Energy budget estimates explicitly using feedbacks

This post was partly motivated by Michael Tobis’s recent post in which he rebuts a doom-monger by pointing out that feedbacks can’t really produce some kind of runaway process in the next few decades, and by my uncertainty as to whether or not energy budget estimate for the ECS properly account for feedbacks. In a simple sense they do (I was just confused) but that doesn’t mean that there aren’t other issues with these estimates.

The standard energy budget method for estimating the transient climate response (TCR) and equilibrium climate sensitivity (ECS) is
where F2x is the change in anthropogenic forcing after a doubling of CO2, ΔT is the change in temperature over the time interval considered, ΔF is the change in anthropogenic forcing over that same time interval, and ΔQ is the system heat uptake rate (i.e., the current energy imbalance). The TCR equation has always made sense to me (if we know how much the temperature has changed for a given change in anthropogenic forcing, then we can estimate how much it should change when the change in forcing is equivalent to that from a doubling of CO2). The ECS equation essentially uses the current heat uptake rate to estimate the warming in the pipeline for the change in anthropogenic forcing, but I’ve always been unsure as to whether or not it properly accounts for the feedbacks due to the warming in the pipeline.

Being a simple physicist, I thought I would redo the energy budget estimate more crudely. All the values I’ll be using are from the top row of Table S2 in Otto et al. (2013)’s Supplementary Information. If the temperature changes by an amount ΔT, then the outgoing flux should increase by
where the 0.62 is an estimate of the surface emissivity. Using Teff = 288 K, and ΔT = 0.75 K gives ΔFout = 2.53 Wm-2. If the heat uptake rate (energy imbalance) is 0.65 Wm-2, then that means that the net change in radiative forcing is 3.2 Wm-2. If the change in anthropogenic forcing over the same time interval is ΔF = 1.95 Wm-2, then that means feedbacks must be providing a radiative forcing of ΔFfeed = 1.23 Wm-2.

If we assume that the feedback forcing depends linearly on ΔT (which seems reasonable in this simple approximation) then we can assume it depends linearly on ΔFout (which is okay for small changes in ΔT). In the absence of feedbacks, a doubling of CO2 would – for equilibrium – require an increase in outgoing flux of ΔFout = 3.44 Wm-2. Therefore, if ΔFfeed = 1.23 Wm-2 when ΔFout = 2.53 Wm-2, the feedback will be ΔFfeed = 1.67Wm-2 when ΔFout = 3.44Wm-2. However, as the temperature rises in response to this feedback, it will produce an additional feedback. As it rises in response to this additional feedback, it will produce another additional amount of feedback. Essentially it is an infinite sum. If F is the ratio ΔFfeed / ΔFout, the net effect of feedbacks will be
If F < 1, then this converges. In the example I've given here, F = 0.49. You can calculate the sum yourself, and should get 1.96. Therefore, at equilibrium, the influence of feedbacks means that the total change in forcing (anthropogenic + feedbacks) should be 1.96 x 3.44Wm-2 = 6.74Wm-2. If I now use the second equation in this post with Fout = 6.74Wm-2, and solve for ΔT, I get 2 K. So, exactly the same as Otto et al. (2013). So, it seems my concern that the energy budget estimate for ECS didn’t properly account for feedbacks was wrong, but I think I’ve shown that you can get the same result by explicitly considering feedbacks (which rather scuppers those who keep claiming that feedbacks have yet to manifest themselves). It does, however, nicely illustrate how the energy budget estimate for the ECS very explicitly assumes that feedbacks are linear.

Having done this, I realised that I can now constrain one more parameter in my two-box model, and that is the relationship between temperature change and feedback forcing. In that model I used Ffeed = 1.8ΔT, whereas this analysis suggests it should be Ffeed = 1.6ΔT. If I use the latter in my two-box model, the only free parameter now is β which I adjust slightly (β = 0.2) so as to recover a good fit. With these parameters, I get a TCR of 1.6 K and an ECS of 2 K (as one might expect).

Anyway, there’s not really much to this post, other than my attempt to check whether or not energy budget estimates properly account for feedbacks (it seems that they do). I should add that I’ve completely ignored any error analysis or uncertainties, so I’m not claiming that these estimates are reasonable values, simply that if I use a consistent set of values I can recover the same ECS if I use the standard energy budget method, my feedback method (which is the same as the energy budget method, but just written out in a way I can understand) and my two-box model. When I did play around with some of the values, the results can change quite substantially which may indicate how sensitive such methods are to the values used, but I haven’t done some kind of detailed study to really address this. Anyway, as usual, feel free to correct my working/thinking, through the comments.

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24 Responses to Energy budget estimates explicitly using feedbacks

  1. So, what do you think about the retraction of Lewandowsky paper?

  2. Rachel says:

    Victor! This is *very* naughty.

    What do you think? 🙂

  3. I think the retraction is the best thing since sliced bread.

    It is an open access journal, so the article is still available. The journal performed the peer review and the retraction was not for academic reasons, so that review is still valid. In other words, the paper is still fully intact.

    On the other hand, the retraction has produced much publicity and therewith helped spread the ideas of the authors. And it showed the kind of people the climate dissenters are and that does not look good.

    A small negative could be that is should not become custom to suppress academic freedom with threats of juridical trouble, but I would not expect a scientist to do what the anti-scientists did. A scientist would ruin his reputation by behaving that way.

  4. Rachel says:


    I hadn’t thought of it in that way. Perhaps it is a good thing. Do you know what it is in that paper that is potentially libellous? The publisher must have had legal advice which suggested they could potentially be sued but if that’s the case, then so can the authors and perhaps also the University. And as you say, the paper still stands.

  5. I read somewhere that Lewandowsky asked his juridical department and they had no problem with putting the article on the university homepage for download. It did not sound like they are that easy to intimidate. And the University is probably more motivated to invest in academic freedom as a small commercial publisher.

    I have not read the paper fully yet, but did not notice anything libellous yet. It is also hard to imagine that an academic would something more damning as simply citing what these people write. There is no need to exaggerate that.

  6. Rachel says:

    This is obviously how one discusses something contentious on a blog: make the post about something completely different, put lots of equations in it and then discuss the contentious topic in the comments.

    If we get in trouble for this, I’m saying you started it. 🙂

  7. I thought the equations were there to have less interference of people that do not like science. 🙂

  8. Steve Bloom says:

    If so, so far, so good. 🙂

  9. jsam says:

    I passed first year Statics and Dynamics by remembering F=ma and you can’t push on a rope. So much for equations.

    As for Lew,

  10. jsam,
    Just to be clear, F here is flux, not force 🙂

    I’ve been trying to avoid discussing the recursive fury paper as it just seems to be a bit too contentious. As I understand it, the criticisms are that they didn’t put some survey up at SkS even though they said they had. They also analysed online comments for consipracy ideation and included the comments in their supplementary information, so people felt that they had been identified as conspiracy theorists. It’s an interesting issue. The comments are public so should be available for study. If you do so, then presumably one should then include the data (or, at least, that’s what people keep saying). So, it does seem different to some kind of private study where the participants would expect anonymity. Having said that, I can see why people would not be happy at being identified as espousing conspiracy ideation. However, given that those who’ve complained the most seem (as far as I can tell) to have done exactly that, I’m somewhat less sympathetic.

  11. ATTP,
    The feedback strength you consider is a totally separate issue from the difference between TCR and ECS. Your approach is based on the assumption that feedbacks are fast enough to be fully present in both TCR and ECS. Similarly you assume that the feedbacks are fully present in the empirical numbers you use in your calculations.

    The feedbacks are fast, if they are only atmospheric feedbacks to the warming of the surface. There’s one weakness in the derivation of the formulas as it assumes implicitly that the relationship between surface temperature change and feedback strenght remains constant. For a while we are almost certainly in a linear range that applies regionally, but different regions warm at different rates and different regions have different regional strengths of feedback. Furthermore the couplings between regions change (e.g., circulation of air between oceans and continents). Therefore the overall feedback coefficient is not constant even if regional feedback coefficients remain constant. This is the point of the recent paper of Shindell. Many seem to think, however, that the effect is rather weak, and might be dismissed without significant error in the results. As far as I can see, there may also be competitive effects to those discussed by Shindell making even the sign of the overall effect dependent on the results of a comprehensive analysis. As far as i have understood neither Shindell nor anyone else has looked carefully on all known effects, let alone estimates of the importance of less known effects.

    A further important issue is the slow feedbacks. Slow means here too slow to contribute with their full strength to the existing relationship between the surface temperature and the energy balance at TOA. Surface albedo feedbacks from both the snow cover and from vegetation belong to slow feedbacks, which are mostly missed by your analysis – and by the definition of ECS given in your post. What’s denoted by ECS in those formulas is often called effective climate sensitivity rather than equilibrium climate sensitivity as it refers to equilibrium only under the inaccurate assumptions that total feedback strength is equal to the apparent feedback strength that can be inferred from observations.

  12. Pekka,

    There’s one weakness in the derivation of the formulas as it assumes implicitly that the relationship between surface temperature change and feedback strenght remains constant.

    Yes, and that was why I mentioned that it was linear, although maybe I should have said constant instead. I agree, this is one of the issues with this simple method.

    This is the point of the recent paper of Shindell.

    Yes, and maybe I should have linked to that in this post. Sometimes there’s a limit to how many caveats one can include in a single blog post 🙂

    A further important issue is the slow feedbacks.

    Again, I agree. The ECS is typically defined as a response to fast feedbacks only. So, in reality, the slow feedbacks are important and aren’t captured by ECS estimates. This is one argument why the low estimates, in my view, are even weaker than people will admit. As I understand it, most of what these low-end estimates ignore (non-linearities, regional variations, slow-feedbacks) all act to increase the equilibrium temperature. So, even if the fast-feedback response is on the low end, the actual response will be higher (I think).

  13. Given we’re discussing the Recursive Fury paper here, this is probably the most balanced view I’ve come across so far.

  14. Anders, you have referral power, the above recursive fury link to Planet 3 is dead and says:
    “Resource Limit Is Reached The website is temporarily unable to service your request as it exceeded resource limit. Please try again later.”

    Clearly not a libertarian webserver. They do not have resource limits. 🙂

  15. Victor,
    It was doing that to me this morning before I added that comment, so not me 🙂 . Seems to work for me now.

  16. Yes, I got though after waiting a long time. Planet 3 mentions one point I forgot above, that the paper is not an enormous contribution to the scientific understanding and that also in that respect the retraction is not bad. The paper is more aimed at the general public and the retraction helps spread the ideas in the general public and shows the weird kind of comments people on the other side make. If I make add a conspiracy theory, conspiracies do exist, maybe the authors encouraged the journal to make the retraction.

  17. Victor,
    Yes, there’s a wonderful irony to this whole situation 🙂

    I think I also agree with Planet 3 that it would have been interesting as an article or a blog post, but probably isn’t really something that one would typically submit for peer-review. I am, however, no expert at the social sciences and, as far as I can tell, Lewandowsky is well-regarded in the field and is regarded as someone who is more rigorous than most, so I can’t say for sure if this is something that would or would not be suitable for peer-review. I did, however, find Michael Tobis’s argument quite compelling.

  18. BBD says:


    And then the lizards launched a DOS attack on the P3 server! This is just the beginning! First they came for the equations, then the journals! I warned you all but nobody would listen!

  19. BBD, just write a paper about my conspiracy ideation.

  20. andrew adams says:

    I am, however, no expert at the social sciences

    From the comments (at least the critical ones) I’ve read about Lew’s paper it seems that everyone is an expert in social sciences (just like everyone is an expert on tree rings).

  21. Steve Bloom says:

    Given we’re discussing the Recursive Fury paper here, this is probably the most balanced view I’ve come across so far.

    Hmm, is that a recommendation? Does balance confer correctness in some manner?

  22. Steve,
    No, just an opinion. I happened to read it and think “fair enough, that sounds reasonable”. I haven’t followed the Recursive Fury saga, partly because it just seemed too absurd to actually bother working out, partly because I couldn’t access the paper to see what the issue was, and partly because most of it happened before I started doing this. Do I think what it presents is about right? Based on what I’ve experience, sure. Conspiracy ideation abounds, as far as I can tell.

  23. Pingback: Another Week of Global Warming News, March 30, 2014 – A Few Things Ill Considered

  24. kap55 says:

    Essentially the same result can be inferred from this graph:

    … where temperature is seen to rise 2.11°C from doubling CO2.

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