Question on whether climate is chaotic or not

[sylas]

Of course climate is chaotic given that the inputs are chaotic.

I tend to use the word "chaotic" to mean arbitrarily small changes in input can lead to large changes in output.

The notion of "tipping" point is related but not quite the same. It refers to cases where a system can slip from one comparatively stable condition to another as you pass a certain threshhold. A system with hysteresis, for example, has tipping points.

The clearest example of tipping points so far in this thread would be the ice ages. The evidence is not completely conclusive, but it is widely considered that ice ages over the Quaternary period, which we can see in the graphs of the thread, are caused by small changes in Earth's orbit; and moving in or out of an ice age occurs as a tipping point is passed, leading to a cascade of changes in the whole climate system that together raise, or lower, temperatures more than one would expect from the orbital changes alone.

The Quaternary contrasts with more stable conditions earlier in the Cenozoic, and one major hypothesis for this relates to the particular arrangements of land masses, which contribute to the conditions that allow for the tipping point. Specifically identified features have been the almost enclosed northern Arctic ocean and the existence of a contrasting case in the South, with a free passage for ocean circulations around the Southern Antarctic oceans.

Look at solar cycles - which are mostly, but not precisely periodice (~ 11 years).

Look at volcanic eruptions - which are relatively random events - some of which have a dramatic impact on climate.

But one can have bounded chaos, which means one cannot predict the trajetory precisely, because one cannot predict the future, besides the fact that there is still much we do not know.

Yes indeed. The 11 year solar cycles are poorly understood, but they are periodic, not chaotic. They also have a comparatively small impact. There may be stronger impacts from longer term and much more mysterious cycles, in which the 11 year cycle may be totally suppressed. This is often proposed as a factor in the "little ice age". Interestingly, the 11 year cycle seems to be particularly slow getting started for the next solar maximum at present.

Volcanic eruptions do have a dramatic impact; though it tends to be in the form of random "spikes" that then die away in the years following an eruption; or the decades following an eruption if it is a big one. The very fact that there is a recovery after an eruption indicates that the climate system itself is not chaotic, even though the input may be unpredictable.

The frequency of eruptions world wide can vary; a period of time with comparatively few major eruptions is sometimes proposed as a contributing factor for the rise in temperatures in the early part of the twentieth century.

The comment about "bounded chaos" is particularly important. Weather is certainly chaotic. Climate, however, is usually defined as the range (or bound) within weather is found. The bound itself is not obviously chaotic at all. It may have tipping points -- as we see suggested in the ice ages -- but the response seems to be much too regular to be truly chaotic, in the normal sense of the word.

In my view, the evidence shows that climate is complex, and hard to predict; that it does have tipping points although it is very hard to identify them precisely; but it is not actually chaotic, as the word is usually defined.

Cheers -- sylas

 

[Astronuc]

I tend to use the word "chaotic" to mean arbitrarily small changes in input can lead to large changes in output.

Chaos refers to predictability, where 'stability' refers to the impact of change on output.

The notion of "tipping" point is related but not quite the same. It refers to cases where a system can slip from one comparatively stable condition to another as you pass a certain threshhold. A system with hysteresis, for example, has tipping points.

Tipping point is different, and the real issue is one of desirable or compatible outcome with respect to the process.

Nature (a physical process) is self-regulating. The real issue with respect to 'climate change' is whether or not any change is compatible with human existence.

If climate change (warming or cooling) is occurring, then one has to consider at what point does it become incompatible with our currently way of living. This matter is a different topic.

However, the question (OP) is about whether climate is chaotic or not, and I believe we have demonstrated that it is.

The 11 year solar cycles are poorly understood, but they are periodic, not chaotic. They also have a comparatively small impact. There may be stronger impacts from longer term and much more mysterious cycles, in which the 11 year cycle may be totally suppressed. This is often proposed as a factor in the "little ice age". Interestingly, the 11 year cycle seems to be particularly slow getting started for the next solar maximum at present.

The Maunder minimum would represent an aperiodic (chaotic) event. And yes - the Sun is unusually quiescent at the moment - yet another aperiodic (and unpredictable) event. Who would have predicted that 11 years ago, let alone last year, or 6 months ago. Of course, it could return to it's more usual trend in the next few days, weeks, months or years. But we just won't know until it happens.

Volcanic eruptions do have a dramatic impact; though it tends to be in the form of random "spikes" that then die away in the years following an eruption; or the decades following an eruption if it is a big one. The very fact that there is a recovery after an eruption indicates that the climate system itself is not chaotic, even though the input may be unpredictable.

Recovery on a different climate trajectory.

Another term for chaotic is 'noisy', and the 'noisiness' can be insignificant (even if not predicatible, but it's nevermind) or it can be significant.

 

[Astronuc]

Time out pending moderation. So save your thoughts.

Thread is re-opened. Please keep posts on-topic, which is about "whether climate is chaotic or not".

Claims and assertions must be supported by evidence from textbooks, scientific journals, and other peer-reviewed sources.

 

[seycyrus]

Of course climate is chaotic given that the inputs are chaotic.

I do not know if this is correct. Do chaotic inputs into a system *always* lead to chaos? Is this proven in chaos theory? This is the question I was trying to get at when I questioned D P.

I wonder about the human body as an example. I believe that certain subsystems are chaotic, but the whole is not...?

Is the current notion that the entire universe is chaotic, but simply riding along in an attractor?

Edit: I am reminded about a certain chapter intro in Hitchhikers guide to the Galaxy.

 

[Borek]

Vanesh already sumarized perfectly what I have on my mind when I hear words "weather", "climate", "chaos". My $0.02 is: single nuclear fission event is unpredictable, mass of the sample large enough after time t can be predicted with a very high precision. So I can imagine situation in which weather is chaotic as it is, but climate is quite predictable if you know how to describe it properly. Whether that's the case I have not the slightest idea.

 

[Andre]

Please keep posts on-topic, which is about "whether climate is chaotic or not".

However, the OP states:

...Their argument is more or less that because climate changes over longer periods they don't need to treat it like a chaotic system (with all the inherent unpredictability that comes with a chaotic system).

But then on the other hand, they claim that a moderate increase in Co2 will cause this "tipping point" to occur causing run-away global warming...

It appears that the intention was to discuss contradictions between chaotic behavior and triggering runaway conditions.

 

[vanesch]

Ok, let us get some definitions right.

Chaotic system: I think the wiki entry on it is pretty good.
http://en.wikipedia.org/wiki/Chaos_theory

A chaotic system is first of all a deterministic dynamical system that has initial conditions of its state one can specify and starts out from there.

The main property is indeed "sensitivity to initial conditions", but there's also another important condition, which is "topological mixing" (the image through dynamics of any open subset comes arbitrarily close to any point of the considered phase space). Sensitivity to initial conditions alone is not really sufficient, although if the phase space is bounded, sensitivity to initial conditions everywhere (usually or always?) also leads to mixing.

Divergent systems are sensitive on initial conditions, but not necessarily chaotic. A run-away system for instance, is not chaotic, because it lacks this mixing property.

So my personal "feel" for a chaotic system is that it "diverges but comes back pretty close but not exactly". I don't know how close that gut-feeling definition is in agreement with what is the right definition of a chaotic system.

Note that these properties are properties of the dynamics of the system, and have nothing to do with any properties of any input signal. So strictly speaking, a system is not chaotic because it receives external "chaotic" signals (you cannot really have "chaotic" signals, you can only have chaotic dynamics and eventually output signals of a chaotic system, but it is hard/impossible to find that out).

But it is worse: a dynamical system can also only be chaotic in certain subspaces of its phase space, and non-chaotic in others.

Related to this discussion are the questions of whether current climate models are chaotic in the range of validity and interest where they are used.

That really shouldn't be difficult to find out, by running those models with different initial conditions. Point is, most of these models are stochastic, so strictly speaking the definition of chaos doesn't even apply.

It is even pretty evident that running climate models cannot be chaotic exactly where they are used, because otherwise they would generate widely different results from run to run and it would not be possible to draw any conclusions from that.

In how much that this has anything to do with the real climate dynamics is very hard to tell. Historical data will probably (my opinion) not be of much use, as we probably don't know the external inputs.

 

[vanesch]

It appears that the intention was to discuss contradictions between chaotic behavior and triggering runaway conditions.

First of all, I don't think any serious climate scientists considers a run-away climate. But even a run-away climate would not be a chaotic dynamics.

A bifurcation in a dynamics is also not chaotic, but it is true that there is a small region of phase space that is then extremely sensitive to initial conditions, namely all the open sets around the trajectory that hits the bifurcation, because they are split in (at least) 2 pieces.

So a "tipping point" (= bifurcation ?) by itself would not indicate any chaos in climate dynamics by itself.

It is very well possible that long-term dynamics of the climate turns out to be chaotic, but that will be on a time scale much longer than what is of interest in the AGW debate.

Note for instance that the solar system is believed to be chaotic (see http://www.sciencemag.org/cgi/content/abstract/257/5066/56 ) but nevertheless allows for extreme precise predictions of orbits over millions of years.

 

[Astronuc]

Tipping point is about stability, and there seems to be some confusion between chaos and stability.

Chaos refers to uncertainty and noisiness in the system behavior.

I wasn't referring to the choatic or random inputs, but the response to those inputs.


Earth's climate is also not a closed system, and not only are the boundary conditions variable (noisy) and non-linear, the system itself is non-linear - but it's bounded somewhat - or at least part of the system is bounded.


Another matter is the definition of choas theory or chaotic system. Traditionally or conventially, "Chaos theory is an area of inquiry in mathematics, physics, and philosophy which studies the behavior of certain dynamical systems that are highly sensitive to initial conditions." What initial condition do we apply to today's climate?

Chaos also looks at system response/behavior to perturbations and noise (periodic and aperiodic), which also include perturbations to boundary conditions or transients, e.g. volcanic eruptions or meteoric impacts, or somewhat noisy inputs like solar-cycles. Add to this long term tectonic drift. The equilibrium shifts. The challenge is to determine why and attempt to predict weather and climate in the long term.

One can look at turbulence in a fluid. Locally the velocity changes constantly, but it is bounded by constraints, e.g. structure or limits on momentum and energy.

 

[Andre]

First of all, I don't think any serious climate scientists considers a run-away climate.

It seems that one of my posts got deleted that introduced somebody like that. Was that illegal?

 

[Monique]

To re-iterate Astronuc's words: let's keep it on the topic of whether the climate is chaotic or not and how that relates to the models Earth scientists use.
The OP explained himself: https://www.physicsforums.com/showpost.php?p=2468484&postcount=13

Off-topic posts were deleted to get the thread back on track.

 

[Astronuc]

One has to look into chaos as applied to multi-input, multi-output (MIMO) systems.

Perhaps the bifurcation is a more rigorously appropriate term.

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family. Examples of such families are the integral curves of a family of vector fields or, the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour.

http://en.wikipedia.org/wiki/Bifurcation_theory

 

[Xnn]

OK; let's focus on the topic.

Wikipedia is not an acceptable source for this site.

So, where is the textbooks, scientific journals, and other peer-reviewed sources
that define a chaotic system?

So far, most post are just about opinion (some more reasoned than others).

 

[vanesch]

Wikipedia is not an acceptable source for this site.

So, where is the textbooks, scientific journals, and other peer-reviewed sources
that define a chaotic system?

We are talking here about elementary definitions of concepts in dynamics that have a clear meaning, so in as far as the Wiki article is serious, it is good enough a source for the clarification of such things, I would say. There is nothing controversial in these "claims" of elementary definitions, like chaotic system or what is a bifurcation. We're in the "let's get the definitions of the words right" stadium.

It corresponds pretty well to what I remember from reading the book "deterministic chaos" http://books.google.fr/books?id=WaEGkJ3XAtEC&lpg=PP1&ots=4tbrm7zFEv&dq=deterministic%20chaos&pg=PP1#v=onepage&q=&f=false which I have somewhere on my shelf but not at hand right now.

 

[Astronuc]

Don't know if this helps.

If the parameter V-m is increased even further, the behavior changes to an apparently random, erratic, and aperiodic waveform. This situation is illustrated in Figure 1.4. Such a bounded aperiodic behavior is known as chaos.

from Nonlinear Phenomena in Power Electronics: Bifurcations, Chaos, Control, and Applications
Soumitro Banerjee (Editor), George C. Verghese (Editor)
http://media.wiley.com/product_data/excerpt/38/07803538/0780353838.pdf

I was thinking IEEE would have a formal definition somewhere, for example

In this article, a nonlinear dynamical phenomena leading to bifurcation and chaos in power systems is explored using a sample power system. After giving an introduction to nonlinear dynamical power systems in section II a basic knowledge to nonlinear dynamics and chaos theory is given. Section III deals with bifurcation theory. In section IV a dynamical power system model has examined. In section V the theories are applied to a sample power system example and various bifurcation and chaotic phenomena are examined.

An application of chaos and bifurcation in nonlinear dynamical power systems
Kuru, L. Kuru, E. Yalcin, M.A.
http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1344840
Intelligent Systems, 2004. Proceedings. 2004 2nd International IEEE Conference
Publication Date: 22-24 June 2004
Volume: 3, On page(s): 11- 15 Vol.3
ISSN:
ISBN: 0-7803-8278-1
INSPEC Accession Number: 8109177
Current Version Published: 2004-10-25


Another issue, and an important one, is - when does weather become climate?

Parts of North Africa (Algeria, Tunisia, Libya(?)) were once the bread basket of the Roman Empire. Now it's mostly desert.

More recently, the Horn of Africa (W. Somalia and E. Kenya) are suffering a third year of drought, but just to the north (N. Kenya and Uganda), they've had devastating floods.

Australia now has persistent drought in the south, and that's a lot different from when I lived there 43+ years ago.

 

[seycyrus]

A chaotic system is first of all a deterministic...

*deterministic* is also problematic in that is can be misconstrued, to imply arbitrary predictability. Certainly the system itself knows it's own dynamic evolution, but that doesn't help out any onlookers.

...Sensitivity to initial conditions alone is not really sufficient, although if the phase space is bounded, sensitivity to initial conditions everywhere (usually or always?) also leads to mixing.

In class,it was taught that neither *usually or always* can be assumed.

...So my personal "feel" for a chaotic system is that it "diverges but comes back pretty close but not exactly".

Eventually.

...So strictly speaking, a system is not chaotic because it receives external "chaotic" signals (you cannot really have "chaotic" signals, you can only have chaotic dynamics and eventually output signals of a chaotic system,

I was wondering about using those outputs of a chaotic system as inputs for another systems. I.E. the sun's output (postulated as chaotic) into the Earth's climate (questioning the chaosticity (woot woot!) of this system)

Thinking about it, I believe that such chaotic inputs cannot drive another system "chaotic". it is either chaotic or not from the get-go. Of course this just begs the question about whether it is correct to consider any systems as independent from another.

But it is worse: a dynamical system can also only be chaotic in certain subspaces of its phase space, and non-chaotic in others.

Difficulty amplified again by the requirement that the attractors be dense.

 

[Astronuc]

Try to define the Climate (system) statepoints, inputs and outputs.

Also try to define the climate system and it's boundaries.

 

[seycyrus]

OK; let's focus on the topic.

Wikipedia is not an acceptable source for this site.

So, where is the textbooks, scientific journals, and other peer-reviewed sources
that define a chaotic system?

So far, most post are just about opinion (some more reasoned than others).

Vanesch has got it right.

1) Sensitivity to initial conditions.

2) In the phase space around the attractors, it has to topologically mix.

3) The orbits are dense.

 

[Astronuc]

Interesting article - Chaos and stability of the solar system
http://www.pnas.org/content/98/22/12342.full

In its scientific usage, chaos is not a synonym for disorder, rather it describes the irregular behavior that can occur in deterministic dynamical systems, i.e., systems described by ordinary differential equations free of external random influences. Chaotic systems have two defining characteristics: they show order interspersed with randomness, and their evolution is extremely sensitive to initial conditions. Extreme sensitivity to initial conditions is quantified by the exponential divergence of nearby orbits. The rate of such divergence is characterized by the e-folding time scale called Lyapunov time. A second characteristic time scale is the escape time, which is the time for a major change in the orbit.

So if one applies the requirement that the evolution (of Earth's climate) is extremely sensitive (or at least sensitive) to 'initial conditions', then climate may not be chaotic. What are the initial conditions?

It does represent a dynamic with both periodic and aperiodic behaviors.

 

[seycyrus]

Interesting article - Chaos and stability of the solar system
http://www.pnas.org/content/98/22/12342.full

Nice find. I wonder what the lyapunov and escape time scales are for the weather/climate. I failed with google.

What are the initial conditions?

I always understood initial condition to be those you stipulate at t0, any t0. Or are you pointing out that we don't even know which variables are required inputs?

 

[Astronuc]

I always understood initial condition to be those you stipulate at t0, any t0. Or are you pointing out that we don't even know which variables are required inputs?

In the context of Earth's climate, I wondering at what point one takes the initial, or perhaps more appropriately, the reference time. The climate has constantly changed - sometimes slowly, sometimes abruptly.

In the context of today's climate, do we look back 100 years, 200 years, 400 years, 1000 years, 5000, 10,000, 100,000, 1 million, 10 million.

Also - what is the system (which defines statepoints) and what are the inputs, and more importantly - what are the 'natural' inputs/perturbations, and what are the 'anthropogenic inputs/perturbations - and what are the relative magnitudes of these inputs/perturbations?

 

[Xnn]

Vanesch has got it right.

1) Sensitivity to initial conditions.

2) In the phase space around the attractors, it has to topologically mix.

3) The orbits are dense.

Okay;

Then how can we answer the question if we don't know if
there are tipping points or how close we are to them?

 

[turbo]

PBS hosts a nice article on weather, climate, and chaos:

http://www.pbs.org/kcet/wiredscience/blogs/2007/10/climate-chaos-and-confusion.html

The explanation of how weather and climate are described by the illustration of Lorentz attractors is a good one, IMO, though greatly simplified, since weather and climate have a LOT of variables.

 

[seycyrus]

Okay;
Then how can we answer the question if we don't know if
there are tipping points or how close we are to them?

We really need for that nonlinear-mathematician friend of a poster to swing by ...Maybe if we are lucky, he will be a "Chaotician" (I cringed when they used that word in Jurasic Park)

Perhaps your question reflects the uncertainty the one scientist in question (Gavin??) meant to reflect upon when he said it "might" be chaotic.

How does one label a natural system as chaotic when one cannot formulate and solve expressions that correctly model the observed behavior? Does being able to model the sensitivity to initial conditions suffice to say that the natural system being modeled is chaotic? Not in a strict mathematical sense.

It's more straightforward for a simpler system. One can write down the equations for a double pendulum and compare the results with a *physical* double pendulum. One notes the behavior of the real beast and says
"Look at that pendulum go! It's all crazy-like!"
and then
"Wow, when I map out the trajectories in phase space solved from my equations of motion they get pretty crazy too! Why, they look just like that swinging contraption!"
and then
"I can show that these equations describe a chaotic system, therefore it seems fair to induce that the *real* double pendulum is chaotic."

But doing it for the weather ...

I'm liking the *might be* answer more and more.

 

[turbo]

I'm liking the *might be* answer more and more.

I think I'd like that answer even more if it were qualified with *on some scales*. Certainly the Younger Dryas event shows that global climate can take some pretty rapid swings. It brought on an ice age in the northern hemisphere, and much warmer temperatures in the southern hemisphere, so the results of that event were divergent N-S.

 

[Astronuc]

Then how can we answer the question if we don't know if
there are tipping points or how close we are to them?

That's the proverbial $64 million or now $64 billion (or maybe it's $6.4 trillion) question.

In order to know how to procede, one needs to understand the path/trajectory and the tipping points/pitfalls. If one makes the wrong assessment and/or wrong prediction, things might get dicier.

 

[Andre]

I was trying to shed some light on that but the post about that got deleted.

 

[Xnn]

Personally, I'm not fond of "We don't know" as an answer.

Here is an article from Scientific America which suggest that the
Earths climate becomes chaotic at around 1000 ppm CO2.
Fortunately, that is not expected to occur for at least 200 years.

http://www.scientificamerican.com/article.cfm?id=impact-from-the-deep

 

[vanesch]

Okay;

Then how can we answer the question if we don't know if
there are tipping points or how close we are to them?

But there are 3 different aspects:

- chaotic behaviour of the inherent dynamics of climate
- eventual existence of bifurcations (tipping points)
- relatively short term (a few centuries) predictability.

They are not related. The system can be chaotic with Lyapunov exponents which are of the order of 1/1 million years for instance, which would mean that predictability over centuries or hundreds of millennia isn't going to be a problem by this chaotic dynamics.

Bifurcations don't mean necessarily chaos, and even less "unpredictability". A beam under compression has a bifurcation point (from a certain stress onwards, it will bulge). That doesn't mean we can't calculate beam deformation.

So the argument "they say there might be tipping points, but then it is chaotic, and hence we can't make any predictions" is full of invalid inferences.

Let's first find out the dynamics already !

 

[Coldcall]

PBS hosts a nice article on weather, climate, and chaos:

http://www.pbs.org/kcet/wiredscience/blogs/2007/10/climate-chaos-and-confusion.html

The explanation of how weather and climate are described by the illustration of Lorentz attractors is a good one, IMO, though greatly simplified, since weather and climate have a LOT of variables.

Actually I've read that blog before and Alexi Tekhasski has some interesting points contrary to the author of that blog. Just read some of the comment below re Lorenz attractor.

As that pro-agw blog shows there is a concerted effort by the some in the agw community, including Gavin Schmidt at RC, who for some odd reason, don't want the climate system labelled as chaotic. However the silly part is that their models are indeed non-linear with all the inherent problems about initial conditions.

My whole point has always been that:

1) I am highly suspicious of any scientist who claims a computer model is based on an "unknown" physics. (in other words, if one can't explain the underying physics how can one be confident of the model?) I think i hold a reasonable position and if we were discussing most other scientific (less trendy)topics i think most would agree with that stance.

2) Any computer model which needs to virtualise the real climate system in a realistic capacity probably needs an infinite amount of variables and factors for which we are today probably aware of only a tiny fraction. The current models are just way too primitive to accomplish that.

3) I don't accept this idea (often stated by some in the agw community) that while these computer models may not be able to predict shorterm climate, they can predict long-term averages. This goes against very fundamental science known and demonstrated in chaotic systems, which states that small inaacuracies in initial condiions grow exponentially and irregularly the longer the clock runs. Remember that climate is longterm weather, no mattter what sort of clever semantic one wants to use to define "climate".

4) The climate is open ended and affected by cosmic phenomenom so the idea we can create and idealisation of the climate in a model is a non-starter.

So I am not arguing that Co2 does not affect the climate or we are not having climate change, or perhaps even some warming depending on what time scale one uses as a boundary.

I'm simply saying that with our current knowledge of the scientific fundamentals which MUST be adhered to in any theory, there is too large an uncertainty and unpredictability factor for claiming anything is settled.

That is how i am sceptical.

PS: The burden of proof is on the computer models to confirm and validate their predictions as with any scientific theory. They have not done so, and in fact, their models did NOT predict the post 98 cooling period. So those models have failed at the first hurdle.