.5 K(W m2 )1
dQ
Ts  Ts (Q,H2O)
dTs Ts
Ts dH2O


 0.5 K(Wm2 )1
dQ Q H2O dQ
0.26 K (Wm-2)-1
Climate sensitivity has doubled with water vapor feedback.
Summary
Sensitivity = response / forcing.
Climate Forcing: change in external
factors that breaks the energy
balance of the climate system
(usually measured in changes in
energy flux density in W m-2 at TOA).
RTOA 
S0
(1   )  T 4  0
4
Climate Response: adjustment of
the climate system in response to
the external forcings (usually
measured as change in surface
temperature, Ts).
Climate sensitivity w/o feedback:


dTs
1
1
2 1


3  0.26 K(W m )
dQ
RTOA /Ts 4Te
Double CO2 forcing:

4 W m-2 -> 4×0.26 ≈ 1 K
Summary
dTs
dQ
Feedback mechanism:
dQ
Increasing CO2
(or whatever causes
the warming)
dTs
Temperature goes up
H2O goes up

dTs
1

 0.5 K(W m 2 ) 1
dQ
RTOA /Ts
Water vapor: a strong
positive feedback, doubling
the climate sensitivity