Troy's Scratchpad

June 13, 2014

Estimating ECS bias from local feedbacks and observed warming patterns–example with GFDL-CM3

Filed under: Uncategorized — troyca @ 10:45 pm

1. Introduction

Typical energy balance methods, such as Otto et al (2013) or Masters (2014), have been frequently used for estimating equilibrium climate sensitivity (ECS) over the instrumental period.  Recently, several studies have raised the possibility that these may be biased, even if one were to precisely pin down the magnitude of the aerosol forcing and the current top-of-atmosphere (TOA) energy imbalance. In particular, the net feedback or radiative restoration parameter (typically represented by lambda) may be different when calculated over the instrumental period vs. the long-term, idealized CO2-only simulations.  One could partition this into two factors:

1) Inhomogenous Forcings. Certain forcings, such as aerosols, may be concentrated in higher-latitude regions, where the net feedback is smaller, thereby “cancelling out” more of the homogenous greenhouse forcing when looking only at the globally-averaged TOA imbalance.  This was raised recently in the Kummer and Dessler (2014) extension of the Shindell (2014) forcing “enhancement” from TCR to ECS (although as discussed previously, the enhancements in TCR and ECS are very different).

2) Time-Varying Sensitivity. Armour et al (2013) suggest that this may result from time-invariant local feedbacks, with only the spatial warming pattern changing over time and triggering these local feedbacks in changing proportions over time.

In fact, if we take the Armour et al (2013) view, both #1 and #2 have the same root cause – the differing spatial warming pattern between the transient, instrumental warming pattern case, and the long-term idealized 2xCO2 scenarios run in the models. 

2. Method 

If you will recall, one of my criticisms of Shindell et al (2014) was that it did not consider the observed spatial warming pattern.  Essentially, when determining forcing enhancements, it relied on the GCM output for:

a)  The ratio of the localized temperature response relative to the global response (essentially the relative heat capacity of each region)

b) The spatial warming pattern in the idealized GHG scenarios

c) The spatial warming pattern over the historical/instrumental period

While a and b may need to be relied upon, it seems as though one could use the information from c during the observed instrumental period, rather than relying on something that GCMs do poorly (regional warming projections, response to aerosols, and horizontal heat transfer).  Moreover, an ensemble of runs from a GCM will not be able to reproduce the exact variability seen in the single “realization” of the instrumental period.  To me,  it seems more likely that GCMs might be more correct and agree better on a and b than they would on c.  But even if they are not, at least it is one less assumption that needs to be made regarding GCM accuracy. 

By analogy, I think there is a reasonable method for determining the bias in ECS when calculated over the instrumental period.  Essentially, for different GCMs, we would calculate

a) The zonal net feedback according to the GCM.  We do this by first calculating the forcing for each zone using the difference in TOA imbalance for the last 25 years of the fixedSST regressions: sstClim4xCO2 – sstClim.  Then, for each zone we calculate the difference in TOA balance between the abrupt4xCO2 experiment and the piControl experiment, subtract the forcing calculated in the prior step, and normalize this by the local temperature increase calculated as the difference between abrupt4xCO2 and the piControl.  Note: this relies on the assumption of Armour et al (2013) that the increase in surface temperature in one region primarily affects the outgoing radiative response locally, rather than over some other area. At the end, we have an n x 1 column vector A,  with n being the number of zones.

b) The spatial warming pattern over the idealized scenario from the GCM that we will use as a baseline.  For example, if one were interested in how inhomogenous forcings may create a bias in energy balance estimates over the instrumental period, one might use the historicalGHG or abrupt4xCO2 scenario over a time period similar to the length of the historical period.  If, on the other hand, one wanted to figure out the combined effect of those inhomogenous forcings + time-varying sensitivity, one would use the spatial warming pattern of a run that had achieved radiative equilibrium after a doubling of CO2.  Regardless, we calculate an n x 1 column vector B, which consists of each of zone’s temperature change, normalized by the global temperature change.

c) The observed spatial warming pattern.  Similar to the step above, we calculate an  n x 1 column vector C, which consists of each zone’s observed  temperature change over the instrumental period, normalized by the total global temperature change.

Finally, we can then calculate what the expected bias in net feedback calculated over the instrumental period will be, according to each model, using the following equation:

Eq. 1 

Where the function “weight” simply multiplies each element in the vector by its area fraction of the globe.

3. Example with GFDL-CM3

First, here is the fixed-SST forcing calculation:


One thing of interest here is that the global forcing actually comes out to 7.2 W/m^2 for 4xCO2, which is more in line with the typical value of 3.7 W/m^2 for a doubling of CO2, and well above what is calculated in Andrews et al (2012) using the regression technique.  However, a look at figure 1 from Andrews et al (2012) highlights why:


As you can see, there is one point with a net imbalance above the intercept at 6 W/m^2 which Andrews et al (2012) takes as the forcing.  Clearly, since the regression is affected greatly by the larger T points and there is significant curvature, the regression method in this case underestimates the forcing.

Moving on, here is the calculation of the zonal net feedbacks:



Interestingly, this seems to differ from the local feedbacks of the CCSM4 model used in Armour et al (2013). However, the primary difference seems to be the peak response at –60 degrees that does not appear to be present in that CCSM4 model.

Next up, here are the normalized temperature responses for the different scenarios in GFDL-CM3, along with the observations from Cowtan and Way (2014).


What is obvious in the GFDL-CM3 historical pattern is the dip in temperatures between 30 and 60 degrees in the northern hemisphere, which seems a pretty clear indication of the aerosol response in that model.  In the observed historical record, this dip appears to be absent, and overall the observed warming pattern seems much more similar to the historicalGHG and abrupt4xCO2 scenarios than the historical scenario, apart from the lack of Arctic warming

Finally, if one where to calculate the feedback bias relative to the abrupt4xCO2 scenario, we find the following ratios:


This essentially isolates the expected forcing “enhancement” bias, as it is baselined against the idealized 4xCO2 run.  In this case, the above ratios are 0.95 (histGHG),  1.13 (historical), and 1.06 (observed).  This suggests that if one used the historicalGHG runs to estimate ECS, there might be a slight overestimate relative to the CO2-only run, as GFDL-CM3 includes the inhomogenous ozone forcing in their histGHG runs.  Were one to trust the relative strength of zonal feedbacks in GFDL-CM3, it suggests about a 6% underestimate of ECS from energy balance methods over the instrumental period due to the inhomogenous aerosol forcings.  However, ideally one could use this method with a variety of GCMs to identify the expected bias among a wider array of models.

Moreover, if one were interested in the bias of “effective sensitivity” (EFS) relative to ECS, one would ideally get a longer run of GFDL-CM3 with a doubling (or quadrupling) of CO2, and see how the warming pattern ended up after it reached equilibrium.  Unfortunately, I am not aware of any such output currently available for this model. 

***Final note: in my calculations, I initially performed the analysis using land and ocean zonal feedbacks and temperatures separately, rather than combined into one.  For GFDL-CM3, this did not seem to make much difference, and I did not readily have available separately gridded land and ocean temperatures from observations.  However, I seem to recall that the response was substantially different between land and ocean in Armour et al (2013), so perhaps it a more wide-ranging survey of GCMs it would be better to separate these out again.

Does anybody else think this a promising method for leveraging historical observations in estimating the potential bias in energy balance estimates?

Code and Data


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  3. Hi Troy,
    You might find your temperature-weighting approach runs into some problems because of the influence of meridonial flux. The individual zonal energy balance equations need to include these terms in order to make sense of the final steady-state solution, since locally, the net radiative flux balance does not go to zero at steady state. Consider n latitudinal zones, say. The local equations can be expressed as:-

    dHi/dt = Net radiative flux (for ith zone) plus heat flow in from zone( i-1) less heat flow out to zone (i+1)
    = Fi – λi * Ti + Qi-1 – Qi+1 for i = 1 to n
    Qi-1 = Qi+1 = 0

    When these equations are summed up, all of the Q terms (i.e. the meridonial heat transfer terms) sum to zero. However, they do have a serious impact on the averaging of local properties. Although the net heat flux, dHi/dt, must go to zero at steady state, the local energy balance is achieved by a balance of meridonial and radiative flux which can be both substantially non-zero.

    The average temperature gain at time t is the areal weighted sum of terms like (Fi + Qi-1 – Qi+1 – dHi/dt)/λi .
    At steady state, dHi/dt -> 0 and the effective equilibrium temperature is given by the areal weighted sum of (Fi + Qi-1 – Qi+1)/λi (Equation A)

    The total change in TOA net flux is equal to the total areal weighted forcing, F. Hence the overall aggregate feedback shown by the model will be equal to the total areal-weighted forcing divided by the temperature given by Equation A.

    In other words, if you want to calibrate two models against each other to assess the degree of bias in lambda, I don’t think that you can do so by just looking at the TOA net radiative flux in each zone.

    Comment by Paul_K — July 6, 2014 @ 12:07 am

    • Hi Paul,

      Thanks for you comments, and sorry I’ve been so slow to respond. I have wondered a similar thing myself, but I am not sure that horizontal heat transfer must explicitly be taken into account in order to assess the bias I am describing here.

      First, I think we are in agreement that theoretically (using the same assumptions as Armour et al) one can diagnose λi for each zone without consideration of Q, right? After all, while Q affects the temperature increase, λi is only concerned with the change in outgoing flux normalized by that temperature increase. I don’t believe I am assuming that there is radiative equilibrium within each zone at TOA, as we are primarily dealing with the change in that outgoing flux, not its value in absolute terms. Moreover, it seems to me that any change in Q (vs, for example, the shorter timescale vs. equilibrium case) would implicitly be included in those different spatial warming patterns that we use for weighting the zonal TOA feedbacks.

      “In other words, if you want to calibrate two models against each other to assess the degree of bias in lambda, I don’t think that you can do so by just looking at the TOA net radiative flux in each zone.”

      I agree with this sentence, but isn’t this the point of then weighting by the normalized temperature increase in each zone for different states? Essentially I am saying “if we take from model X the spatial pattern of T at some point (either equilibrium or CO2-only idealized scenarios) and the spatial pattern of TOA feedbacks, what does the observed spatial warming pattern imply about the bias in lambda according that model?”. I suppose I don’t understand what assumptions I am making here that are not present in Armour et al (2013)?

      Regardless, I will try to whip up some simplified demo with only 2 zones and some value of Q that either changes or remains constant, to see if I can satisfy myself whether failing to account for Q is significant or not, unless you have something similar already handy?



      Comment by troyca — July 10, 2014 @ 12:52 pm

      • Troy,
        Thanks for your response. Having considered the matter further (in light of your response), I withdraw my original objection.
        My main reservation (now) is that the approach rests on the assumption that local feedbacks are invariant between model cases for the same model. I do not know how robust this assumption is.

        Comment by Paul_K — July 11, 2014 @ 4:27 pm

      • Hi Paul,

        I too am unsure about the assumption…primarily I wonder whether the feedbacks are indeed time invariant for each zone, which is the primary assumption of Armour et al. I have been planning on running a few quick tests on this soon…

        Comment by troyca — July 12, 2014 @ 10:14 am

  4. Troy,
    I’ve just realised that part of what I wrote above doesn’t scan. When I wrote Qi-1 = Qi+1 = 0, what I should have said was that Qi-1 = 0 when i = 1 and Qi+1 = 0 when i = n; these are just the end-member boundary conditions. Sorry for any confusion caused.

    Comment by Paul_K — July 8, 2014 @ 3:26 am

  5. Please see [1] N. Schaller, et al of Knutti’s groups article ‘the sensitivity of modeled energy budget and hydrological cycle to CO2 and solar forcings , Earth system dynamics -4, 256-266, 2013, and
    [2]Environ. Res. Lett. 9 (2014) 031003 (7pp) doi:10.1088/1748-9326/9/3/031003
    Perspective Implications of potentially lower climate sensitivity on climate projections and policy
    Joeri Rogelj, Malte Meinshausen, Jan Sedláˇcek and Reto Knutti

    These try to explain the possibility of lower effects from CO2 by increased water vapor – kept in atmosphere with a longer residence time than 9 days [Trenberth used 8 days in his ’97 GEB, – to fix latent heat at 78 or 80 w/m2 – non-moving value.. while Graeme Stephens calls it 88+/-10 W.m2 [2012 article on GEB in Nature]..

    Comment by Neale R Neelameggham — August 3, 2014 @ 10:19 pm

  6. We can see in the chart that the temperature is rising slowly so what do we expect in terms of global warming?
    We all see the changes outside so we need facts to stop this trend not only words.
    It is time to learn a few tips that can help us stop the temperature to reach higher values.

    Comment by Angela — March 5, 2015 @ 3:44 am

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