## linear vs non-linear

Andrew Montford has a post about the errors in Lennart Bengtsson’s rejected paper. I haven’t seen the paper, and nor – I think – has Andrew Montford. All we have access to is a copy of one of the reviewer’s reports. What Andrew takes issue with is this comment from the reviewer

Even more so, as the very application of the Kappa model (the simple energy balance model employed in this work, in Otto et al, and Gregory 2004) comes with a note of caution, as it is well known (and stated in all these studies) to underestimate ECS, compared to a model with more time-scales and potential non-linearities (hence again no wonder that CMIP5 doesn’t fit the same ranges)

Andrew’s issue is that

What struck me – a humble blogger, a mere accountant, a grubby scribe, as my detractors are occasionally wont to say – is that the Kappa model is not actually used in Otto et al (or indeed Gregory 2004). So here we have an expert reviewer who seems to be less familiar with the details of the relevant studies than I am.

Well, I’m slightly confused, as I thought that the reviewer was referring to models based on this kind of energy balance formalism

$N = F - \alpha \Delta T,$

where N is the system heat uptake rate, F is the change in forcing, and $\alpha$ is the climate sensitivity paramter (in Wm-2K-1). One can rewrite this – as in Otto et al. (2013) – to estimate the transient climate response (TCR) and equilibrium climate sensitivity (ECS) – which should probably be called the effective climate sensitivity

$TCR = F_{2x} \Delta T/F,$
$ECS = F_{2x} \Delta T/ (F - N).$

All the the reviewer is pointing out is that these are linear equations and that $\alpha$ is essentially a constant. Hence this formalism doesn’t capture – as the reviewer points out – potential non-linearities.

Normally I might just assume that Andrew Montford was simply wrong, but Nick Stokes – in the comments – seems to partially agree. Am I missing something here? As I see it, all the papers mentioned do essentially use simple energy balance models (which I assume are often referred to as a $\kappa$ model), therefore the reviewer is essentially right; but maybe there’s some subtlety that I’ve missed. Having said that, I would argue that it’s all somewhat irrelevant, since his point was more about potential non-linearities – that these models can’t capture – rather than specifically about the name of the model used.

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### 24 Responses to linear vs non-linear

1. Maybe I’ve worked this out. It might have to do with whether or not the energy imbalance is zero at the reference time. If it’s not, then some of the warming during the period considered is because of this pre-equisting imbalance. Hence, when computing the TCR (I think) I should compensate for this warming in the pipeline. I don’t think it has any effect on the ECS, as this can be compensated by using the difference between the energy imbalance today and that during the reference period. Not sure that this makes Andrew Montford’s criticism any more valid though, as it is simply a small correction to a standard energy balance estimate – the linear vs non-linear issue still exists.

2. Tom Curtis says:

Anders, the formula

1) TCR = F(2x)*dT/F
with is invalid. The quantity defined by the right hand side of the formula is actually the effective climate sensitivity.

To understand the difference, we can imagine two scenarios, one (scenario A) over 70 years with CO2 increasing at a compounding rate of 1% per annum and no other changes of forcing. In the second (scenario B), Forcing rises to +10 W/m^2 over fifty years, retains that value for 100 year, then falls back to the same forcing as that in scenario A over 10 years, ie, 3.7 W/m^2. Transparently the temperature at the end of the scenario will be much warmer in B than in scenario A, and according to formula (1), that means that the world under consideration has a different TCR depending on the forcing history – which is absurd.

In Forster and Gregory (2008) we have:
2) TCR = F(2x)/(alpha + kappa),
where alpha is the climate sensitivity parameter (F(2x)/dT(2x)) and kappa is “ocean heat uptake efficiency”.

That means that in scenarios in which formula (1) actually does estimate the TCR (such as scenario A, by definition of TCR):
dT/F = alpha + kappa

It seems to me that simple energy balance models, when used to estimate the TCR must assume that condition to be true. As alpha will not vary by scenario, that means they are making very strong, hidden assumptions about the evolution of kappa. The reviewer may have just been making that fact transparent by his name for the models, and indeed it may be standard practice to do so. Certainly neither Montford, Nic Lewis, nor Nick Stokes are members of the relevant expert community, so their opinion as to whether the labeling is a mistake or not is moot.

Of course, it may be that kappa is not defined relative to a scenario (is effectively constant across time), in which case the energy balance model estimate of TCR are assuming that:
dt/F = alpha + kappa plus some narrowly defined temporal evolution of forcings. It may also have been that the reviewer has simply made a mistake. Montford, however, has not shown that; and still less has he shown that energy balance model estimates of TCR are free of assumptions about linearities.

3. Tom,
I was taking the TCR equation from Otto et al. (2013). I can see, though, that the evolution can influence ΔT and hence such estimates will depend on the evolution of the forcing.

I guess I’m still slightly confused, as I’m busy working through some of these equations and they all seem to be essentially different ways of writing the same thing. Unless, as you say, there is some hidden assumption about $\kappa$. Although, I do now understand where the $\kappa$ comes from, which I hadn’t before.

I guess an issue is that Otto et al. were trying to use observed changes in system heat uptake rate, temperature, and forcings to estimate the TCR and ECS. Gregory and Forster seem to use these simple models to determine the best values for $\kappa$ and $\alpha$. I can see that they’re slightly different, but fundamentally they seem to be based on the same assumptions.

4. Tom,
Maybe I’m starting to get this, but you can probably correct me if I’m wrong. If you want to use a basic model to evolve the temperature in response to a forcing you can use

$C d \Delta T/dt = \Delta F - \alpha \Delta T$,

where C is the heat content of the oceans (normally assumed to be constant, even though it probably isn’t). This has the advantage of the left-hand side going to zero when the system reaches equilibrium. Alternatively, one can simplify it and use

$\kappa \Delta T = \Delta F - \alpha \Delta T$

This is easier, but has the disadvantage of the left-hand side not going to zero when the system reaches equilibrium. However, if one’s goal is to simply run a model like this and then compare with some observational data, this might be fine as one will get estimates for $\kappa$ and $\alpha$ and hence can estimate climate sensitivity.

In this formalism,

$\alpha = F_{2x}/T_{eq} = F_{2x}/ECS$

Once one has determined $\kappa$ and $\alpha$ one can then write

$ECS = F_{2x} / \alpha = F_{2x} \Delta T / ( \Delta F - \kappa \Delta T ) = F_{2x} \Delta T / ( \Delta F - N )$

which is essentially, the Otto et al. formalism. So, unless I’m being silly, the Otto et al. method is implictly using a $\kappa$ formalism even if this isn’t explicit.

5. Tom Curtis says:

Anders, Gregory and Forster write:

“One way to describe the ocean’s role is as thermal inertia, with N = CdDT/dt, where C is a constant heat capacity [Frame et al., 2005]. For forced climate change on
multidecadal timescales, the effective heat capacity C is greater than that of the ocean ‘‘mixed layer,’’ and is actually not constant [Keen and Murphy, 1997; Watterson, 2000], because the ocean is not well-mixed, and the vertical profile of temperature change is time-dependent. An alternative description is N = kDT, where k is the ‘‘ocean heat uptake efficiency’’ [Gregory and Mitchell, 1997; Raper et al., 2002]. This formulation views the deep ocean as a heatsink,
into which the surface climate loses heat in a way analogous to its heat loss to space. It permits the influences of climate feedback and ocean heat uptake to be compared, since a and k have the same units. Like climate sensitivity, this formulation of ocean heat uptake is a model-based result. It is evident that its validity is restricted; it cannot be correct for steady state climate change, because N -> 0 as DT approaches its equilibrium value, so the efficiency of ocean
heat uptake must decline. The formulation was proposed only as a description for a system in a time-dependent state forced by a scenario with a fairly steadily increasing forcing.

(Note: a = alpha, k = kappa, and D = delta)

Thus they explicitly express the identity implicit in your final formula, but state that it only holds in “…a scenario with a fairly steadily increasing forcing”. That is not a limitation of the Otto et al (or similar) formulations, although without steady changes in forcings, estimates of the energy balance are not currently accurate enough to provide reliable estimates for calculating climate sensitivity as done in Otto et al (IMO). At least, the wild fluctuations in the estimates with changing baselines are more likely due to inaccuracy in estimates of energy balance than of temperature or forcing.

In any event, with the restriction on kappa, it is not obvious that energy balance models could be classified as “kappa models”. On the other hand, that may be the general term used for both Gregory and Forster style models and Otto et al style models among experts studying climate sensitivity, or experts studying climate models who also look at climate sensitivity. Just because it represents a formal error does not mean an a particular view has not been immortalized in a name, as everybody interested in the greenhouse effect should know. Absent comment from members of the appropriate research community we have no way of knowing whether the reviewers usage standard, a common but minority usage in the community, or idiosyncratic, and todate no experts from the relevant communities has commented (and, no, Nic Lewis is not an expert in that field, although he may eventually become one).

Personally I consider the far more interesting question is not whether the reviewers usage was idiosyncratic (or if you prefer, a mistake). He, after all, clearly indicates the type of models he has in mind so there is no question of miscommunication. The interesting question is whether or not he is right in saying that “… it is well known (and stated in all these studies) to underestimate ECS, compared to a model with more time-scales and potential non-linearities” Otto et al write:

“Both equations (1) and (2) assume constant linear feedbacks and (2) further assumes that the ratio of ΔQ to ΔT for the observed period is the same as that at year 70 of a simulation in which atmospheric CO2 levels increase at 1% per year, which is approximately
the case over the past few decades if we exclude periods strongly affected by volcanic eruptions.”

Evidently the reviewer was right, in his substantive claim and this focus on “kappa models” is simply a distraction.

As a side note, I do not think the near constant increase in forcing over the past few decades (exlcluding volanism) is all that relevant when you are making TCR estimates over the period from 1860 to more recent decades. Nor do I think you can simple ignore the volcanism.

6. Tom,
Thanks.

In any event, with the restriction on kappa, it is not obvious that energy balance models could be classified as “kappa models”.

Yes, okay, I agree.

As you say, whether or not he is formally correct about these being kappa models, he then clarifies what he means and seems quite correct in mentioning that these models only apply to constant linear feedbacks. Unless I’m missing something, though, isn’t the constant ratio between $\Delta Q$ and $\Delta T$ essentially an implicit $\kappa$ formalism – for the period until CO2 doubles, at least?

7. Tom Curtis says:

Yes, it is. However that is only relevant to TCR, whereas the reviewers comments were technically about ECS.

8. Tom,
Yes, I agree. Having said that, I’ve been assuming that noone would actually run a true kappa model beyond the point at which CO2 doubles, since that it is presumably when the problem with energy conservation becomes more of an issue. My understanding is that you use the kappa model to compare the model values with either other model values or with observations, but over a period shorter than – or the same as – the period over which CO2 doubles. Having done that, the best-fit value of $\alpha$ can then be used to infer the ECS. So, it still seems that there’s some complementarity between a kappa model and what is done in Otto et al., for example.

Something else I realised is that presumably the whole point of a kappa model is to avoid integrating. If you rewrite

$C d \Delta T / dt = \Delta F - \alpha \Delta T$,

(which needs integrating) as

$\kappa \Delta T = \Delta F - \alpha \Delta T$,

then you can avoid integrating as everything depends only on ΔT. It, therefore, seems to me that any energy budget model that doesn’t involve integration can be recast as a kappa model. Again, maybe I’m still missing some subtlety.

9. Nick Stokes says:

Tom Curtis says: May 17, 2014 at 9:13 am
“Certainly neither Montford, Nic Lewis, nor Nick Stokes are members of the relevant expert community”

True for M and me. But Nic Lewis is a co-author of Otto et al, the paper in question.

My understanding is that κ is a proportionality between ocean heat uptake, or maybe total influx, and ΔT, and Gregory&Forster 2008 considered a case where it is constant. I don’t think Otto et al do that, but I could be wrong. I thought it was clear what the referee meant, and he was right, but his usage of the term “kappa model” might not be correct.

10. BBD says:

A storm in a teacup?

11. Tom Curtis says:

Nick Stokes, I use two definitions of expert, one more general, and one particular to scholastic endeavor.

The more general one comes from a paraphrase of a quote by Weiner Heisenberg:

““An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them.”

Perhaps the best paraphrase is, “An expert is somebody who knows all the well established blunders that can be made in their subject, and how to avoid them.”

By that definition Nic Lewis is not an expert on climate sensitivity. The most basic blunder you can make in that field is to focus exclusively on one line of evidence to the exclusion of all others. That is not just a blunder Lewis makes, but his strategy.

The second, more particular definition of expert is that an expert must be capable of novel, publishable research in the field free from blunders, and be well read on the relevant literature.

Nic Lewis satisfies the first condition, as demonstrated by his co-authorship of Otto et al; but his knowledge of the literature on climate sensitivity is singularly focused around Otto et al type methods. So, again he is no expert, although he is potentially becoming one.

I will happily concede that he is more expert than either Montford or you in that area; just as I will happily concede that you are more expert on the global instrumental temperature record than either Montford or Lewis, with out being willing to concede that you are an expert on the subject.

12. Steve Bloom says:

Otto et al. author order: Alexander Otto, Friederike E. L. Otto, Olivier Boucher, John Church, Gabi Hegerl, Piers M. Forster, Nathan P. Gillett, Jonathan Gregory, Gregory C. Johnson, Reto Knutti, Nicholas Lewis, Ulrike Lohmann, Jochem Marotzke, Gunnar Myhre, Drew Shindell, Bjorn Stevens & Myles R. Allen

Hmm, sixth author along with ten others. I’m not very impressed by that placement. What did he actually contribute to the paper?

13. Tom Curtis says:

Steve Bloom, I believe he contributed the basic idea of the paper from his unpublished paper of which he was sole author. While definitely the least expert of the authors (from among those whose names I recognize), his contribution was not inconsequential.

14. Nick Stokes,
Thanks for the comment. Given that I didn’t really know why they were referred to as $\kappa$ models until Tom pointed it out, I’m very clearly not expert.

I agree that it’s fairly clear what the referee meant. If you consider the Otto et al. ECS equation

$ECS = F_{2x} \Delta T/ (\Delta F - \Delta Q)$,

then I think – and this is kind of what I was getting at – one can recast this as

$\Delta Q = \kappa \Delta T$,
$ECS = F_{2x} \Delta T / (\Delta F - \kappa \Delta T).$

So, I think one could argue that the Otto et al method is essentially a kappa model in which you fit to only two data points (the reference period and the final period). Of course, in the way they’re using it, you wouldn’t bother doing this, but I think I can see why some might regard these as all variations of the $\kappa$ model. Of course, it is possible that the referee simply used the wrong terminology, but – as you say – we all know what they were getting at.

15. Isn’t the non-linearity in the spectral frequency range of the forcing F?

A power spectrum of energy density that is in the infrared has a different effect than an equivalent density in the visible and above. The fact that incoming high-frequency power (sunlight) is transferred into a low-frequency heat (infrared) on the rebound makes all the difference in the world. According to Planck’s Law, temperature has to rise quite a bit to make up for the weak energy carried by an infrared photon. That is a non-linear transform of an extreme variety. Not as extreme a non-linearity as the operation of an infrared laser, but on the continuum in that direction.

This totally screws up being able to deal with the system in the simple way that Nic Lewis desires. It turns him into a dragon-slayer equivalent — he misses an important bit of physics and thus makes a mess of things. Or is this intentionally misreading/misleading on his part?

You know the way around this? Don’t deal with F directly. Just stick with changes in effective GHG, represented by the changes in CO2 observed empirically. If you do that, you get TCR of around 2C and ECS of around 3C.

And if you want to understand the land/ocean dichotomy in TCR, then do something like this:’
http://contextearth.com/2014/01/25/what-missing-heat/
There is more than one way to skin a cat and deal with the uncertainty in the forcing numbers.

[Mod : Unnecessary characterisation removed] perhaps it is Nic Lewis’s intent to use overly simplistic physics as a primitive club to wield against a more comprehensive analysis.

… or else I just may be missing something.

16. WHT,
You may have a point, but in this case as I understand it, it refers to the following equation

$\Delta Q = \Delta F - \lambda \Delta T = \Delta F - (\sigma T^3 - W) \Delta T$,

where $W$ is the feedback response (Wm-2K-1). The standard energy balance formalism assumes that $W$ doesn’t depend on $\Delta T$, but I think GCMs suggest that it does – otherwise how do they match our recent warming while still having an ECS of ~ 3K.

17. Steve Bloom says:

Thanks, Tom. My view, FWIW, is that to be considered expert you need to be more than a one-trick pony, even if your particular trick has clear utility, which this one does not.

18. ATTP,
The mean value linearization via S-B is straightforward– that is dF ~ d(T^4) ~ 3T^3 dT as you have shown.

But not so simple is the realignment of the photonic spectrum which is the Planck’s Law part of S-B.

As you say, we really have to add the whole ball of wax at this point — Modtran with altitude lapse rate adjustments, vaporization of water, etc to understand the warming observed.

19. WHT,
Indeed, there are many reasons why the standard energy budget formalism is a little too simple. What I find odd is that – I think – in any other field, a simple calculation – like the energy budget one – that was similar to but not the same as more complex calculations, would typically be regarded as a nice consistency check. Here, it’s seems to be used to argue against the more complicated and detailed calculations.

20. Andrew Dessler says:

My main comment is that these peer reviews are not written for public consumption or for the lawyerly nitpicking that the skeptical blogosphere excels at. Regardless of whether this one statement is right or not, the review seems competent and, in conjunction with the second review, I think the editor was fully justified in rejecting the paper. It is sad that, as a frequent peer reviewer myself, I now have to start considering as I write it how my review could be selectively quoted if it were publicly released.

21. Andrew,
I agree. The review is simply meant to be something that helps the editor to make a decision about a paper. I may wrong, but I don’t think that future generations will thank the pedants of today.

As an aside, one of the most frustrating reviews I ever carried out was one where the authors starting nit-picking at minor errors in my review (I’d done a basic back-of-the-envelope calculation to show that their result was probably wrong and left something out that didn’t change the point I was making). I pointed out in my response that technically I was reviewing their paper, they weren’t reviewing my report 🙂

22. Fred Moolten says:

Hi ATTP – I’m coming to this discussion late, but my take is that the reviewer was correct regarding ECS but that “kappa” was not the ideal term to use in his criticism. One could argue that the energy balance model in Otto et al uses kappa implicitly, since in theory there should be some kind of reciprocal relationship between kappa and alpha. However, in Gregory and Forster, this is not apparent and so it would have been better simply to refer to “the energy balance model”, which assumes a constant alpha (i.e. a linear relationship between temperature change and change in radiative restoring), whereas alpha may in fact decline with time and warming, yielding a higher ECS than estimated by the energy balance model. My preference is to use the term “effective climate sensitivity” (EFS) for the latter model, and ECS for paleo or feedback-based estimates. None of the above actually estimates temperature change following a CO2 doubling, since they all disregard long term feedbacks involving ice sheets, vegetation/dust, and the carbon cycle.

23. Fred,
Yes, I agree that it would probably have been better to simply refer to is as “the energy balance model” – which the referee did do parenthetically. Your latter point is, however, what the referee was ultimately getting at.

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