Thanks for the link , it was interesting .
I am not familiar with “Tropical convective variabilities” so couldn’t follow much the physics that they were doing nor the physical implications .
However they confirm what I already said about ENSO : “In current ENSO modelling the tropical convective variability is considered only as white noise forcing that has no memory by itself . The current finding suggests that a fundamentaly different type of stochastic forcing may be required .”
Pity that they do a bit too much hand waving at the end by saying that the spatial averaging is no problem and 1/f stays preserved by it .
I doubt it very much and if it is true , then it is far from being trivial .

Dan

Why do you say that chaos arises only from numerical solutions of ODEs ?
It is true that it is how it became historicaly popular but there are results on physical systems (3 body problem) where Poincarré has proven already somewhere around 1900 that the planet’s orbits were chaotic (without using the word) and he didn’t use any computer .
Also the whole class of Rayleigh Benard flows are chaotic what is an experimental proof that N-S solutions can exhibit chaotic behaviour and there is a ton of chaotic systems in fluid dynamics .
Ruelle & Takens have proven a theorem that turbulence is chaotic in some cases .
The problem with numerical methods is that they make the things more complicated and could even be able to make appear chaos or randomness there where there is none .

The whole problem is then to be sure how to separate numerical artefacts from the real behaviour of the system .
Of course with Navier Stokes you will never know – as you know nothing about the solution(s) , you can’t prove that the numerical simulation converges to something unknown and the Teixeira paper shows that making only the steps smaller and smaller is not sufficient .
If the system is chaotic we already know that there can’t be uniform convergence anyway so it is a kind of Catch 22 .

I agree that this does not mean the assumptions are, by definition, valid. The difficulty is without them, we sort of end up with “can’t prove anything”.

I think “prove” is too strong a word for AGW anyway. “Build confidence in” is about the best you’ll manage. And for that, you need to have some confidence in your assumptions as well as your calculations. I don’t have a great deal of confidence in the red noise assumption. But that’s l’il old me That may not cause you to reach for the worry beads.

On that post, I’m discussing CFD modeling of systems with Navier Stokes. There is no “white noise, pink noise, red noise” issue involved.

OK, I might have been talking at slight cross purposes here. I was referring to complex non-linear systems in general – and I was careful at the end to replace the concept of stochastic noise with (deterministic) unforced variability There are many examples of non-ergodic systems that are actually fairly simple to model; Vit Klemes circular cascade of reservoirs; high DC current through a carbon composite resistor; Per Bak’s self-organised criticality (sandpile experiment). These are non-chaotic, non-ergodic systems which have f^-1 (which I should really refer to as 1/f) spectral dependency in their unforced behaviour. If you want to model the unforced behaviour of these systems, pink noise is a valid mechanism for doing so; averaging or ensembles are not. For some systems you can do this, some you can’t, and have to take a different approach. Chaotic systems are another issue altogether, and have their own set of issues. These are fundamental issues that nobody really seems to be getting their teeth into in the climate community at the moment, as Dan notes.

I say no-one is looking into it: from the modelling side, this seems true. From the statistics side some are (e.g. those cited above), and some from the observation side (e.g. below); Tom, you may like aspects of this paper (I’ve tried to link it on climate audit but got bounced by the spam filter!)

Tropical Convective Variability as 1/f Noise, Yano, Fraedrich and Blender, Journal of Climate 2001 Vol.14

This does not mean simplifying assumptions are valid.

I agree that this does not mean the assumptions are, by definition, valid. The difficulty is without them, we sort of end up with “can’t prove anything”.

Once again, I caveat this by saying in terms of testing the IPCC, testing them by their own standards is a valid simplification.

That’s the way I mean to use valid. Of course, I do still make some statistical tests to make sure the data themselves are not wildly inconsistent with those assumptions.

From your earlier post:

In any case, whether we think of things as chaotic or not, it’s known that trying to “average” the random features, and substitute parameterizations for the averages behavior has worked less the splendidly in the past.

This is only true for ergodic time series.

On that post, I’m discussing CFD modeling of systems with Navier Stokes. There is no “white noise, pink noise, red noise” issue involved. We try to solve conservation of mass, momentum and energy. Some sort of averaging is done on the continuum equations to come up with other “averaged” continuum equations. Unfortunately, when you average terms like <vu> terms like <v><u>+<v’u'> show up. Now you need more equations. People resort to parameterizations for <v’u'>. (Or write more equations and parameterize the higher order terms.)

This method sometimes does ok and sometimes just isn’t so great. So, then people try other methods, and so on.

The GCM’s read like methods that kinda-sorta worked, but kinda-sorta didn’t in the past.

My approach is pretty much to test using their own assumptions, and then the real weather we got. Otherwise, it seems to me no hypothesis test is possible.

Yes and No. (I did say I like to have my cake and eat it, didn’t I?) It is very difficult to determine which model is most appropriate through purely statistical means (e.g. red vs. pink vs. another). This does not mean simplifying assumptions are valid. One possible approach is to assess the consequence of different models and comment on them – the approach taken in Cohn and Lins, as an example. This is no panacea, but it helps us to understand some of the uncertainty associated with the assumptions we make.

Once again, I caveat this by saying in terms of testing the IPCC, testing them by their own standards is a valid simplification. Falsification here is a falsification of their results by their own assumptions. Kinda difficult to wriggle out of that one. That said, if you failed to falsify, it does not preclude falsification through different means, e.g. by calling into question the assumptions that they make. Going further, it is more difficult to make the latter kind of criticism stick. Your method of falsification makes a powerful case.

On the pink vs. red vs. white: Whether or not the weather is red, pink, white, or something else, it is clear that the measurements of weather almost contain some white noise.

I agree that there are multiple noise processes going on in the measurement data, and one would expect a white noise component. But for trend analysis, the biggest problems occur from the low frequency components, not the high frequency components – and if pink noise is present, it will dominate the low frequency components of the model. Simple polynomial orders tell us f^-1 will dominate over f^0 at some cut off frequency and below.

Another way to look at it: say I’m determining a trend from 100 years of data. I have noise with periodicity in the 5-20 year range, and noise with periodicity 500-2000 years range. The lower frequency noise has ten times the amplitude. Which of these will mess up my trend analysis the most? I would argue, in this case, the low frequency components. Yet the assumptions we make (pink / red) have little effect on the 5-20 year noise band (which may be dominated by the white noise component anyway) but have a huge effect on the probable amplitude of the 500-2000 year noise band. Yet we have insufficient data to determine what the “right” noise level is at these low frequencies. The devil is buried in the detail of the assumptions.

From your earlier post:

In any case, whether we think of things as chaotic or not, it’s known that trying to “average” the random features, and substitute parameterizations for the averages behavior has worked less the splendidly in the past.

This is only true for ergodic time series. Pink noise series – as just one example – do not exhibit ergodicity. There is a neat quote from Prof Demetris Koutsoyiannis (highlighted by UC at ClimateAudit) who describes pink noise series as time series that “forget their mean”. I’ve used the term “pink noise” here, but of course “pink unforced variability” would be more appropriate

Postscript: in principle, whilst I disagree that the red noise assumption is valid, I think Lucia’s analysis has some validity – because it tests the IPCC results using their own assumptions. The IPCC are, in effect, hoisted by their own petard. That said, if Lucia’s test found the IPCC results passed, I would point out that the red noise assumption is too strict. In other words, I like to have my cake and eat it

My approach is pretty much to test using their own assumptions, and then the real weather we got. Otherwise, it seems to me no hypothesis test is possible. I conclude– as you do– that unless we make the sorts of simplifying assumptions I make, (and the IPCC makes) to do any sort of hypothesis test. Using the sorts they use should avoid contention, but it doesn’t seem to do so.

On the pink vs. red vs. white: Whether or not the weather is red, pink, white, or something else, it is clear that the measurements of weather almost contain some white noise.

It should never be forgotten that the data are measurements and measurements nearly always contain “instrument noise”. The agencies themselves admit the existence of this noise. It seem unlikely that errors associated with measurement have long lasting autocorrelations for the following reason:
Suppose the “bucket” method is used to measure water temperature. Errors happen because different people drop the bucket different distances, wait longer or shorter to measure the temperature etc. The measurement error in the “bucket” measurement done today is likely to be independent of the error yesterday. Similar things happen for every measurement.

These tend to average out over all measurements, but it never zeros. So, this component of the error is “whitish”. You can step through all the component of actual errors in measurement, and see that the measurement uncertainty is likely to have a strong white component.

( GISS with its extrapolation over the poles and adjusting station temperatures for average of the region may end up with a red componenet to the measurement uncertainty.)

But, in anycase, the noise due to measurement errors will have a different spectral character from the weather noise. Weather noise is larger, but, based on reported values at GISS and Hadley, and intercomparison of the instruments, measurement noise is not insignificant.

f.Ex a paper I linked somewhere says that ENSO could be pink aka a mixture of white and red noise .

I agree with the sentiment (ENSO variability – and many other climatic parameters – could have a pink spectral response), I would also add that certain caveats apply when modelling pink noise through a combination of red and white noise processes. For the purposes of this discussion, I am assuming white noise has spectral dependency f^0, pink noise spectral dependency f^-1 and red noise f^-2.

Clearly, on its own, white noise (e.g. gaussian i.i.d.) is a poor model for pink noise; red noise (e.g. simple Markovian process) can be made to fit better, but obvious discrepencies still occur (cf. Tamino’s efforts). It is possible to get a better “fit” with merged red and white. This gives you additional degrees of freedom, and each part maps to a different part of the spectrum; the white noise fits to the shallower high-frequency tail of the pink noise, and the red noise maps to the low frequency components.

However, the fit is just made to the data available. If a mix of red and white noise (e.g. Lucia’s ARMA approach) is used, you get a good fit within the limits that you can estimate for the data you have. Unfortunately, you have essentially no data for the lower frequency terms (i.e. oscillations at some multiples of the data length). In this respect, the analysis inevitably ends up extrapolating the red noise term into the lower frequencies – but this is a problem, because red and pink noise extrapolate differently.

The result of this is that when using a red and white mix to test a pink process, a substantial inflation of significance occurs in estimates of things affected by low frequency variability – trends being an obvious case! In fact, under pink noise assumptions, even the late 20th century warming fails significance tests (e.g. Cohn and Lins 2005).

The failure of climate science to properly address these topics – which as you note, date back nearly 70 years – is a disgrace.

Postscript: in principle, whilst I disagree that the red noise assumption is valid, I think Lucia’s analysis has some validity – because it tests the IPCC results using their own assumptions. The IPCC are, in effect, hoisted by their own petard. That said, if Lucia’s test found the IPCC results passed, I would point out that the red noise assumption is too strict. In other words, I like to have my cake and eat it

Chaotic response is fundamentally known only from numerical solutions of discrete approximations to the continuous equations. It seems reasonable then to first determine, eliminate beyond any doubts, that the observed response is not due to properties of the chosen numerical solution methods.

I think this is not so. Chaotic response has been observed in flows near transition to turbulence. There are also analytic solutions that proceed from a stability analysis that results in a particular flow, to then doing a stability analysis on that and so on. We can see that chaos happens in flows.

Of course, these aren’t directly climate.

I’m not really “into” describing things as chaos. But random-seeming behaviors certainly can arise in systems involving the NS — or other non-linearities. These behaviors aren’t simply artifacts of the numerics.

In any case, whether we think of things as chaotic or not, it’s known that trying to “average” the random features, and substitute parameterizations for the averages behavior has worked less the splendidly in the past. The circa ’70s and ’80s transport models that were chockful of these types of parameterizations were useful but needed to be used with great care. (This often meant tuning constants to the specific flow of interest or hunting for closures that worked for flow “a” but not flow “b”.)

It’s difficult to believe that models could be bang on accurage when the documentation in the literature reads like it using ’80s commercial package software methods (e.g. Fluent, Flow3d etc.). This is not to say they aren’t useful. The 80s commercial software was useful. But, useful doesn’t mean accurate.

” … (we know that climate is somewhat chaotic) … ”

So far as I know, ’somewhat chaotic’ is an undefined concept. Can you point us to reports and papers in this area?

Chaotic response is fundamentally known only from numerical solutions of discrete approximations to the continuous equations. It seems reasonable then to first determine, eliminate beyond any doubts, that the observed response is not due to properties of the chosen numerical solution methods. Accurate numerical solutions of systems of nonlinear continuous equations are notoriously difficult. The process is filled to overflowing with subtle pitfalls. Mathematical analyses of the properties and characteristics of the continuous equation systems is also notoriously difficult.

So far as I have been able to determine, both these steps have been skipped over in the case of GCM models, methods, and codes. That the trajectories of the dependent variables obtained in calculations show chaotic response is an untested hypothesis. These steps do represent very difficult work, but that is no reason to avoid them.

A somewhat rigorous approach to develop deep understanding of mathematical models of physical phenomena and processes includes the following steps. (1) Development of the final form of the continuous model equations (generally ODEs, PDEs and some algebraic equations). (2) Determine the characteristics of these. And by characteristics I mean determine if the equations are elliptic, parabolic or hyperbolic. Sometimes there are surprises discovered at this step. (3) Determine the initial and boundary conditions that lead to a closed system of continuous equations and a well-posed problem. (4) Develop the discrete approximations to the continuous equations. (5) Analyze the discrete approximations to determine consistency, stability, and convergence of the proposed numerical solution methods for the discrete approximations. Sometimes there are surprises discovered here (see Step 2).

For all but the most trivial problems, there are usually lots of iterative loops within each step, and around several subsets of the steps. Some might require development of software to successfully carry out the step. Analyses of numerical solution methods, for example, might in fact require that software be developed to determine that the proposed methods are stable. Numerical solution for the linear growth factors, for example, in the case of almost all real-world problems of interest is generally required. The type of continuous equations, as determined in Step 2, is an important consideration relative to maintaining conservation of mass and energy for example, in the discrete approximations. Parabolic equations, for which the dependent variables are coupled throughout the solution domain, present special problems relative to conservation of mass and energy conservation at stationary and non-stationary interfaces.

All these processes should be completed, and the results understood in depth, before planning, designing, development, and coding of the software for the model begins. To gather up some continuous equations, throw some discrete approximations onto these, and wrap everything up with some coding, and then obtain the properties of the model from the generated numbers is simply not correct

Tom, I got some of them.

I’m don’t think climate models assume gaussian. I’m not even sure which formalism they use to deal with any “off-average” behavior due to the NS

Thanks .
I am sure that they do and I have it from the head modeller Schmidt himself .
In the W.Briggs blog he has written :

But there is another component – e – the internal variability – which is chaotic and depends on exactly what the weather is doing. The atmospheric component of ‘e’ is only predictable over a few days, while for the ocean part, there might be some predictability for a few months to a few years (depending on where you are).

As Schmidt is apparently not familiar with the chaos theory , he has always been using the concept “chaotic” improperly .
In his mind it means – stochastic and “cancelling out over some (not nearer defined) time period” .
I agree that “gaussian” is a technicality that is not necessary – any probability density function that is symmetrical wrt the mean would do .
But then with such a hypothesis the Gaussian imposes itself as the most convenient and relevant mathematical form .
Besides I have several papers dealing with ENSO modelling that explicitely use gaussian “noise” .
I am sure you would find tons of that if you wanted to look nearer in the “modelling” litterature .
All those papers that I have read have one thing in common – the (gaussian) noise hypothesis is considered self explaining , the authors don’t seem to realize that introducing randomness in otherwise perfectly deterministic processes is an EXTREMELY strong assumption .
Things that Kolmogorov and Lorenz (peace to his soul) were very well aware of and argued with painful carefulness basing on continuous equations describing the process they were studying .

Then it is also possible to take the problem the other way round and there is a whole corpus of litterature that takes time series of observables as such and tries to look if there can be detected some colored signal – f.Ex a paper I linked somewhere says that ENSO could be pink aka a mixture of white and red noise .
This approach is good to detect definitely deterministic and non or slightly chaotic systems but it is very difficult to differentiate deterministic chaos from a purely random process by exclusively statistical methods .
So if somebody who wants to do that , he has to do the legwork like Kolmogorov did and do what Dan is saying – looking hard at the continuous equations AND their numerical approximations that one uses in a model AND deduce from there under what conditions (if any) a statistical field theory could be legitimated .
If that is not done and it isn’t , we stay at the level 0 of reasoning – “Hey I collected some data over some arbitrary time period and if I remove some fit from it , I get a thing that looks more or less random . Let’s say that the fit is the explanation and that anything else is random for any time interval and let’s move on .”

And, I haven’t started on the sea level data, though running after every Chicken Little peeps is getting old. They just fail to mention the whole IPCC TAR and AR4 evaluation of limitations of sea level data, an analysis I thought was rather thoughtful.

Now that I’m done figuring out how cyanide dilutes, I’ll be able to blog tomorrow!

Stefan Rahmstorf, author of the initial post, criticised David and also Roger Pielke Jr for their misspelling of his (Rahmstorf’s) name. According to Rahmstorf, ‘this is … an indication of the care someone takes in getting things right.’

Wow. Stefan must be having a hard time finding real things to criticize!

The misspellings by David and Roger were in blog postings, but the name of one of Rahmstorf’s RealClimate colleagues -Caspar Ammann – was misspelled in his (Rahmstorf’s) contribution (‘Anthropogenic Climate Change: Revisiting the Facts’) to the important edited collection ‘Global Warming: Looking Beyond Kyoto’ (Ernesto Zedillo, eds., Brookings Institution Press and Yale Centre for the Study of Globalization, 2008). The error is in footnote 41 on p. 52, in which ‘Caspar M. Amman (sic)’ is cited as a co-author of a Comment published in ‘Science’ in 2006.

Is this an indication of the care someone took ‘in getting things right’? As yet RealClimate hasn’t published a post I sent 15 hours ago, drawing attention to the error. And I suspect that they won’t: teacher musn’t be seen to have made mistakes.

I’m don’t think climate models assume gaussian. I’m not even sure which formalism they use to deal with any “off-average” behavior due to the NS. But, I do agree they way the modelers who blog discuss things, they don’t seem to have much probability-statistics type understanding. (And I don’t just mean hypothesis testing.)

In my opinion, all the observed responses should be assignable to the equations that make up the complete models. If a physical significance is to be attached to ‘chaotic response’ then the first step is to eliminate, by deep investigations into the mathematical properties of the models, the possibilities that what is being seen are purely artifacts of the numerical methods. And by mathematical properties of the models I mean the mathematical properties of the continuous equations, their discrete approximations, and the numerical solution methods applied to the discrete equations.

Corrections and additional clarifications appreciated.

Dan you say what you say because you did a considerable work in deterministic chaos emerging from non linear ODEs and PDEs .
That is not the case for 99% of people dabbling in climate “models” .
As you can always calculate an average of ANY observable , the most naive representation of ANY observable is to say that it is equal to [X] + noise(t) where [] is the average of the observable and noise(t) is the difference between X(t) and .
Here X can be anything – a velocity , some surface or a volume integral of pressure or temperature , the frequency or amplitude of ENSO etc .
So far apart from saying that it is naive , there is not much to add and it is a tautology anyway (an always true statement containing no additional information) .
Serious things begin when people add that noise(t) is random , gaussian , “cancels out” etc .
In the matter of fluid dynamics since Kolmogorov 60 years ago we know that it is not true in general .
Noise(t) is neither random nor gaussian nor “cancels out” in the general case .
However it is possible to construct a statistical asymptotic field theory (similar to statistical thermodynamics) for CERTAIN cases where additionnal hypothesis can be made .
Namely homogeneity and isotropy .
In those cases this theory shows that for very small scales the statistics have a universal form and allow to make statistical statements about energy dissipation .
Obviously the theory dramaticaly fails when the hypothesis are wrong and that is the domain of low dimensional chaos that you mention and Komogorov of course also knew that the theory was not valid for every general case .

The climate “modellers” are still in the pre Kolmogorov era where they ALWAYS do the hypothesis that in ANY conditions and for ANY not understood observable one has
X(t) = [X] + gaussian noise(t) .
Why Gaussian ?
Because it “cancels out” so you don’t need to bother about details that you don’t understand anyway and only calculate averages what you are able to do .
Of course if that unfounded , naive assumption should be justified , they would have to do exactly what you say and what Kolmogorov has done 60 years ago – start from the continuous equations AND their numerical approximations and prove that there are specific conditions and scales (both spatial and temporal) where a statistical asymptotic field theory could make sense .
Everybody who knows a bit about chaos also knows that such a climate theory cannot be founded on statistics themselves alone because in most cases it is impossible to make the difference between the deterministic chaos that obeys to no statistics and will always surprise by “unprobable” brutal behaviour changes after a certain time and a gaussian noise where the probability of changes far from the average is negligible .
They look alike until they don’t

In the current state of climate “modelling” there is sofar not even a hint that the people have understood that “noise” does not necessarily cancel let alone begun a serious theoretical (aka properties of continuous equations) work on this matter .

P.S
You should put the Teixeira paper as a permanent link on your blog because I find it extremely enlightening for the understanding of chaos , convergence and continuous equations in physics .

All physical phenomena and processes require driving potentials in order to take place.

Agreed. However, the problem comes (I think) in that the reaction to drivers is chaotic itself. There’s no doubt that we don’t understand unforced variability, but we don’t have to understand it perfectly to estimate climate sensitivity.

“unofrced variability” includes things (like ENSO) that aren’t really “weather”. We have a very hard time predicting ENSO, but this is improving.

The GCMs cannot calculated the weather because they do not contain models for important weather phenomena and processes, the spatial (and maybe the temporal) resolution is not sufficiently refined, and the injection of empirical data into the calculations is not carried out.

Not sure what you mean by “important weather phenomena.” GCMs do simulate ENSO and other oceanic and atmospheric oscillations. Some are well simulated. Some are not. Some change in unexpected ways, as the Southern Annular Mode did in response to ozone depletion. Coarseness is a problem in some GCMs. The big issue is that GCMs are rarely initialized and so cannot predict short term. Moreover, the most accurate an important output of GCMs is climate sensitivity. This is done with ensemble means. Since the unforced variability from individual models is canceled out, it actually makes determining CS easier.

The assignment of “unforced variability” to particular events is done more on observation (we know that climate is somewhat chaotic) and theory (There is no forcing known that can explain observations). The current “cooling” is still consistent with what we know of unforced variability; however, a few mores years would demand a reassessment of the forcings or the extent to which unforced variability can have an effect.

I rambled a bit there.

From what I have read so far, “unforced variability” seems to refer to the ‘chaotic’ nature of calculated numbers as seen in applications of GCMs. Additionally, I remain somewhat confused that this seems also to be assigned to weather.

The GCMs cannot calculated the weather because they do not contain models for important weather phenomena and processes, the spatial (and maybe the temporal) resolution is not sufficiently refined, and the injection of empirical data into the calculations is not carried out. To assign the observed ‘chaotic response’ to weather doesn’t make sense if the GCM models don’t have weather phenomena and processes in them and don’t make weather calculations.

For me, assignment of an observed response to things not included just doesn’t follow. This seems to be another of the hypotheses that are simply attached to the behavior seen in the results produced by GCM applications.

In my opinion, all the observed responses should be assignable to the equations that make up the complete models. If a physical significance is to be attached to ‘chaotic response’ then the first step is to eliminate, by deep investigations into the mathematical properties of the models, the possibilities that what is being seen are purely artifacts of the numerical methods. And by mathematical properties of the models I mean the mathematical properties of the continuous equations, their discrete approximations, and the numerical solution methods applied to the discrete equations.

Corrections and additional clarifications appreciated.

I don’t seer the big deal. It gets tiresome saying “unforced variability” over and over.