A Stategy for Predicting Climate Sensitivity Using Satellite Data

Daniel B. Kirk-DavidoffUniversity of MarylandDepartment 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

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M = ΔT / ΔF

M = g(τ )dτ0

τ L∫

M = U(τ )/U(0)dτ0

τ L∫

where U = L−1 T0

L∫ ( t +τ )T ( t)dt

The Fluctuation Dissipation TheoremAs 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 forcing

Climate change

Time over which U is

different from zero

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.

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Derivation

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dx

dt= Bx+ ξ

C(τ )C(0)−1 = exp(τB)

B = τ ln[C(τ )C(0)−1]

C(τ )C(0)−1dτ = exp(τB)dτ0∞∫0

∞∫

−B−1 = C(τ )C(0)−1dτ0∞∫

Starting from a simple stochastic differential equation:

It’s not hard to see that the lag autocorrelation should fall off exponentially:

From which it follows that:

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:

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−B−1 = R xξ0τ L∫ (τ )dτ

where

R xξ (τ ) = x( t +τ )ξ( t)dt0∞∫ / ξ( t)ξ( t)dt0

∞∫

Variations on FDT

2. Our variation:

1. Cionni et al.’s variation:

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−B−1 = R xξ (τ )dτ0τ L∫ / Rξξ (τ )dτ0

τ L∫

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 surface and atmospheric heat capacity Ts, Ta surface and atmospheric temperature albedo Stefan-Boltzmann constantatmospheric longwave emissivity base emissivity0 forced emissivityS atmospheric short wave absorptivityS00 solar constantS0 variable insolationf f feedback parameters for albedo and longwave emissivityA AR1 noise, scaled to zero mean, standard deviation 1.cA AR1 noise parameterr Random noise, flat distribution 0-1

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CsdTs

dt= γ(TS2 − TS )+

1

4S0(1−εS )(1− α )− σTs

4 +εσTa4

Cs2dTs2

dt= γ(TS − TS2)

CadTa

dt=

1

4S0ε s(1+α )+εσTs

4 − 2εσTa4

ε = ε0 + fε (Ta − Tε )

α = α0 + fα (Ts − Tα )

ε0 = ε 00 + c2A

S0 = S00 + c1A

A( i +1) = cAA(i)+ r( i)

A = A −1

2(1− cA )

⎡

⎣ ⎢

⎤

⎦ ⎥ 1− cA

2

Surface, atmospheric Energy budgets.

Generation of AR1 noise

Feedbacks on albedo and emissivity

Forcing of emissivity and insolation

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.

ReferencesBell, 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.

Haskins, R.D., R.M. Goody, L. Chen, 1997: A statistical method for testing a general circulation model with spectrally resolved satellite data. J. Geophys. Res., 102:16,563–16,581.

Kirk-Davidoff, D.B., 2005: Diagnosing Climate Sensitivity Using Observations of Fluctuations in a Model with Adjustable Feedbacks. Submitted to J. Geophys. Res.

Leith, C.E., 1975: Climate response and fluctuation dissipation. J. Atmos. Sci., 32: 2022–2026.

AcknowledgementsThis 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