By James Annan and William Connolley
In this post, we will try to explain a little about chaos theory, and its relevance to our attempts to understand and forecast the climate system. The chaotic nature of atmospheric solutions of the Navier-Stokes equations for fluid flow has great impact on weather forecasting (which we discuss first), but the evidence suggests that it has much less importance for climate prediction.
Chaos is usually associated with the sensitivity of a deterministic system to infinitesimal pertubations in initial conditions (the full definition is a bit more difficult: see technical bit at the end). The identification of chaos in atmospheric systems is due to an accidental discovery by Lorenz in 1961. Using a greatly simplified model of the atmosphere, he restarted a computation from part-way though a previously-completed run. However, for the initial conditions, he used a printout that only had 3 figures of precision, compared to the 6 used internally by the computer. The outputs of the two runs initially appeared indistinguishable, but then diverged and became wholly decorrellated. So it was an atmospheric model which provided some of the first insight into the ‘chaos effect’, thus teaching us something quite profound about nature
In fact, this type of behaviour had already been identified and studied more than 60 years earlier by Poincare, in the form of the “3-body problem” of celestial dynamics. 2 stars (or planets etc) in orbit around each other will each follow a regular ellipsoidal trajectory around their joint centre of mass. However, when a 3rd (or more) body is thown into the mix, their future trajectories may be highly sensitive to the precise initial conditions. One extremely useful result of chaos theory is the design of complex orbits that enable spacecraft to travel great distances in a fuel-efficient manner, by analysing the Earth, Sun and spacecraft as a 3-body system (eg see the articles here and here).
Back in atmospheric physics, chaotic behaviour is a highly-studied and well-understood phenomenon of all realistic global models, arising directly from the nonlinearity of the Navier-Stokes equations for fluid flow. So any uncertainty in the current atmospheric state, however, small, will ultimately grow and prevent accurate weather forecasts in the long term. This is the sort of thing that is easy to show with numerical models of the atmosphere. Simply perturb a reference run, and see what happens. So long as the perturbation is not rounded out by the limited numerical precision of the model, it will invariably grow.
Here is an example using the HADAM3 model. One standard run was performed, and then another run was started where the pressure in a single grid box was changed by 10-10 (one part in 1015 of the model value) For a bit more about this experiment, see here. The first graph shows how the RMS difference in sea level pressure increases over time, and the second graph shows the evolution of the spatial pattern of differences. The perturbation rapidly saturates the highly local convective mode in the tropics, before more slowly spreading to the much larger mesoscale differences that matter to weather forecasters (note the scale changes). If this model resolved hurricanes, then their appearance and paths would be completely uncorrelated in the two runs – a classic example of the “Butterfly effect“.
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The x-axis is time in days: 0-5 (top); then 0-15 (mid); then 0-89 (bottom). By day about 25, and definitely by day 60, the difference between the runs has saturated: their weather is totally different, so no further growth in RMS occurs. The 5-10 day oscillations in the last month are what you’d expect to see from weather evolution. The y-axis is in Pascals. 100 Pascals is an hPa, ie 1 mb. Standard weather charts tend to plot pressure in contours of 4 hPa, so in real weather terms the diffs are sort-of negligible out to about day 15 (although this is a global value, so locally there will probably be bigger values). |
The top pic is about day 4 (in the not-much-happening phase). The middle, day 15 (in the exp growth). The bottom, day 31 (saturated). Note that the pics have a different contour interval. By days 15/31 we’re into “real meteorology” and hence the MSLP field is most different in extratropics, as it always is (its tropical dynamics, folks). |
Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat overstated. Chaos is defined with respect to infinitesimal perturbations and infinite integration times, but our uncertainties in the current atmospheric state are far too large to be treated as infinitesimal, and furthermore, all of our models have errors which mean that they will inevitably fail to track reality within a few days irrespective of how well they are initialised. Nevertheless, chaos theory continues to play a major role in the research and development of ensemble weather prediction methods.
Although ultimately chaos will kill a weather forecast, this does not necessarily prevent long-term prediction of the climate. By climate, we mean the statistics of weather, averaged over suitable time and perhaps space scales (more on this below). We cannot hope to accurately predict the temperature in Swindon at 9am on the 23rd July 2050, but we can be highly confident that the average temperature in the UK in that year will be substantially higher in July than in January. Of course, we don’t need a model to work that out – historical observations already give strong evidence for this prediction. But models based on physical principles also reproduce the response to seasonal and spatial changes in radiative forcing fairly well, which is one of the many lines of evidence that supports their use in their prediction of the response to anthropogenic forcing.
Fortunately, the calculation of climatic variables (i.e., long-term averages) is much easier than weather forecasting, since weather is ruled by the vagaries of stochastic fluctuations, while climate is not. Imagine a pot of boiling water. A weather forecast is like the attempt to predict where the next bubble is going to rise (physically this is an initial value problem). A climate statement would be that the average temperature of the boiling water is 100ºC at normal pressure, while it is only 90ºC at 2,500 meters altitude in the mountains, due to the lower pressure (that is a boundary value problem).
We can demonstrate this sort of climate response clearly in the Lorenz model, or any more complex climate model. Perturbing the initial conditions gives a completely different trajectory (weather), but this averages out over time, and the statistics of different long-term runs are indistinguishable. However, a steady perturbation to the system can generate a significant change to the long-term statistics. Here is some output from a run of the Lorenz model in which a change was applied half way through. At time t=0, the parameter “r” (which relates to an idealised thermal forcing) is changed from 26 to 28. When viewed in close-up detail, the trajectory looks qualitatively similar before and after the change, but in fact the long-term statistics such as the mean value of z, and its 95% range, are changed. In this simple model, the steady perturbation changes the climate in a highly linear manner – increasing r again to 30 would add the same change on top of that shown for 26 to 28, and r=27 would sit half-way between the cases shown. Of course, these results cannot be directly extrapolated to the real climate system, but they do disprove the common but misguided claim that chaotic weather necessarily prevents meaningful climate prediction. In fact, all climate models do predict that the change in globally-averaged steady state temperature, at least, is almost exactly proportional to the change in net radiative forcing, indicating a near-linear response of the climate, at least on the broadest scales. The uncertainty is in the steepness of the slope, which is what “climate sensitivity” describes.
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It was thought until relatively recently that chaos provided a substantial practical challenge to so-called “optimal” model tuning and climate prediction with state-of-the-art climate models, since it generally prevents the use of one of the most powerful and widely-used optimisation and estimation procedures (an adjoint – eg Lea et al 2002). The obvious alternative method of exhaustively searching parameter space requires a huge number of model simulations, and the Climateprediction.net project is pursuing this approach. However, one of us recently showed how another efficient estimation method known as the ensemble Kalman filter can be applied to this type of problem (various applications here). This is certain to remain an active area of research for some time to come.
The climate of a model can be easily defined in terms of the limit of the statistics of the model output as the integration time tends to infinity, under prescribed boundary conditions. This limit is well-defined for all climate models. However, the real world is slightly messier to deal with. The real climate system varies on all time scales, from daily weather, through annual, multi-year and decadal (ENSO), Milankovitch, glacial-interglacial cycles, plate tectonics and continental configurations, right up to the ultimate death of the Sun. The average temperature, and all other details of the climate system, will vary substantially depending on the time scale used. So how can we talk meaningfully about “the climate” and “climate change”? Well, although there are interesting scientific questions to ask across all the different time scales, the directly policy-relevant portion is on the multi-decadal and centennial time scale. It is quite clear that the perturbation that we are currently imposing is already large, and will be substantially larger, by up to an order of magnitude, than any plausible natural variability over this time scale. So for the policy-relevant issues, we generally focus on the physical atmosphere-ocean system, sometimes with coupled carbon-vegetation system, and treat the major ice sheets, orbital parameters and planetary topography as fixed boundary conditions. It’s an approximation, but a pretty good one.
[Technical para. The phenomenon of chaos can be formalised through the use of “Lyapunov Exponents” (LE) and their associated Lyapunov Vectors (LV). The leading LE can be defined as the limit of the maximal time-averaged logarithmic growth rate of the distance between two nearby model states, as the integration interval increases without bound. The leading LV gives the direction in phase space (which depends on the specific initial state, in contrast to the LE which is a fundamental constant of the system) for which this maximum growth rate is attained. Chaos is indicated by a leading LE greater than 1, indicating that initially similar model states diverge over time. See also Chaos theory]
Re 49 Buckets again.
No it is not sensitive to initial conditions. The “trajectories” in the bucket analogy do not diverge. When the water reaches a certain level the bucket tips over. Change the angle a little and the level will change a little. Change the initial amount of water a little and the time it takes to tip changes a little. Small errors in the angle, water level or fill rate only result in small errors in predictions of when the bucket will fall. Errors will not grow exponentially.
I am only belaboring the point to clarify what chaotic means and to point out that a system can have multiple stable states without being chaotic.
I will defer to others to argue over whether the climate is chaotic or not.
I believe there is evidence of bucket-like mechanism working in the past (Younger Dryas). I don’t know what could possibly constitute evidence as applied to the future.
BTW, I’m not claiming authorship. The idea (can’t really call it a model) is not mine at all. The actual toy models exhibiting chaotic behavior also exist but it’s hardly news for you.
Re: 51.
No analogy is perfect. But even in this framework you can have exponential (or worse) error growth. Consider two similar realizations of the random process. In one case, the bucket get enough water to tip over. In the second, due to small errors in the initial conditions and/or calculation of the total amount of water collected over time, the bucket holds steady and never tips (let’s say there’s a small hole in the side of the bucket to allow for a long term quasi-equilibrium). Conceptually, you can make it as subtle as butterfly effect.
[Response: You are confusing non-linearity that allows two stable states with chaos. In a chaotic system, there are always perturbations that grow exponentially. In your bucket example, there is one one special state where that occurs. If you want a relatively simple chaotic system, try an inverted double pendulum with an up-and-down vibrating axle. -gavin]
Re 29, 37, 46 (DO events – chaotic?)
From the latest Nature:
Follow the Sun
Dansgaard-Oeschger events are rapid climate fluctuations that occurred during and at the end of the last ice age with remarkable regularity: there were 23 such events between 110,000 and 23,000 years bp, with a periodicity of 1,470 years. Identify the source of this cycle, and it should be possible to say what triggered these events. The Sun had been excluded as a possible cause because of the lack of a 1,470-year spectral contribution in records of solar variability. Despite this, Braun et al. present a new hypothesis that convincingly explains the 1,470-year period as the net result of two well known solar cycles, the DeVries and Gleissberg cycles, with periods of 210 years and 87 years.
Letter
Possible solar origin of the 1,470-year glacial climate cycle demonstrated in a coupled model
Holger Braun, Marcus Christl, Stefan Rahmstorf, Andrey Ganopolski, Augusto Mangini, Claudia Kubatzki, Kurt Roth and Bernd Kromer
Although William Connelly and James Annan know who I am, others may welcome this brief biography. I am an amateur earth system scientist who stumbled on the cause of the rapid climate change at the end of the Younger Dryas. This has consequences for paleoclimatology reaching well beyond the Holocene, but when I sought help from professional scientists in taking my ideas further I was dismissed with the comment that â??If you make such exaggerated claims then no wonder you are not believed.â?? As a result, I have pursued the investigations on my own, but have been loath to discuss them widely for fear, typical of a rank amateur, that they would be stolen. However, there is no point in hiding my light under a bushel any longer and besides, if I am correct, then a new rapid climate change is imminent and communication of my ideas is urgent.
The idea for the end of the Younger Dryas is quite plausible, despite originating from a comparison with the climate of the planet Venus. Venus can be thought of as being in a runaway state driven by the greenhouse gas carbon dioxide. Of course, it is not running away at present, because clouds of sulphur dioxide are capping the solar flux that the planet’s surface receives. My idea is that the Earth is in a similar state, but both the runaway greenhouse gas and the clouds that cap the temperature, are formed from water. (Thus the climate is in a chaotic runaway state and hence the weather appears to be random, but what we are really seeing is white noise.)
During the Younger Dryas, the sea ice spread out of the Artic ocean as far south as Ireland. This increased the planetary albedo causing cooling which together with the sea ice sealing off the ocean decreased the water vapour content of the atmosphere. This lack of water vapour acted to cool the planet further. When this extension of the Arctic sea ice melted, the change in albedo would have caused warming. Then an increase in water vapour would have lead to a runaway warming, only capped when sufficient clouds had formed.
The sea ice would melt rapidly because it would be even, floating on a level sea, and it would be affected by the positive feedback from the increased greenhouse effect of water vapour. The initial cause of the melting is probably solar warming since solar flux was increasing in the NH due to the Milankovitch cycles during the Younger Dryas. Another possible reason for the ice to melt is an increase in salinity due to the effect of the fresh water from Lake Agassiz being dispersed, and the North Atlantic losing fresh water by evaporation which was carried by the Trade Winds over the Isthmus of Panama into the Pacific Ocean.
However, there is the third possiblity; that Stefan is correct. It may be that the extent of the Arctic ice varies randomly, and if it retreats far enough one summer, it will trigger a threshold event, what I call a runaaway positive feedback. Moreover, white noise produced by chaos is indistinguisable from a random signal, so Stefanâ??s stochastocity could well be my chaos :-)
This runaway greenhouse effect of the water vapour cuts both ways, and can produce rapid warming from the collapse of ice sheets and rapid coolings from the sudden appearance of ice sheets. This fits the shape of D-O cycles. The colapse of a sea ice sheet causes a warming that melts the surface of the continental ice sheets leading to an extended rapid warming. This is reversed by the renewed slow growth of the continetal ice sheets once the land has dried and snow evaporated from the ice free sea form new ice sheets. Eventually, the cooling produced by the albedo from the growth of the land ice causes the sea ice to reform, and at that point a rapid cooling results. This cold stage lasts until the sea ice again melts and the cycle restarts. On occasions, the land ice forms ice shelves, which surge into the sea. As these icebergs melt, the fresh water they are adding to the North Atlantic allows it to freeze and thus a Dansgaard event can trigger a rapid cooling.
These ideas are similar to a paper by Gildor and Tziperman presented to the Royal Society in 2002 â??Sea-ice switches and abrupt climate changeâ?? http://www.journals.royalsoc.ac.uk/link.asp?id=n1q5pbnpf467cljy . Eli Tziperman was also a coauthor of another paper relevent to this – Li et al. â??Abrupt climate shifts in Greenland due to displacements of the sea ice edgeâ?? GEOPHYSICAL RESEARCH LETTERS, VOL. 32, L19702, doi:10.1029/2005GL023492, 2005.
If the Tiamat Hypothesis that water vapour is in a runaway state is correct, then there follows three important consequences. First, the Arctic sea ice sheet will melt suddenly, and it will cause a runaway warming. Instead of the rapid cooling and a return to an ice age shown in the film â??The Day After Tomorrowâ??, we will experience a rapid warming. The climate will change to that of the Eemian Interglacial with a speed similar to that which ended the Younger Dryas â?? three years.
Secondly, by the principle of uniformitariansism, this runaway state of water vapour should have existed throughout the history of the Earth. If so, it would explain the faint young sun paradox, the rapid warmings at the end of Snowball Earth, and rapid climate changes during the Cretaceous when sea level changes exposed or flooded carbonate platforms thus altering global albedo, and triggering runaways. For me these claims are evidence that the Tiamat Hypothesis is correct.
The third conclusion is that the current GCMs are wrong. This has already been pointed out by Professor Pielke Snr. I have amazed myself by finding the error, and later discovered that Professor Pierrehumbert was already suspicous of the same algorithm. In Section 3.5 of his â??First Course in Climateâ?? he wrote â??This shows that something is wrong with the slab atmosphere model.â?? http://geosci.uchicago.edu/~rtp1/geo232/Notes.pdf The GCMs use Schwarzschildâ??s equation to calculate radiative heat flow through a slab, however Schwarzschild wrote that the equation only applied where Kirchoffâ??s law of blackbody radiation was true. The Earthâ??s atmosphere does not radiate â??pure thermal radiationâ?? ie continuous blackbody radiation. It absorbs and emits pressure broadened lines. This means the slab models is wrong and that surface temperatures rise linearly with greenhouse gas concentration, rather than logarithmetically as believed at present. Water vapour increases non-linearly with temperature and hence it produces a non-linear greenhouse effect.
Just in case someone thinks that all this makes sense and tries to steal my ideas I will now add the following to make them completely unbelievable :-) Witn this new non-slab model, the greenhouse effect is concentrated at the surface of the Earth and in the boundary layer. This explains why the MSUs have not shown the warming expected in the troposphere. The warming that is measured there is due to the increase in aerosols, mainly the Asian Brown Cloud, and this explains why there is very little warming of the troposphere happening in the southern hemisphere.
Re 53:
Well, I’ve just begun reading up on D-O events, but, per usual, it doesn’t stop me from having an opinion. I think there are two issues here: one is, are they chaotic? the other is, will they ever be predictable? Gavin is asserting that they do not meet the definition of “chaos”. And far be it from me to question this. But perhaps the more important question is, even if D-O events are not chaotic, will they ever be predictable? And here Sashka’s bucket example is potentially informative. If D-O events do follow the “bucket” example, or something similar, could they ever be usefully predicted (note that, to be useful, the prediction would probably need to be able forecast a flip to within a century or better – not a millenium).
Sashka – note corrected spelling of your name!
Re: 53
Gavin, I have to repeat that the bucket is not stand-alone. It’s a part of the complex system with multiple response time scales. The bucket is not chaotic itself but it potentially lends the chaotic mechanism to the larger system.
In a chaotic system, there are always perturbations that grow exponentially.
You may be right but I’m not familiar with such a theorem. Conceptually, if there is an unstable (exponential divergence of trajectories) part of the phase space throough which every trajectory passes infinite number of times then the rest of the space could be locally stable.
Re: 56
Even if D-O events are perfectly regular and therefore predictable you still need to explain why they started and ended. Is it a chaotic mechanism or not – we don’t know. Thanks for the spelling :)
Getting my two cents worth in at this late point, I would like to support Ray’s comments. The issue is just signal vs noise, ie, internal variability vs externally-forced variations. Integrate some coupled atmosphere-ocean-land model forward in time, with no volcanoes, no variations in solar flux or CO2, etc. The model will have some variability on all time scales, incuding the centennial time scale of central importance in this discussion, not just for global temperature but, say, for precipitation on regional scales, Is this larger or smaller than the size of the signal we expect from the CO2 increase. Period. There is no need to distinguish “climate” from “weather” or to bring in the technical definition of “chaos”.
I don’t have as much confidence in the ability of our current climate models to simulate the magnitude of centennial-scale variability as do some of your other contributors. My main concern is that the ocean models used for climate studies do not simulate meoscale eddies, from which one can conceive of an “inverse cascade” to larger space and time scales. We have very little experience with integrations of eddy-resolving ocean models on centennial time scales. The ocean models we use are relatively laminar. Another concern is with interactive vegetation and its potential for generating some metastable transients, especially in semi-arid regions. But, agreeing again with Ray, it is the stability of the Holocene climate record that is the key piece of information here. For global mean temperature, say, it does not look like the models are missing anything important, and, consistently, no one has come close to generating internal variability comparable to the doubled CO2 signal. Rainfall in the Western US, for example, given the “megadroughts” seen in tree rings, might be a different story. Roger Sr focuses on these regional responses; we should be careful when addressing his concerns not to switch the discussion implicitly to signal-to-noise for the global scale. When Stefan argues that D-O events are not “chaotic”, the issue is really whether they are part signal (responding to some solar periodicity) rather than noise. The question is not whether stochastic resonance (Stefan’s preferred model for this phenomenon) is, by someone’s technical definition, “chaotic” or not.
Re #58 If you view the climate as a chaotic system then it is possible to explain how D-O events happen and to predict future ones. Here I am defining a D-O cycle as two D-O events, one warming and one cooling. And I am saying that these runaway events are caused by the positive feedback from the greenhouse effect of water vapour. Moreover, I am saying this does not conflict with Stefanâ??s view. It is just another way of looking at it, but it does give a little more insight into the mechanisms.
In the paper Held, I.M. and Soden, B.J. (2000) â??Water Vapour Feedback and Global Warmingâ??, Annual Review of Energy and the Environment 25 441 â?? 75 it is explained that if the value of βH2O , the measure of water vapour feedback, were greater than unity, the result would be a runaway greenhouse. You go on to write â??It is of course self evident that the Earth is not in a runaway configuration.â?? But this is not so during D-O events. The point to realise is that a runaway cannot go on forever. It is just a transitory state between two stable states. It eventually reaches a stable state just like the one that Venus is in today. Moreover, since a system will be repelled out of any unstable state, then the system will spend most of its time in stable states such as the Holocene. This is similar to saying that it moves from one strange attractor to another, and that the time spent moving from one to the other is much shorter than the times spent in either of them.
The first question is what stops this runaway from boiling the oceans dry, and the answer is the clouds. But, you argue, the clouds do not increase linearly with an increase in water vapour. That is true, and the climate system heats up until the clouds do increase. We are talking about a non-linear system here. Interestingly, I have just had that confirmed today when I read Kevin Trenberthâ??s slides. In slide 5 he explains how during summer the heat cannot escape from the North Atlantic and the result is hurricanes. http://www.ametsoc.org/atmospolicy/documents/October252005KevinTrenberth.pdf
Of course it is not even as simple as implied above. I have not mentioned ice, which plays a complimentary role to the clouds. For instance, when the Arctic sea ice melts, the planet will heat up until the climate has switched into a new state where the lost albedo of the ice is replaced by new albedo from clouds. Nor have I explained why it is that the GCMs do not show this behaviour. That is partly explained in a brief paper I have posted on the web at; http://www.abmcdonald.freeserve.co.uk/brief/brief.pdf
With regard to Gregory Lewis’s remark (# 6), what can look like multiple basins of attraction in a chaotic model may well be only one. Indeed, this is exactly the case with the Lorenz model of 1963, the first one to raise the possibility of chaotic dynamics in a climate model. The “Lorenz attractor” has two distinct “wings,” (the whole thing sort of resembling a butterfly), each of which appears to converge on a point, but does not succeed in doing so. As it approaches one point it then jumps over to the other wing and approaches its center point for awhile until it jumps back. These are not separate basins of attraction, but merely sub-parts of a single basin. However, there are these sudden jumps that look like cross-basin jumps and that are not mere discontinuities from sensitive dependence on initial conditions such as the famous butterfly flapping its wings in Brazil.
For some papers on chaotic dynamics in economics-climatic models and nonlinear dynamics, visit my website at http://cob.jmu.edu/rosserjb.
Barkley Rosser
Re: 59
If you view the climate as a chaotic system then it is possible to explain how D-O events happen and to predict future ones.
I don’t understand how predictability follows from everything else that you say. Or, in fact, from anything at all.
[Response:
If you decide that D-O events are chaotic, then all you’ve done is decide that you *can’t* predict them. Which I suppose would be interesting in itself. However, D-O events are unlikely in the holocene. Predictability might follow if you had a more mechanistic understanding of D-O events – William]
If you take “climate” to mean a statistical description of weather or of the output of the “climate system”, then speculating as to whether or not “climate” is chaotic represents a category error. Chaos is a property of the output of a dynamical system. The “climate system” as Roger describes it (although choosing to call it “climate”) is a dynamical system and it is reasonable to ask if it’s output is chaotic. “Climate”, as most people understand it, is not a dynamical system and doesn’t have output that can be described as chaotic or not.
Jim Dukelow
Stephan wrote in one of comments:
“But to the point: yes, in theory the climate system could show chaos on longer time scales. But my null hypothesis is that it doesn’t, until I have seen actual evidence that it does. And I have not. I have seen no data that show any clear evidence for chaotic behaviour. As Richard Alley has shown in a couple of papers, the ice core data of DO events are entirely consistent with stochastic resonance – which is not chaos but arises from a simple threshold process (“flicking of a switch”) in the presence of noise. I am not aware of any publication refuting this and showing evidence for chaos in these data.”
and:
“Can you point me to a scientific paper that provides evidence for DO events being chaotic?”
I think you get it backwards. It is known that the ocean is a turbulent chaotic system, atmospheric models (derived as a center manifold of a more general Navier-Stockes boundary problems) have also chaotic behaviors in certain areas of parameters. Therefore, it is a given that the whole Earth system is a coupled set of chaotic subsystems, which makes it a chaotic by definition. There is an opinion that every essentially large multidimensional system is chaotic, so, as a modern physicist, you should start with this assumption first. Of course, the spectrum of fluctuations of various variables is wide, and amplitudes of some slow components may be small. You are certainly free to abstract various sub-systems and sub-models to obtain some short-term predictability, but it is now YOUR responsibility to prove that you can NEGLECT all the remaining variables, not the opposite.
More, it is a common nature of a hyperbolical system (mathematically chaotic) to have areas in its phase space that are behaving in a simple deterministic way (locally “stable”), and have only few small hyprbolically-behaving areas. In fact, the normal areas dominate the space, vastly. BTW, success of your own model of stochastic resonance is a maniferstation of the existance of a hyperbolical point in the whole system. The stochastic resonance is just one of simplified models for more general hypebolic behavior of a dynamic system.
Cheers,
– Alexei
[Response: I think you are missing the point. While the atmos may be chaotic, its long-term averages aren’t – at least not in any meaningful sense – William]