Transcript 1. Climate sensitivity

Climate feedbacks: calculating from models
Want to consider effect of variations in:
a) water vapor; b) clouds; c) sea-ice; d) snow cover; etc..
For
ith climate
variable:
So feedback factors:


ciT  R
j, ji
R  di

T

i j, ji dT
R  i
fi  0
 
i j, ji T
(1)
(2) comes from climate
model integrations.

(1) come from off-line radiation calculations.
(2)
Climate feedbacks: estimating from models
From suites of GCMS:
Individual feedbacks
uncorrelated among
models, so can be
simply combined:
Soden & Held (2006):
f  0.62;  f  0.13

Colman (2003):
f  0.70;  f  0.14

• How does this uncertainty in physics translate to uncertainty in climate sensitivity?
Uncertainty: it all depends on where you are.
T for 2 x CO2 (oC)
Can show:
T  G T0
T
T ~ G2  T0  f
T

f
f
• Uncertainty in climate sensitivity strongly dependent on the gain.
Climate sensitivity: the math
Let pdf of uncertainty
in feedbacks hf(f):
T(f) 
Also have:
So can write:


Assume Gaussian h(f):

Gives
hf (f)

hT (T) 
T0
1 f
 T 
df
T0
0
hT (T)  hf (f) 


h
1


f
2
d(T) T
 T 
2 
 

1
1 f  f 

hf (f) 
 exp  


2

 f 2
  f  

2 
 

1
T0
1 1 f  T / T0 

 2  exp  


2

 
 f 2 T
f
 

Climate sensitivity: the picture
for:
f  0.65
 f  0.14

• Skewed tail of high climate sensitivity is inevitable!
Climate sensitivity: an envelope of uncertainty
250,000+ integrations, 36,000,000+ yrs model time(!);
Eqm. response of
global, annual mean
sfc. T to 2 x CO2.
6,000 model runs,
perturbed physics
Slab ocean, Q-flux
12 model params.
varied
• Two questions:
1. What governs the shape of this distribution?
2. How does uncertainty in physical processes translate into uncertainty in
climate sensitivity?
Climate sensitivity: GCMs
• GCMs produce climate sensitivity consistent with the
compounding effect of essentially-linear feedbacks.
Climate sensitivity: Observations and Models
• h T (T) works pretty well.
1. Climate sensitivity: can we do better?
• How does uncertainty in climate sensitivity depend on f?
1. Climate sensitivity: can we do better?
f, f
T
0.65, 0.3
2 to 4.5 oC 4.5 to 8 oC > 8 oC
29%
14%
13%
0.65, 0.2
43%
18%
12%
0.65, 0.1
55%
20%
8%
0.65, 0.05
95%
5%
• Not much change as a function of f
0%
science
is
here
need to
get here!
1. Climate sensitivity: can we do better?
f, f
T
0.65, 0.3
2 to 4.5 oC 4.5 to 8 oC > 8 oC
29%
14%
13%
0.65, 0.2
43%
18%
12%
0.65, 0.1
55%
20%
8%
0.65, 0.05
95%
5%
• Not much change as a function of f
0%
science
is
here
need to
get here!
1. Climate sensitivity: can we do better?
• Combination of mean feedback and uncertainty at which a given climate sensitivity
can be rejected.
• Need to get cross hairs below a given line to reject that T with 95% confidence
Climate Sensitivity: estimates over time
Climate sensitivity  Equilibrium change in global mean, annual mean temperature
given CO2 2 x CO2
1. Arrhenius, 1896
2. Moller, 1963
3. Weatherald and Manabe, 1967
4. Manabe, 1971
5. Rasool and Schneider, 1971
6. Manabe and Weatherald, 1971
7. Sellers, 1974
8. Weare and Snell, 1974
9. NRC Charney report, 1979
10. IPCC1, 1990
11. Hoffert and Covey, 1992
12. IPCC2, 1996
13. Andronova & Schlesinger, 2001
14. IPCC3, 2001
15. Forest et al., 2002
16. Harvey & Kaufmann, 2002
17. Gregory et al., 2002
18. Murphy et al., 2004
19. Piani et al., 2005
20. Stainforth et al., 2005
21. Forest et al., 2006
22. Hegerl et al. 2006
23. IPCC4, 2007
24. Royer et al., 2007
• Why is uncertainty not diminishing with time?
4. Paleoclimate speculations?
• What if feedback strengths change as a function of mean state?
Eocene
Proterozoic
• Dramatic changes in physics are not necessary for dramatic
changes in climate sensitivity!
6. Climate at a point: the effect of random noise
Can always expect random noise
in the climate system:
What is the effect of this noise on the behavior of the system?
For time dep. eqn with
random noise, dRf:
dT  dF dS 
C
  
T  dR f
dt dT dT 
Where

C0
1  f i
i
rearranging

dT  T  dR f
 
dt

C
restoring ‘force’  a relaxation back to
equilibrium (T'=0)
(see review)

Random forcing
drives T' away
from eqm.
6. Climate at a point: the effect of random noise
Discretize eqn
into time steps, t:
dT Tt  Ttt

dt
t
Equation becomes

  a0  t
Tt a1Ttt

current
state

memory of
last state
random
noise

where
t is white noise
| dN| t
t
a0 
, a1  1
C



t
a1  exp( ) if   t 



- How does the system’s response to noise vary as a function
of the memory,  = (,C, fi)?
6. Climate at a point: the effect of random noise
= 1 yr
The effect of varying 
on the response of T’
to forcing by noise.
= 5 yrs
note multi-decadal
excursions here
= 25 yrs
note century-scale
excursions here
Time (yrs)
The greater the memory, the longer the timescale of the variability
6. Climate at a point: the effect of random noise
Climate is defined the statistics of weather.
(i.e. the mean and standard deviation of atmospheric variables)
Therefore a constant climate has a constant standard deviation
(i.e., even in a constant climate there is variability)
Crucial point:
In paleoclimate, if the proxy variable (i.e., glacier, lake, tree,
elephant, etc.) has long memory, say, ~25yrs, that proxy will
have long timescale (i.e., centennial) variability even in a
constant climate.
7. Climate at a point: power spectrum of response to noise
Time series
amplitude
Quick intro to power spectra:
they are an alternative way of describing a time series
Power spectrum
log(power)
time
log(period=1/frequency)
Gives power (energy) at each sine wave frequency that makes
up the time series (analogous to spectrum of light)
7. Climate at a point: power spectrum of response to noise
Power spectra of response to
noise as a function of 
As  there is more
damping at high
frequencies
log(power)
=1 yr
=5 yrs
=25 yrs
log(period=1/frequency)
-The long  is, the ‘redder’ the power spectrum,
i.e., the greater the relative amount of long period variability.
7. Climate at a point: power spectrum of response to noise
Eqn for this power spectra
 
1 2f
=1 yr
2
- Can show 50% of power
in the spectrum occurs at
periods which are greater
than 2 x .
log(power)
P(f) 
a0
2

=5 yrs

=25 yrs

log(period=1/frequency)
Crucial point.
Long period variability is driven by short timescale physics.
7. Climate at a point: power spectrum of response to noise
Why the factor of 2?
i) Hand-wavy analogy with pendulum:
Physical timescale  
Oscillation period

l
g
l
2
g
ii) Real reason:

Real time-behavior
~ exp(t / )
2
Projected onto sines and cosines ~ exp(it),  
period

8. Example: the Pacific Decadal Oscillation
Dominant pattern of sea surface temperatures in North Pacific
8. Example: the Pacific Decadal Oscillation
Power spectrum of PDO index
Lots of variability at multi-decadal time scale
But….
Best fit is a red noise process with a  of 1.20.3 years
i.e., indistinguishable from an annual timescale.
8. Example: the Pacific Decadal Oscillation
PDO index
Random noise & 1.2yr memory
Random noise & 1.2yr memory
[n.b. note the apparent long timescale variability even with short memory]
Examples from the modern climate
The NAO
0
The PNA
0.1 0.2 0.3 0.4 0.5
Frequency (/yr)
Frequency (/yr)
The PDO
Small print:
In general, feedback strengths and sensitivity are functions of
the mean state:
For a black-body
F  T4
We linearized into A+BT:
A  T4, B  4T3
[Stefan-Boltzmann law]
Our basic climate sensitivity: 0=1/B, is a sensitive function of T.

Feedback analyses are completely linear, can be trouble when

strongly nonlinear behavior is being studied.
Feedback analyses rely on defining an appropriate reference state,
against which to test models/observations. You have to be careful
that the same reference state is being use when comparing
feedbacks.
2. Nonlinearities in climate feedbacks.
From basic analysis:
dR
R 
T  O(T2 )
dT
But can take
quadratic terms…
dR
1 d2R 2
3
R 
T 
T

O(T
)
2
dT
2 dT

Basic analysis:

With quad. terms…

1
G
1 f
1
G
T df
1 f 
2 dT
• How might f’s change with mean state?
2. Nonlinearities in climate feedbacks
a) Stefan-Boltzmann relation:
- longwave radiation ~ T4.
- flw ~ 1/(4 T3)
- negative feedback gets stronger with increasing T.
b) Clausius-Clapeyron relation:
- moisture content increases, so water vapor absorption bands fill up.
- fwv~ 1/ esat  d(esat(T))/dT
- positive feedback becomes weaker with increasing T.
Can show:
f~0.02 for 4oC climate change
Other nonlinear interactions:
clouds = fn(sea-ice); snow albedo = fn(clouds); wat. vap. = fn(clouds),
snow = fn(clouds); land sfc. albedo = fn(precip.); etc., etc.
Easy to imagine a few nonlinear interactions giving f~0.1 ...
3. Are we stuck with a skewed distribution?
• Could feedbacks vary enough with mean state
to remove skewness?
1
G
T df
1 f 
2 dT
Have that:
To remove skewness
would require:

df
2f

dT
T
In other words would need f ~ -1.2 per 1oC

(much stronger
than is possible)
• Skewed climate sensitivity distributions are intrinsic to system.
More curves changing playing with the uncertainty in climate feedbacks
More curves changing playing with the uncertainty in climate feedbacks