Great article, Michael. I'm beginning to get it. (The name Hofstadter keeps floating through my mind.)

I agree that chaos in weather doesn't imply chaos in the climate: there can be good effective theories for long-term phenomena. But it doesn't imply the opposite, either.

There is just weak evidence that there is a simple theory of the long-term phenomena and even weaker evidence that it is the first fashionable theory that someone likes to promote.

To other readers: Lubos Motl is a physicist and a prominent critic of the climate science consensus.

Lubos, I appreciate you starting off with a point of agreement. In return I will agree that the argument made here does not prove that climate is (or isn't) predictable. You will note that I clearly asserted as much.

My effort was intended primarily to introduce the idea of a trajectory through state space, and secondly to refute the incorrect and common argument that an inability to predict weather implies an inability to predict climate.

I agree with you that there is not a simple theory of long-term climate phenomena. Indeed, I doubt that such a theory is possible in the scientific sense of the word "theory". I think that there will never be a Newton or an Einstein of climatology that pulls everything together, simply because there are so many pieces to the puzzle.

There is no mathematically rigorous theory of climate change on all time scales. There is established physical oceanographic and meteorological theory, less established glaciology, and quite a muddle on biogeochemistry. Of course there is some argument on how much this applies to shorter time scales, but certainly in the long run the astrophysics of the sun dominates the picture. Also tectonics plays a role in the long run.

This vast confluence of phenomena may be horrifying to a pure physicist. Those of us who are attracted to the field find it beautiful and motivating.

Since we agree that a powerful unifying theory of all this is unlikely, should we give up and say nothing is knowable? I think that it is incorrect to argue that since we do not know everything we must act as if we knew nothing. If it were, we would never consult medical doctors.

In fact there is a consensus, but there is not a theory. Nobody claims there is a theory in the sense you mean. The serious question is the amount of skill that informs the consensus. I am going to try to elucidate that here.

As to your final clause, I know that English is not your first language, but I can't entirely parse it. It seems a bit inclined to stir up controversy on matters other than science. Please take such controversies elsewhere; I am trying to create an outreach site, and we should show people the best of science.

I would welcome you following along in this series and acknowledging points of substance where we agree and criticizing where we disagree as long as you refrain from bringing your political inclinations into the foreground.

Let us provide a model of scientific discourse rather than yet another platform for hostility and suspicion.

I want this to be a blog about climate science, not about climate change science, but I'm aware that a lot of the interest and the discussion will be about climate change.

Readers who haven't heard much about climate change except from journalists and politicians could do worse than to ehear about it from scientifists. These guys are the real deal; I've heard of many of them and I have great respect for all the ones I've heard of. (I especially hold Isaac Held in high esteem.) Anyway, have a look:

http://www.earthsky.org/article/50989/20-scientists-speak

OK, so let me see if I've got this straight? The probable number and intensity (and mean error) of Pineapple Expresses this winter is weather (prediction). The long-term probability of Pineapple Expresses is climate. But when a statistical correlation between ENSO conditions and Pineapple Express probability is observed, is that climate or weather? Do proposed mechanisms for the correlation fall under climate study or weather study?

The distinction between climatology and meteorology is not as distinct as the distinction between climate and weather. Not all climatologists are meteorologists (some are glaciologists, oceanographers, hydrologists, geochemists...) but arguably all meteorologists are climatologists.

That said, your question (about a correlation) strictly falls under climatology as I defined it. Correlations are statistical, hence climatological reather than meteorological.

Is the prediction of the next El Nino cycle weather or climate? Now that's a harder question.

To stick to the clarity of my definitions, it can't be called climate, because it's a single event. It's not really weather prediction, though, because the event is happening in the ocean. Ocean time scales are longer than atmospheric time scales, so the dynamic predictability limit is longer.

An El Nino prediction is therefore more like oceanographic weather prediction than like climate prediction, but unfortunately for the clarity of my presentation sometimes it's called a "climate prediction" for historical reasons.

OK, so if we find a much stronger correlation between the number and intensity of Pineapple Expresses in a single NH winter and the maximum retreat of Arctic sea ice the following summer than we do between average global temperature and such ice retreat the question of whether the number/intensity is the proximate cause of Arctic sea ice retreat is a climatological question? Is this limited to statistical studies or does it extend into mechanism?

What I'm driving at is the ongoing perturbation. The chaos in the Lorenz model above is only dependent on the initial condition, but suppose you extend it by assuming that every time the dot tops out one of its variables may be subject to a new rounding error, depending on the decay of radioactive particles? That would convert the system into a truly random chaotic "weather", but the climate would remain predictable, right?

Would that map cleanly onto the earlier example of Pineapple Express to Arctic Ice that I used above?

AK, I find your question peculiar. Is there anything to it or is it a complete hypothetical.

I am not making a distinction between climatology and meteorology. I am making a distinction between climate prediction and weather prediction. The whole mathematical foundations differ in these cases.

Mechanistic and statistical studies of such a correlation could in both cases be considered meteorological and climatological.

The Lorenz system is deterministic, and typically full-blown supercomputer climate models are deterministic.

Whether the actual world weather system is deterministic is a somewhat inconsequential question. In the end you could always invoke quantum theory and say there must be a random component to reality.

The argument I presented here does not depend on whether there is a random component to the model. In other words the answer is that in the Lorenz case, the climate would indeed remain stable and hence predictable.

I don't fully understand what you are driving at with your last question, though.

Hi Michael...

My question was hypothetical in detail (i.e. Pineapple express, etc.) I just picked a case that didn't seem implausible to my limited knowledge. What I'm trying to understand is how climate modeling deals with "internal" perturbation (which may be the wrong term for it, I don't know).

In my reading about chaos theory (primarily Kaufmann) I've mostly seen systems that were completely deterministic where perturbations were from outside the system. In the weather, as I understand it, there is continual perturbation from things that aren't part of the atmospheric system. (A butterfly flapping its wings, where a tractor is in a field when a t-storm is developing, etc.)

If I'm understanding correctly, the climate would be the basin of attraction that the weather rides in, just as the "butterfly" in the picture above represents the basin of attraction that the dot's path rides in. So I was trying to use the analogy to model "internal" perturbations to the weather with random "jogs" to the dot as it travels.

The question about mapping had to do with the limits of the analogy: could the random "jogs" to the dot's path be analogized to random (or pseudo-random) variations in number and intensity of Pineapple Expresses.

I think you've answered my basic question when you mentioned the difference in mathematical foundation between climate and weather prediction. I'm going to have to chew on it for a while.

Thanks.

Michael, you wrote: "In other words, I've shown that chaos in weather doesn't demonstrate chaos in climate."

No, you just have shown that chaos in one particular highly-refined and abstracted mathematical model demonstrates a structurally-stable attractor. What you elected to not recognize is that Lorenz attractor is not weather, and infinitely-long averages of this Lorenz attractor is not Earth climate. True, the Lorenz attractor has been derived from a Galerkin expansion of gas dynamics model of an atmosphere, but it is only a zero-order approximation of an infinite series of complex base functions, which itself (the expansion) is valid only under certain narrow assumptions and other simplifications. It is known from theoretical and experimental research in hydrodynamics that even turbulent motion on quasi two-dimensional flows like the thin atmosphere alone leads to inverse cascade of eddies leading to longer time scales. In addition to this, atmospheric models usually assume a lot of factors as constants, which is not true in real climate. To get more realistic (but still highly abstracted) model, consider that the coefficients in the Lorenz equations (which are held constant in your example) are in turn a result of another Lorenz attractor operating on a 1000+ year scale, which state variables are, say, polar ice coverage, average albedo, and some factor from deep ocean circulation. Then everything you wrote in the header is not true. In short, you failed to demonstrate that subsequent approximations of the weather model are converging and producing the same stable long-term behavior, a frequent methodological mistake in most (if not all) studies on climate models.

I have shown that chaos in the dynamics of a system is not sufficient to demonstrate chaos in the statistics of a system.

I have not demonstrated whether climate is or isn't chaotic. I have admitted as much.

I have shown that it isn't sufficient to stipulate that weather is chaotic in order to prove that climate is chaotic. The details are particular to the system, and the system isn't well-specified in any case.

I don't think this is the place for a very asbtruse conversation. You have already shown yourself in our prior encounters to be very mathematically sophisticated, more than myself, but in our past conversations you have missed the point of how to apply mathematics to think about climate.

I hope you will follow along in the history of climate science that I am presenting here. You may get more of an idea of where the ideas came from and the nature of our knowledge.

Judging from your email address you are in the same city as myself. Why don't you look me up and we can discuss matters over coffee?

Michael,
We continue to have communication problems. You again reiterate your "slippery-slope" chain of logic: {climate == statistics of weather} => {weather == Lorenz attractor} => {Lorenz attractor has stable statistics}=>{climate is stable and predictable, QED}. First, as I already said, weather is far from being fully represented by Lorenz attractor. Second, 30-year average (==climate) is not the same as infinite statistics of attractor's invariant measure: if you perform time-limited sliding averages of Lorenz variables, they will still fluctuate and be chaotic. Also, not every "strange" attractor has simple statistics as Lorenz. Please look at the following research summary by Daan Crommelin,
http://homepages.cwi.nl/~dtc/pubs/thesis.pdf
which I consider as a very good example of research. Please note that even the 10-dimensional Crommelin model (which is much more realistic than Lorenz truncations) exhibits chaotic fluctuations on at least 250-years time scale. Please also note that this model is still severely truncated and does not include coupling to variety of other, much slower changing variables (like trends in ice coverage or shape of oceanic conveyor), which certainly would extend time scales on the long side of the spectrum. Therefore, I don't see any reason to "follow along in the history of climate science that I[you] am presenting here", because the history clearly does not represent contemporary knowledge about behavior of complex systems. Nor am I interested in "where the ideas came from and the nature of our knowledge", because the nature is obviously grossly underestimated, and even well-known ideas of nonlinear dynamics are misapplied.

I certainly appreciate your invitation to discuss matters over a cup of coffee. Are you suggesting that you may publicly change your position after being presented with more elaborate explanations and examples? Then I am all for coffee ;-)

Regards,
- Alexi

I try to be an honest man and will withdraw errors when need be. You can see an example here:

http://gristmill.grist.org/story/2007/11/11/174117/52

wherein I withdraw an assertion about the carbon emissions of electric vehicles.

However, you are continuing to argue against a position that I didn't take.

> We continue to have communication problems. You again reiterate your
> "slippery-slope" chain of logic: {climate == statistics of weather} =>
> {weather == Lorenz attractor} => {Lorenz attractor has stable
> statistics}=>{climate is stable and predictable, QED}.

I do not say that. I say that the argument:

weather is chaotic AND climate is the statistics of weather IMPLIES climate is unstable and unpredictable

is refuted by the observation that:

Lorenz system is chaotic AND Lorenz sytem has stable statistics.

That is, the chaos of weather is insufficient to prove the chaos of climate.

I did not in fact make a claim about chaos in climate.

I hope we can clarify what exactly each other thinks by personal conversation.

Michael,

Ok, I might be jumping into critique of the implications from your statement. However, even your statement alone is still false.

The argument "weather is chaotic AND climate is the statistics of weather IMPLIES climate is unstable and unpredictable" is not refuted by the Lorenz system example. As I already tried to formulate, apparently not very clearly, climate is not "statistics" of an attractor. When you say, “Lorenz system has stable statistics”, you imply infinitely long averages, since only then the “statistics” will be stable. Climate, according to most definitions, is a limited-time (30-years) running average of weather. Running average is just a regular, smooth function of state variables. Any reasonable function of chaotic state variables is still a chaotic function. True, it will have somewhat smaller range of variation and slower frequency spectrum, even in the Lorenz case, but is not it true that climate variations are somewhat smaller than weather variations?

For example, if you consider some empirical data from, say, paleo reconstruction of Earth climate,
http://www.scotese.com/climate.htm
application of your concept of “stable statistics” would result in an observation that Earth climate has a stable mean of approximately 17C, nothing more. I am sure your infinite statistics is not the right tool to assess the climate variability issue.

In short, you are re-defining the climate as infinitely-long statistics, while it is not. That's where the mistake is.

Cheers,
- Alexi

P.S. I consider this as an important public discussion, so I would prefer to keep it open.

I would prefer to talk to you in person to determine whether you are amenable to reason.

Our correspondence in the past does not show great promise. Based on past experience, neither I nor other informed people cannot convince you of much of anything about climate in text exchanges. The effort would only serve to confuse a general audience, which is the opposite of what I want to do here.

Now, specifically, "climate according to most definitions" has nothing to do with 30 years; the 30 year mean is a convention to help us draw maps and has no special meaning. I concede that the definitions of "climate" are various. The one I used in this essay is perfectly legitimate and appropriate to the point I want to make to a broad audience, a poit which you are nitpicking rather than refuting.

I do not take the scotese site to be presenting a valid paleoclimate record, but that's beside the point of the direction you are pursuing. A more informed presentation on that time scale, like this one:

http://www.globalwarmingart.com/wiki/Image:Phanerozoic_Climate_Change_Rev_png

would still show broad variations. Those are forced variations, though, and have essentially nothing to do with the chaotic fluid dynamics of the atmosphere. I should think you were in a position to understand this even if my audience is not.

I'm not interested in your sniping and tearing at my or others' attempts to communicate. I have offered you an honest good faith conversation in person. If we make some sort of progress at mutual understanding I would be willing to converse again publicly.

I do not offer you a place to waste my readers' time nor my own.

Readers wishing to examine the past history of conversation with Alexi are referred to http://tinyurl.com/264dln