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Quantifying the Sources of Inter-Model Spread in Climate Sensitivity. Peter Caldwell, Mark Zelinka , Karl Taylor, and Kate Marvel LLNL. Prepared by LLNL under Contract DE-AC52-07NA27344. LLNL-PRES-661065. Motivation.

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Quantifying the sources of inter model spread in climate sensitivity

Quantifying the Sources of Inter-Model Spread in Climate Sensitivity

Peter Caldwell, Mark Zelinka, Karl Taylor, and Kate Marvel


Prepared by LLNL under Contract DE-AC52-07NA27344.



  • Equilibrium climate sensitivity (ECS, the global average equilibrium surface temperature change after doubling CO2) is one the most important measures of global warming

    • since most climate impacts scale with ECS

  • Isolating the source of ECS differences between global climate models (GCMs) is a natural way to reduce uncertainty about global warming

    • This was done in Dufresne + Bony (JClim, 2008, hereafter DB08), who found that cloud feedbacks dominate inter-model spread

      Here we point out issues with their approach and re-evaluate their conclusions

Db08 method
DB08 Method:

Goal: Write ECS as a sum of terms representing contributions from each feedback process. This can be done as follows:


net feedback, λ=Σλi


Inter-model mean forcing

Planck feedback

In essence, Planck ΔT is calculated first and the remainder of ECS is partitioned amongst remaining feedbacks with size proportional to relative magnitude

Problems with db08 method
Problems with DB08 Method

  • It is arbitrary – why multiply by λPl/λPlinstead of λ/λ or some other quantity?

  • Even if var(λi)=0, var(Ti)≠0 because all T contributions are scaled by var(1/λ)

An alternative approach
An Alternative Approach

using identity:








  • Neatly (and exactly) partitions forcings and feedbacks

  • var(Ti)=0 whenever var(λi)=0


  • Constant term makes scheme unsuitable for looking at inter-model AVE

Data sources
Data Sources

  • Radiative response to Planck (Pl), water vapor (WV), lapse rate (LR), shortwave (SW) and longwave (LW) clouds, and surface albedo (alb) changes are computed using radiative kernels

  • Responses are broken into feedback and forcing components using the slope and y-intercept (respectively) of model response to abrupt quadrupling of CO2 following Gregory et al (2004)

  • ECS is the ratio of net forcing and feedback terms calculated above

  • Analysis are based on 17 CMIP5 models with SW, LW, and net clear sky kernel errors of < 10%

Problem with db08 interpretation
Problem with DB08 Interpretation

DB08 consider inter-model standard deviation in each Ti (see figure), but formally

so covariances between feedbacks should also be considered!

Fig: Normalized inter-model standard deviation of ECS partitioning. Reprinted from DB08.

Covariance between water vapor/lapse rate is well known (e.g. Cess, 1975) and covariance between other λi has been noted recently (Huybers, 2010; Mauritsen et al, 2013)

Visualizing the covariance matrix
Visualizing the Covariance Matrix

  • Covariance matrices for DB08 and new methods are shown to the right

    • entries are normalized by var(ECS) so matrices sum to 1

    • color also just reflects magnitude

    • since covariance matrix is symmetric, bottom triangle is removed and upper triangle is doubled.

Fig: Covariance matrices for both ECS partitionings. Details are listed above.

Covariance matrix results
Covariance Matrix Results

  • DB08 and the new partitioning give very different answers

    • DB08 terms are smaller due to division by λPland cancelation between covaryingλiand λ

    • Cld feedback is actually not dominant in DB08!!!



Fig: Covariance matrices for both ECS partitionings.

Covariance of raw feedbacks
Covariance of Raw Feedbacks

  • Variance in λ and ECSare not the same thing

    • they shouldn’t be since ECS=-F/λ

    • but degree of dissimilarity with DB08 is alarming

Fig: Covariance matrices for ECS partitionings(right) and for λ=Σλi (left)

But vial says forcing isn t important
But Vial says Forcing isn’t Important!

Since Vial doesn’t separate forcing variability from λ variability:

Vial eq 24:

Divides by λPlinstead of λ

Uses F instead of F


Fig: After partitioning ECS into components listed on x-axis, Vial computes Inter-model standard deviations and normalizes by the standard deviation of ECS. This is Fig. 6 from Vial et al. (2013)

Getting rid of off diagonal terms 1
Getting Rid of Off-Diagonal Terms (1)

  • Combining terms which covary helps ‘orthogonalize’ the covariance matrices

    • DB08 and Vial et al (ClimDyn, 2013) merge WV+LR and SW+LW

    • combination doesn’t solve the problem

    • Combining terms we expect to cancel simplifies interpretation but reduces information content… a tradeoff

Fig: Covariance matrices as on previous slides, but with WV+LR and SW+LW cld feedbacks combined

Getting rid of off diagonal terms 2
Getting Rid of Off-Diagonal Terms (2)

Held and Shell (JClim 2012, hereafter HS12) note that relative humidity (RH) tends to stay constant as climate changes, but water vapor feedback is computed as the change in specific humidity

  • this results in big WV increases compensated by big temperature increases at the surface (Pl) and aloft (LR)

  • Defining WV feedback using specific humidity results in a physically unrealizable (supersaturated) planet

  • Computing WV, Pl, and LR based on changes in RH decouples moistening and warming feedbacks

Impact of hs12 definitions
Impact of HS12 Definitions

  • covariances involving WV and LR are reduced (as expected)

    • but off-diagonals are still non-negligible

  • DB08 and new partitioning methods agree better under HS12

    We think HS12 is better because it:

  • Decreases differences between methods

  • Reduces the importance of covariance terms

  • Makes physical sense

Fig: Covariance matrices shown previously (left) and using HS12 definitions (right)

So what is the source of ecs spread
So What is the Source of ECS Spread?

corr(λSW, λnet)2=0.69 (i.e. λSWexplains ~70% of λnet spread)so λSWshould play a dominant role in any reasonable partitioning of ECS

Fig:ECS calculated as –F/λnet (in blue) and as calculated using multi-model averages for all feedbacks except λSW


  • There is no unique way to partition ECS

    • different methods yield different conclusions

    • using feedbacks directly gives very different impression than using DB08

  • Attributing inter-model spread in ECS requires consideration of covariances,not just variances

    • Combining feedbacks with significant covariance (e.g. WV+LR and SW+LW) reduces but does not remove the importance of covariances

  • Inter-model spread in forcing is substantial, in contrast to Vial’s conclusion

  • Using constant RH (the HS12 approach) rather than constant specific humidity (as typical) is a better way to compute λWV, λLR, and λPl

    • it makes physical sense

    • importance of covariance terms is reduced

    • it reduces differences between ECS partitioning methods

  • Because λSW explains ~70% of inter-model spread in λnet, we conclude that it must be the dominant source of ECS spread regardless of the details of how ECS is partitioned

Include comparison against db08 and vial papers here
Include comparison against DB08 and Vial papers here?

  • covariance matrices using the same CMIP3 models DB08 used

  • covariance matrices using the same CMIP5 models Vial used

    • mention the specific conclusions each paper makes in their abstract/etc and talk about whether we believe them.

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