A Stategy for Predicting Climate
Sensitivity Using Satellite Data
Daniel B. Kirk-Davidoff
University of Maryland
Department of Meteorology
dankd@atmos.umd.edu
Talk Structure
•Background on the Fluctuation Dissipation Theorem
•Experiments using a Toy Model
•Model Description
•Results
•An additional complication
•Preliminary model-data comparison exercise
•Conclusions
The Fluctuation Dissipation Theorem
As discussed in Leith (1975), the Fluctuation Dissipation theorem states
that the infinitessimal impulse-response tensor g(t’) is equal to the lag
covariance matrix of the response lag t’, divided by its variance.
Climate change
M  T / F
Typically, the FDT is used to
derive macroscopic
properties of a system from a
theoretical model of its
statistical properties. Here,
the hope is that by
measuring rapid fluctuations
over a relatively short time,
we can derive the long-term
climate sensitivity of a model,
or of the real world.
L
Climate forcing
M   g( )d
Time over which U is
0
L
different from zero
M   U( )/U(0)d
0
L
where U  L1  T (t   )T (t)dt
0
Derivation
Starting from a
simple stochastic
differential equation:
dx
 Bx  
dt
It’s not hard to see that the
lag autocorrelation should
fall off exponentially:
C( )C(0)
From which it follows that:
1
 exp(B)
1
B   ln[ C( )C(0) ]

It turns out, though, that we
get better results by
integrating both sides of the
second equation,since this
averages over a lot of noise:


1
 0 C( )C(0) d   0 exp(B)d

1
 B   0 C( )C(0)1 d

Variations on FDT
1. Cionni et al.’s variation:
B
L
  0 R x ( )d
1
where


R x ( )   0 x(t   )(t)dt /  0 (t)(t)dt
2. Our variation:
B
1
L
L
  0 R x ( )d /  0 R ( )d
We next construct a half-dimensional toy model, with only surface and
atmospheric heat budgets, and a gray atmosphere.
•The model can be run very quickly (10 seconds for 20 years of model time
on a laptop computer under MATLAB).
•The model sensitivity can be varied by making either the atmospheric
emissivity or the surface albedo functions of temperature, vaguely analogous
to a water vapor or ice-albedo feedback, respectively.
•We force the model with AR1 noise applied to either the solar constant or the
emissivity (anologous to CO2 forcing), and compare sensitivities derived using
the Fluctuation-Dissipation Theorem with the true climate sensitity, easily
found by running the model to equilibrium.
Cs, Ca
Ts, Ta




0
S
S00
S0
ff
A
cA
r
surface and atmospheric heat capacity
surface and atmospheric temperature
albedo
Stefan-Boltzmann constant
atmospheric longwave emissivity
base emissivity
forced emissivity
atmospheric short wave absorptivity
solar constant
variable insolation
feedback parameters for albedo and longwave emissivity
AR1 noise, scaled to zero mean, standard deviation 1.
AR1 noise parameter
Random noise, flat distribution 0-1
dTs
1
Cs
  (TS2  TS ) S0(1  S )(1 ) Ts4  Ta4
dt
4
dTs2
Cs2
  (TS  TS2 )
dt
dT
1
Ca a  S0 s (1  ) Ts4  2Ta4
dt
4
   0  f (Ta  T )
Surface, atmospheric
Energy budgets.
Feedbacks on albedo and emissivity
  0  f (Ts  T )
 0   00  c2A
Forcing of emissivity and insolation
S0  S00  c1A
A(i 1)  cA A(i) r(i)

1 
2
Generation of AR1 noise
A  A 
 1 cA
 2(1 cA )
Equilibrium Climate Sensitivity for a
range of parameter values
Model Response to AR1 Solar Forcing
Model
Response for
Various Heat
Capacities
• For sufficiently small heat capacity or sufficiently long
time series, FDT-based methods gives excellent
predictions of the relative magnitude of model sensitivity.
•The length of the time series necessary for an accurate
prediction of sensitivity is comparable to the model’s
equilibration time scale for a given heat capacity.
Preliminary model-data
comparison
• We look at a forced (1% /year increase in
CO2) run of NCAR CCSM 2.0
• Use FDT to derive local sensitivity using CO2
data.
• Compare to “sensitivity” derived from surface
temperature and TOA solar forcing.
• Compare this to result for NCEP data.
• Future: use IR radiances from multiple
channels of AIRS data.
Conclusions
•The FDT or related measures based on lag-covariances give accurate
predictions of model sensitivity for a broad range of feedback and forcing
types.
•The length of the time series required for accurate computation of model
sensitivity increases with the time scale for the approach to equilibrium,
though this relationship becomes complicated when multiple surface heat
capacities are involved. Thus these measures are likely to be useful as a
short-cut to evaluating a model’s climate sensitivity.
•However, our results confirm that lag covariances are intimately connected
to climate sensitivity. This suggests that metrics involving lag covariance
of surface temperature and TOA radiative fluxes could be a very powerful
metric by which to compare models and data, and thus to estimate the
climate system’s true sensitivity to radiative forcing.
References
Bell, T.L., 1980: Climate sensitivity from fluctuation dissipation: Some simple model tests.
J. Atmos. Sci., 37: 1700–1707.
Chou, M.-D., M. J. Suarez, X.-Z. Liang, M. M.-H. Yan, 2001. A thermal infrared radiation
parameterization for atmospheric studies. NASA Technical Memorandum 104606,
vol. 19, 65 pp. Available at (http:// climate.gsfc.nasa.gov/ chou/clirad_lw).
Cionni, I., G. Visconti, and F. Sassi, 2004. Fluctuation dissipation theorem in a general
circulation model. Geophys. Res. Letts., 31:L09206, doi: 10.1029/2004GL019739
Emanuel, K.A., 1991: A scheme for representing cumulus convection in large-scale models.
J. Atmos Sci., 48: 2313-2335. Model code updated by the author in 1997, available at
ftp://texmex.mit.edu/pub/emanuel/CONRAD.
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Kirk-Davidoff, D.B., 2005: Diagnosing Climate Sensitivity Using Observations of
Fluctuations in a Model with Adjustable Feedbacks. Submitted to J. Geophys. Res.
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2026.
Acknowledgements
This work was inspired by conversations with John Dykema, Jim Anderson, Richard
Goody and Brian Farrell. It was made possible by start-up funds provided by
the University of Maryland