Since I’ve discussed climate sensitivity here quite a lot, I thought I would highlight a recent paper, by Knutti & Rugenstein, called feedbacks, climate sensitivity and the limits of linear models. It’s a very nice, and readable, summary and – as the title suggests – focuses somewhat on the linear models that are a favourite of Nic Lewis, for example. The basic message, as the abstract says, is
Our results suggest that the state- and forcing-dependency of feedbacks are probably not appreciated enough, and not considered appropriately in many studies. A non-constant feedback parameter likely explains some of the differences in estimates of equilibrium climate sensitivity from different methods and types of data. Clarifying the value and applicability of the linear forcing feedback framework and a better quantification of feedbacks on various timescales and spatial scales remains a high priority in order to better understand past and predict future changes in the climate system.
It also says:
…it becomes clear that ECS and TCR are rather limited characterizations of a much larger and interactive system. Other feedbacks such as vegetation, chemistry or land ice are now included in some climate models as their relevance is better understood. Some feedbacks operate on very long timescales that are determined by the internal dynamics of the system, and their response is not proportional to temperature.
which reminded me of this Michael Tobis comment, where he essentially argues that focusing on a simple metric, like climate sensitivity, ignores a great deal of important complexity. Climate sensitivity might give us some broad brush idea of the magnitude of the change, but it tells us little of what will happen where we live, which will – of course – differ from place to place.
The interesting issue at the moment, however, is that different estimates for climate sensitivity can produce quite different results. For example:
… some but not all recent studies on the twentieth-century warming find rather low ECS values (median at or less than 2°C) [17–19,21]. Climate models show a large spread in ECS, with the spread half as big as the actual value. The highest uncertainty can be attributed to the cloud feedbacks (traceable to certain cloud types and regions), and the lapse rate feedback [50–53]. But all comprehensive climate models indicate sensitivities above 2°C, and those that simulate the present-day climate best [54–57] even point to a best estimate of ECS in the range of 3–4.5°C.
The paper then goes on to discuss the basic linear model, largely represented as
where 
it has been suggested that the non-constancy in the global
Anyway, I’ve actually said more than I meant to. The paper itself is very accessible, so I would recommend those who are interested in this, and who would like to know more, to go ahead and read it.
...and Then There's Physics 
This is based on an excellent talk that Reto Knutti gave to the Royal Society in London in December 2014. An mp3 is here:
https://royalsociety.org/events/2014/feedback-climate-system/
I think he showed pretty strong evidence that we should be very careful to make consistent calculations. For example, you might calculate climate response from a linear model applied to observations and then calculate it from some model experiments where you quadruple atmospheric CO2.
These can both contain useful information but it doesn’t mean that the two numbers should be directly compared. You should first do the same calculations on both the models and the observations.
For one thing, temperature depends on “energy” to the fourth power under the Stefan-Boltzmann equations which has been proven to be true in every instance they have ever been tested on.
It actually takes 16.4 W/m2 of forcing to increase Earth’s surface temperature from 288.0K (14.9C) to 291.0K (17.9C). CO2 doubling only provides 3.7 W/m2 so the feedbacks must produce a further 12.7 W/m2 of additional forcing. Just adding some real calculations to the debate.
Paul: isn’t the relevant temperature to use in that calculation the temperature at the equilibrium emission altitude?
So the temperature change you’re considering is from 255 K to 258 K is more like 11.4 W m-2 in total, with 3.4 W m-2 from CO2 leading to 8 W m-2 from feedbacks, which is pretty consistent with most evidence we have for climate response.
“where N is the system heat uptake rate, F is the change in forcing, \Delta T is the change in temperature, and \lambda is the feedback parameter. These linear models assume that \lambda is constant, but the paper discusses in quite some detail why this may not, and probably isn’t, the case ”
The funny thing is that some skeptics ( example willis eschenbach) have been criticized for making this same argument.
Steven M. – “The funny thing is that some skeptics ( example willis eschenbach) have been criticized for making this same argument.
Pretty sure I wouldn’t characterize it as ‘funny’ – ironic perhaps, maybe even sad. The difference of course being that climate scientists will use these linear models knowing their limitations. They will take the whole of the relevant evidence (GCMs, paleo) into account. People like Willis look at a changing ECS as a loophole to claim ever lower ECSs – with one goal in sight: deny the earth will warm to anything approaching levels we’d find culturally or economically disastrous. These claims ignore the rest of the evidence. I can’t find anything funny about it – except maybe in the sense that you have to laugh or otherwise you’ll cry.
Steven,
Are you sure it was the same argument?
Paul,
Mark has pretty much covered it, but you do need to distinguish between surface fluxes and TOA fluxes.
The simple linear model you quote is not the only useful linearization – I think the biggest problem is not nonlinearity per se, but trying to lump the state of the earth into a single parameter. For small perturbations you certainly expect linear response, but that can be a complex response function with a spectrum of time constants rather than the single response time the lumped model suggests. And e paper you quote talks about the spread of the response between different systems and geographic regions so it’s clear the issue is mainly a single parameter memory vs the reality of a huge number.
Arthur,
Yes, I think that is a good way to put it. Isaac Held’s post makes essentially that point.
I had thought of bringing this up in this post, but I was aiming for brevity 🙂
Effing one dimensional models. We don’t even use them for Titan anymore.
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