Non-linear feedbacks

I thought I would just briefly mention a recent paper by Bloch-Johnson, Pierrehumbert & Abbot called Feedback temperature dependence determines the risk of high warming. As I understand it, the basic idea is to consider what would happen if the feedback response has a temperature dependence. If the feedback response is linear, then you can estimate the climate sensitivity, \lambda, at any time using

\lambda = -\dfrac{\Delta F}{\Delta T}.

Credit : Figure 1 from Bloch-Johnson et al. (2015)

Credit : Figure 1 from Bloch-Johnson et al. (2015)

However, it is clear that feedbacks do have a temperature dependence. The Planck response (dF = 4 \epsilon \sigma T^3 dT) is clearly temperature dependent. What we don’t know – given all the feedbacks – is the strength of this possible non-linearity. What Bloch-Johnson et al. do is simply assume that the climate sensitivity has a non-linear term,

-\Delta F = \lambda \Delta T + \alpha \Delta T^2.

The figure to the right shows how different values of \alpha influences the response to a change in forcing. The dashed line is the linear response. Negative \alpha values reduce the response, while positive ones increase it. There are also combinations of \alpha and \lambda that could lead to runaway warming. Most GCMs, apparently, suggest that the linear approximation works well. There are some – as shown in the right hand panel above – that do, however, indicate a non-negligible \alpha value.

Credit : Figure 3 from Bloch-Johnson et al. (2015)

Credit : Figure 3 from Bloch-Johnson et al. (2015)

As shown in the figure above, the paper also considers the impact of a possible non-linearity on observationally-based estimates of climate sensitivity. Negative values reduce both climate sensitivity and the range, while positive values do the opposite. Something to bear in mind, though, is that most observationally based analyses assume feedbacks are linear, and so – by definition – cannot be used to determine if they’re not.

Anyway, that’s all I was going to say. As I understand it, the point of the paper is not to suggest that feedbacks will be non-linear, but to illustrate the impact of them being non-linear. Additionally it illustrates that such a non-linearity would not be evident in observationally-based studies. In a sense, it seems to be a Black swan type of argument. If feedbacks are non-negligibly non-linear, then this would not yet be evident, but could result in the probability of high climate sensitivity being much greater than we currently think. This is especially true if we do continue to follow a high emission pathway, which could ultimately much more than double atmospheric CO2.

Update : I had an email from Ray Pierrehumbert with some additional context. I’ve added it below. Bear in mind that the figure is simply illustrative of how a bifurcation might work, not from some kind of actual calculation.

Although the nonlinear term can be quite important even for mid-range IPCC type climate sensitivity, when you go out on the fat tail (say, 8C per doubling) then the nonlinearity becomes not just a modification of the story, but the WHOLE story — unless the world manages to limit radiative forcing to very small values. So, consideration of fat tails and nonlinearity/bifurcations are inseparable. Worse, when there is a bifurcation, the local analysis can tell you that you jump but it doesn’t tell you where you land — could be just a transition to a state a few degrees higher, but could be a Venus-type runaway (not that I think the latter is likely, but it can’t be settled based on the kind of local analysis people usually do). In other words, not just a black swan, but potentially a whole flock of black swans.

Figure from Ray Pierrehumbert

Figure from Ray Pierrehumbert

This sketch may be a useful visualization. The vertical axis represents climate state (think global mean temperature) the horizontal represents control knob (think CO2) The arrows represent two possible jumps, compatible with the same local behavior. As one gets close to the fold, the linear analysis becomes increasingly meaningless.

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27 Responses to Non-linear feedbacks

  1. There’s a subtlety to this whole non-linearity issue. This paper considers the impact of the feedbacks having an explicit temperature dependence. However, it is also likely (as pointed out in the Isaac Held post I link to here) that there will be regional variations that mean that a globally averaged feedback response could have an apparent temperature dependence. For example, if polar regions warm more rapidly as the system approaches equilibrium, then an observationally-based study that considers only the initial part of a warming period, will underestimate climate sensitivity even if the feedbacks aren’t actually non-negligibly non-linear. I hope I’ve explained that properly.

  2. If I get you right, aTTP, what you’re saying is that, when compared with previous estimates, this paper suggests there’s the potential for more uncertainty with regards to eventual warming. This suggests that we need to be even more cautious than we aren’t being because we’re heading into a greater unknown.

    Maybe our species will end up being named by some intelligent successor in future times ‘Homo Sapiens Inprudens’.

  3. I just realized how closely my recent comment

    is related to the post of Isaac Held that you refer to.

    I like indices that tell about temperatures in regions where most people live and prefer excluding very sparsely populated regions that add disproportionally to the “random” variability in the calculated average temperatures or have other issues that make the value less well defined. The areas that I would exclude for those reasons include Arctic, Antarctic and parts of Siberia. (Thus I like HadCRUT better than the alternatives.) Problems that these regions have include variable extent of ice cover and the extreme variability in surface temperatures caused by very common states of temperature inversion. Finland is Northern enough for making it clear, how unstable the surface temperature is during a winter time inversion.

    It’s, of course, important to study also areas that I would exclude from the temperature index, but the indicators used for those areas might be based on something else than the surface temperature. Any single index has it’s problems. In science there’s no reason to restrict analysis to a single index (and it’s seldom done).

  4. John says:

    Reblogged this on jpratt27.

  5. John,
    Yes, I think that’s roughly the situation. This is arguing that there could be fat tails, so it’s not just a high impact, low probability event, it could be a high impact event that is more likely than we currently think,


    Any single index has it’s problems. In science there’s no reason to restrict analysis to a single index (and it’s seldom done).

    Yes, I agree. However, there is always a balance between trying to present nice simple metrics that might not be ideal and introducing so much complexity that it’s hard to explain the overall picture.

  6. Eli Rabett says:

    Pekka, you are assuming that what happens in the sparsely populated areas has no effect on the heavily populated areas, also you are making an interesting argument wrt urban heat islands.

  7. Arthur Smith says:

    Just a note here – the Bloch-Johnson et al. figure 1 showing climate model nonlinear behavior is under the condition of an initial 32x CO2 forcing, so a very large forcing change – I expect almost any realistic model would be nonlinear at that level of change. Their figure 3 which you show is displaying climate model sensitivities under much less forcing – a 4x CO2 impulse (the bottom axis shows the delta T under 4x CO2 for a range of CMIP5 models).

  8. Eli,

    No I don’t, but the direct influence of the temperature is local. The indirect effects may be large, but they cannot be described well by the temperatures. I prefer using the temperature index for the temperature and discussing the other phenomena using measures that describe them like sea level of ocean acidification.

    The other point that I have is that the index should be defined in a way that minimizes noise that may hide trends.

  9. Arthur,
    Yes, that’s a good point. One thing I think they were trying to also suggest is that currents GCMs probably can’t properly model such large changes anyway. They will probably be non-linear, but can they properly represent these non-linearities? As I understand it, a thrust of the paper is to argue that if we do increase anthropogenic forcings substantially, then these non-linearities will become non-negligible. Whether or not that means a small change from the linear expectation, or a large one, is what’s currently unknown.

  10. “What Bloch-Johnson et al. do is simply assume that the climate sensitivity has a non-linear term,”

    Of course they did.

    Why not assume that term is alpha * T ^ 100 ???

    What feedback physically is dependent on the square of temperature?

    Soden and Held 2006 identify the major feedbacks as:
    Water Vapor ( largest )
    Clouds ( very uncertain )
    Albedo ( smallest positive )
    Lapse Rate ( negative )

    Which ones are we suggesting are dependent on the square of temperature change?

    What happens if that non-linearity is the square root rather than the square of temperature change?

  11. TE,
    Because if it’s non-linear, the next order is \Delta T^2, not \Delta T^{100}.

    Consider the Taylor series of the Planck function.

    F(T + dT) = F(T) + \dfrac{dF}{dT} dT + \dfrac{1}{2} \dfrac{d^2F}{dT^2} dT^2 +....

    We know that F(T) = \epsilon \sigma T^4, so you can actually solve this to get dF

    dF = F(T + dT) - F(T) = 0 + 4 \epsilon \sigma T^3 dT + 6 \epsilon \sigma T^2 dT^2.

    So, if we want to keep the higher order terms in the Planck response, it has the form of \alpha dT^2. That would be true for any feedback. Of course, the coefficient of the higher order terms could be zero, but they do consider that possibility.

  12. Pingback: “Feedback temperature dependence determines the risk of high warming” | Hypergeometric

  13. This is the non-linear impact of multiple doublings of CO2, inducing the water vapor rise along with it:

    Here is an alternate differential form

  14. John Hartz says:

    ATTP: Please translate your OP from physics-speak into plain English.

  15. Pekka.

    I could not agree more. well put.

  16. BBD says:

    John H

    The paper is about the impact of ‘non-linear’ feedbacks on climate sensitivity. If the strength of the feedback response is always directionally proportional to the change in forcing, it is considered ‘linear’. If instead it increases disproportionally because it is boosted by rising temperatures, then the feedback would be ‘non-linear’. This is something that we would not see now, at lower temperatures. It is something that would not show up in so-called ‘observational’ estimates of sensitivity, biasing them low.

  17. Yes, that’s a good point. One thing I think they were trying to also suggest is that currents GCMs probably can’t properly model such large changes anyway.

    I’ve sometimes wondered at what point in climate forcing do non-linearities invalidate the extrapolations and calibrations I understand are sometimes done to make models calculable with reasonable mesh sizes and in reasonable times. In particular, when are historical calibrations of less use and ab initio physics all that’s left to be done, as incredibly difficult as that is. I don’t expect we are there now, but this may occur some time in the future. Did it occur during the Siberian coal-burning setup at the end of the Permian?

  18. John Hartz says:

    BBD: Your explanation is consistent with my understanding of the OP.

    The unanswred question that I have is:

    If temperature feedbacks are indeed nonlinear, will “Climate Sensitivity remain constant or will it change over time?

  19. Kevin O'Neill says:

    “The areas that I would exclude for those reasons include Arctic, Antarctic and parts of Siberia.”
    “The other point that I have is that the index should be defined in a way that minimizes noise that may hide trends.”

    Given arctic amplification is *expected* – then wouldn’t excluding the arctic work to increase noise at the expense of the trend?

  20. BBD says:

    John H

    If temperature feedbacks are indeed nonlinear, will “Climate Sensitivity remain constant or will it change over time?

    Sensitivity will change as it warms. It will increase.

  21. Andrew Dodds says:

    BBD –

    Or it may be that the range of possible stable climate states is not continuous. Which would at least explain why arriving at a narrowly defined value for ECS is probably impossible,

  22. BBD says:


    Yes, that’s certainly possible and climate behaviour across the Cenozoic could even be seen as suggestive that this is the case. Hyperthermals, the Oi-1 glaciation, late Oligocene warming, Mi-1 glaciation and the Plio-Pleistocene glacial cycles all hint at inherent instability and thresholds.

  23. Brian Dodge says:

    “The study of past warm climates may not narrow uncertainty in future climate projections in coming centuries because fast climate sensitivity may itself be state-dependent, but proxies and models are both consistent with significant increases in fast sensitivity with increasing temperature.”
    State-dependent climate sensitivity in past warm climates and its implications for future climate projections; Rodrigo Caballero and Matthew Huber;

    I also wonder about rate dependence; would the response curve of significant climate variables(surface & lower troposphere temperature, sea surface temperature, ice volumes/extents/seasonality, sea level) be nonlinearly different for different rates of CO2 increase, and how long would it take for the curves to converge. If you compared an instantaneous CO2 doubling to scenarios taking 100 or 500 years, would there be overshoots, lags, black swans that have policy implications? The paleoclimate proxies indicate thousands of years for collapse of major ice sheets. Larsen B in past high CO2 regimes may have been slowly nibbled away as temperature rose, but the current rate of change suported surface ponding of meltwater and catastrophic collapse. If the rate of melting from a faster rise in seawater temperature at the edge of major ice sheets drives the calving front into areas where the ground slopes inland, can paleoclimate response preclude major nonlinearities? If the change in speed of ice flow towards the calving front lags the rate at which ice removal at the front is changing, then the front will retreat into the ice sheet. If the temperature rises slowly, the rate of calving, offset by an increase in the rate of ice flow, could maintain a constant calving front. If the temperatures rise quickly, the calving front could move into areas where the dynamics change. E.g.
    “The collapse of the Western Antarctica ice sheet is already under way and is unstoppable, two separate teams of scientists said on Monday.”
    “But the researchers said that even though such a rise could not be stopped, it is still several centuries off, and potentially up to 1,000 years away.”

    What if it’s 100 years away?
    “Growing evidence has demonstrated the importance of ice shelf buttressing on the inland grounded ice, especially if it is resting on bedrock below sea level. Much of the Southern Antarctic Peninsula satisfies this condition and also possesses a bed slope that deepens inland. Such ice sheet geometry is potentially unstable. We use satellite altimetry and gravity observations to show that a major portion of the region has, since 2009, destabilized. Ice mass loss of the marine-terminating glaciers has rapidly accelerated from close to balance in the 2000s to a sustained rate of –56 ± 8 gigatons per year, constituting a major fraction of Antarctica’s contribution to rising sea level. The widespread, simultaneous nature of the acceleration, in the absence of a persistent atmospheric forcing, points to an oceanic driving mechanism.”
    Would you bet your beach house that it doesn’t also point to a Larsen B style Black Swan (birds of black feather?)?

  24. John Hartz says:

    For a brief, plain-English summary of the recent paper by Bloch-Johnson, Pierrehumbert & Abbot paper and its implications see:

    How the harm of climate change could explode exponentially down the road by Ryan Cooper, The Week, June 4, 2015

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