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The chaos of confusing the concepts

Posted on 22 January 2010 by Jacob Bock Axelsen

Guest post by Jacob Bock Axelsen

One skeptic claim is that if weather is chaotic, then surely the climate must also be chaotic because "climate, of course, is very-long-range weather". This view seems to logically render all skeptic non-human-causes of global warming worthless, since all predictions would be equally futile. As everybody knows, weather is both chaotic and strongly seasonal. Air temperatures undergo significant daily variations, at times varying unpredictably by more than 20°C in a given location within a few days. Climate indeed varies non-linearly, but this has not prevented scientists from making good predictions. Here, we peek into chaos and why climate is rather predictable over both decades and centuries

In 1963 Edward Lorenz was studying the patterns of rising warm air in the atmosphere. It was known at the time that air could start to move if it came into contact with a warm object. The properties of air are such that it expands a lot when heated, it is a good insulator and it flows with relative ease - technically speaking, it has a high Rayleigh number. Think of a large hot air mass in the atmosphere as rising like a hot air balloon in the shape of a mushroom cloud (!). Based on known hydrodynamics Lorenz derived a set of simplified equations for this movement and found something amazing. For certain values of the parameters, the overall movement of the atmospheric air was oscillating unpredictably (Lorenz 1963):

Lorenz equations, oscillation air patterns
Figure 1: The simplified Lorenz phase space equations of air convection (top) and some numerical solutions (below, separated for clarity).

A major concern was that for small changes in starting conditions the system would always have an unpredictable outcome. This was the discovery of deterministic chaos and we knew there and then that we would never know the weather more than 10 days ahead without using disproportionate computing power with very little pay-off. Naturally, scientists knew about chaos from studying turbulence, which is not both smooth (deterministic) and unpredictable.

When Lorenz looked closer at his graphs, something exciting happened. A peculiar regularity emerged when he plotted the curves against each other: they were attracted to something never leaving a boxed volume. It was strange, because it was not a simple shape, but rather an entire subspace of points strangely smeared into three dimensions.

Strange Attractor
Figure 2: The numerical solutions of Lorenz' equations plotted in the same coordinate system. The curves are functions of time, so imagine them as a roller coaster tracing an invisible object called a strange attractor.

Even stranger, the structure is occupying not just a two dimensional surface but something which is more than two dimensional and less than three dimensional: it is two-dimensional plus a fraction. It exists in a so called fractal dimension. It never truly intersects itself thus committing every trajectory to infinite solitude. The object was therefore aptly dubbed a strange attractor.

A closer look reveals where the unpredictability arises. The blue and the magenta curves are closely following each other for a time. Suddenly the magenta curve takes a wild hike and quickly finds itself far away from its companion curve. This is known as sensitivity to initial conditions, which is seen in everyday weather.

Demonstration of chaotic effect
Figure 3: Illustration of deterministic chaos. Imagine two systems started at slightly different initial conditions. They will follow each other closely for some time, but within a short time our ability to predict them breaks down (front and side view of the Lorenz attractor).

This is what skeptics claim must happen in climate as well. Beyond variations in the solar cycles, airborne particle numbers and volcanic eruptions, most variability in the climate is due to changes in heat transfer between the atmosphere and the oceans. Water has a smaller Rayleigh number than air making it hard to obtain fully developed chaos in ocean currents. However, oceanic climate indices (the heat transfer) can exhibit small-scale chaos as we shall see below.

Ocean cycles. North Atlantic Oscillation, East Atlantic pattern, West Pacific pattern
Figure 4: Ocean cycles. North Atlantic Oscillation (blue), East Atlantic pattern (green), West Pacific pattern (red) (NOAA)

Looking at three of the leading climate indices we recognize the same problem that Lorenz faced. Therefore, the climate system as a whole does exhibit some inherent minor unpredictability. However, even though the El Niño of 1998 was a whopping 10% of the total heat content anomaly since 1950, it passed quickly. The greenhouse effect, on the other hand, works throughout the period.

Returning to the temperature record (normalized to the period 1950-1980) we can now put names on things. Chaotic influences from oceans and volcanoes etc. makes both weather more unpredictable and creates the unpredictable part of the 'wiggles' around the average trend in climate. The climate is definitely non-linear, but also not chaotic in this plot.

GHCN & HADISST1 global temperature record + moving average performed with lowess filter
Figure 5: GHCN & HADISST1 global temperature record. The moving average is performed with a lowess filter.

On the timescale of decades, every planetary object has a mean temperature mainly given by the power of its star according to Stefan- Boltzmann’s law combined with the greenhouse effect. If the sources and sinks of CO2 were chaotic and could quickly release and sequester large fractions of gas perhaps the climate could be chaotic. If the Earth's orbit was affected by a moon of comparable mass to the planet itself, then the orbit and thus the Solar forcing could cause some amount of chaos in the climate. As it turns out the Moon is relatively small and actually stabilizes the rotation axis, which is considered a favorable condition for life on Earth.

To sum up, the weather is chaotic because it can run free, climate is on a leash. Pull the leash hard enough and the climate responds.

11-year moving average of global temperature with Lorenz attractor superimposed
Figure 6: Artistic rendering of the rising global temperatures carrying a small, slightly growing Lorenz attractor up along with it.

Jacob Bock Axelsen is MSc in biophysics and PhD in complexity studies, both from the Niels Bohr Institute, Copenhagen. He has worked twice as a visiting scientist in USA and subsequently as a postdoctoral fellow at the Weizmann Institute of Science. He is a contracted physicist at Centro de Astrobiologia in Madrid. His interest in climate science is purely non-professional.

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Comments

Comments 1 to 42:

  1. Dr. Spencer Weart's "The Discovery of Global Warming" has a truly excellent chapter devoted to the foundations and evolution of climate modeling, particularly general circulation models:

    http://www.aip.org/history/climate/GCM.htm

    Note to doubters, skeptics, etc: Even if you're not prepared to join the mainstream, Weart's book covers this topic warts and all. If you're looking for weaknesses, you need to read it. Don't take on faith what you can pull from a primary source.
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  2. Excellent. Clear, concise and easy enough for averyone to understand.
    One question, the growth of the Lorenz attractor in the last picture is an observed feature or is just an example?
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  3. That's a good question, Riccardo. At first, I thought it was just an illustrative example, but a growth in the Lorenz attractor might represent larger chaotic variation in weather (i.e. more weather extremes). Obviously, whether this occurs or not is an important question when discussing the effects of long-term shifts in climate averages.
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  4. @Riccardo
    Thanks for your positive comment.

    The picture is just artwork - or an example, as you put it. The weather attractor would be huge on this scale, so I scaled it down to match the fluctuations instead. I hope you still like it.
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  5. Excellent post! Thanks!
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  6. Amazing post!!

    I remember reading about this in the book "Chaos" by James Gleick years ago and playing around myself with the bundled software doing fractals and strange attractors.

    You can download the shareware version of it here and source code too.
    http://www.cs.sjsu.edu/~rucker/chaos.htm

    From memory...I also remember reading that Lorentz discovered it by accident (like all great science) when he was in a rush or there was a power cut and he had to start his weather model with less decimal points in it and was very surprised by the print out the next day.
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  7. This is not so. Weather/climate is chaotic on all time scales.

    Chaotic climate response to long-term solar forcing variability
    A. Bershadskii 2009 EPL 88 60004 (5pp) doi: 10.1209/0295-5075/88/60004
    http://www.iop.org/EJ/abstract/0295-5075/88/6/60004

    A slightly earlier (6 Jul 2009 19:10:25 GMT) version of the same paper can be found at arXiv.org
    http://arxiv.org/abs/0903.2795
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  8. I'm not sure if it applies, but I thought of one analogy reflecting the phenomenon that climate on its scale is more predictable than weather.

    Consider the position at time 0 of a specific N2 molecule in a closed room filled with air, vs the temperature at time 0 in the same room. After one hour, it would be much easier to predict the temperature in the room than it would be to predict the position of the molecule. I assume that from a physics point of view one could argue that the two phenomena described is essentially the same, but on different scales, which you may also say about weather and climate.

    However my academic background is far from both physics and meterology so I'm not sure that my analogy is a particularily good one.
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  9. Berényi Péter writes: "This is not so. Weather/climate is chaotic on all time scales."

    Hmmmm. Are you suggesting we can't make long-term predictions of trends in temperature, precipitation, etc.?

    Because where I am now (New England, USA) it's currently -16 C.

    I am highly confident that six months from now, the outside temperature will be in the +20 to +30 C range here.

    I can make that prediction confidently because although the day-to-day weather is chaotic and unpredictable, there is a physical process ("the seasons") superimposed on that short-term variability.

    Likewise, with climate change, we know that increasing CO2, CH4, N2O, CFCs, etc. in the atmosphere will superimpose a warming trend on the global climate. We can't predict the details of the weather on a specific date in 2050 (any more than we can predict them for a given day in New England this summer), but we can predict that on average the climate will be warmer.
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  10. Excellent post, concise and very readable. Thanks!
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  11. @Berényi Péter
    I just browsed through your reference, and I have some rather informal comments.

    The paper bases its statements solely on power spectral analysis of timeseries data. This gives you resonance frequencies (non-chaos) and characteristic timescales (indicative of chaos). No strange attractors, no direct correlations or actual physical processes.

    For instance, the paper claims that turbulence is present on millenium scale in less than an order of magnitude in a bounded frequency region. Even for power spectra I would say this is weak evidence.

    Turbulence in heat transfer over millenia cannot exist within the solar system, which the paper also states. The requirements for turbulent Navier-Stokes dynamics in advection is simply destroyed by viscocity and dissipation. That is also why climate models work well based on average radiation assumptions on the structure of the atmospheric energy budget.

    The paper claims that the source could be the turbulent galactic electron density field modulating cosmic ray fluxes. However, research indicates that this mechanism is too weak to cause major climate change:

    http://www.agu.org/pubs/crossref/2009/2009GL037946.shtml
    http://www.skepticalscience.com/cosmic-rays-and-global-warming.htm

    The final claim is that CO2 power spectra give indications of chaos during the last 500 million years. Keeping the regularity of ice age cycles in mind, I would personally need more proof to accept more than quasi-linear responses.

    I hope this was useful.
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  12. Nice article..

    It's worth restating that treatment of chaos requires a careful consideration of exactly what one means by the term in any particular instance. The notion that climate might be chaotic in the same sense that weather is, not really suportable.

    The example of ice age cycles just referred to is a good one. During the last several hundreds of thousands of years the earth underwent glacial-interglacial-glacial transitions which had profound effects on climate regimes - there was nothing chaotic about the broad properties of these transitions and the climate state transitions that these induced. The transitions were paced by earth orbital cycles, and in each glacial/interglacial the earth transitions were driven towards new equilibrium states having characteristic, (and rather reproducible through several cycles over hundreds of thousands of years) global temperatures, atmospheric CO2 concentrations, ice sheet coverage, sea levels etc.

    And in the general state, climates and their responses to variations in forcings are non-chaotic. Of course there may be stochastic elements to forces that vary these states. For example the "transient" temperature rises and falls in the N. hemiphere in glacial periods during Dansgaard-Oeschger and Heinrich events seem to be due to periodic ice discharge fom the Arctic ice sheets, and temporary slowing or stopping of the thermohaline circulation. These may be contingent/stochastic events (i.e. essentially non-predictable), but local climates likely responded in a well-defined manner according to the resulting changes in local ocean/air heat transport; atmospheric moisture contents etc.

    The idea that climate is something like the long term accumulation of weather is a silly concept that is presumably raised so as to give the impression that climate is hopelessly non-predictable given that one can't predict weather. As Ned has said, this isn't true.

    The relationship between weather and climate is, of course, more sensibly defined the other way round; i.e. weather is the day to day variation in seasonal atmospheric parameters (temperature, wind, precipitation) within a given climate regime...
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  13. Ned,

    One can roughly say there are two classes of chaotic system, deterministic and non-deterministic. The behavior of the latter is the same as a random system. However a deterministic chaotic system isn’t the same as a random system. All deterministic system can in principle be predicted. Therefore saying weather or climate is chaotic is not the very same thing as actually claming it can not be predicted to some degree of certainty, they claim is only that it may be hard to predict, how hard is another matter thou.

    Climate is affected by regular cyclic phenomena and random event, these can be seen as ramps and step pulses to the system. Any system, linear, chaotic or random will respond to such pulses and such response can in principle be measured or filtered out from a time series of measurements. In a linear system this filtering is trivial, but for a chaotic system the process is non-trivial, a detected pulse may very well be a false positive in such system, this is very hard to say with out knowing anything about the history of the system itself. A linear system response to a pulse is easy to predict, but for a chaotic system one can not in general do this, except within small time scales. Chosen time scale short enough even the behavior of a random system can be predicted, but the error will soon grow to large to get any meaning full prediction out from it.

    The difference in predicting a linear system vs. a nonlinear lies in the rate of error growth. Linear system has a much smaller growth rate, and therefore we can make the prediction over longer times series with high confidence, while this is not the case with non-linear system. That errors in prediction grows by time is an inherent property of any simulations. The task in making a good prediction is to try make a system in which the growth of error is as small as possible.

    When you say you can predict that the weather in the next 6 month will be 20 to 30 C degree, then you are NOT basing this on the system itself but on a pulse respond to a ramp signal you know exist. However, still there is very tiny and small possibility your prediction may turn out wrong, but it is so unlikely to happened that you are not regarding it, which you probably do perfectly right in. But never less, it less to weather being non-chaotic and more to weather being affected by a ramp signal that you are able to do such prediction.
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  14. Very interesting stuff, thanks Jacob.
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  15. re your comment #13 batsvensson; i.e.

    "When you say you can predict that the weather in the next 6 month will be 20 to 30 C degree, then you are NOT basing this on the system itself but on a pulse respond to a ramp signal you know exist. However, still there is very tiny and small possibility your prediction may turn out wrong, but it is so unlikely to happened that you are not regarding it, which you probably do perfectly right in. But never less, it less to weather being non-chaotic and more to weather being affected by a ramp signal that you are able to do such prediction. "


    Surely Ned is basing his prediction on "the system itself". It's got nothing to do with "weather being non-chaotic" (we know weather has significant chaotic elements), it's to do with the likely range of weather events being bounded by a rather well-defined climate regime, and a highly predictable seasonal variation essentially based on Newtonian physics.

    As you say, there is a tiny and small ("tiny" and "small"?!) possibility that Ned's prediction may turn out to be wrong. There are two main reasons why this might be the case:

    (i) The variability in the weather encompasses the possibility of rare extreme excursions out of the expected range within a particular climate regime;

    (ii) a contingent event (volcanic eruption; extraterrestrial impact) might intervene.
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  16. It's great to see the chaos subject raised especially from someone with a PhD in complexity studies. Hopefully you will indulge some questions from someone who knows less.

    First of all, I don't see what you have actually demonstrated. Why is it a demonstration that climate is not chaotic by showing a graph of the temperature parameter in climate over 120 years. And how do you know that that particular graph is not actually chaotic?
    For example, I could plot a graph of the motions (the 2 angles) of the double pendulum for a period and say "see it's not chaotic".
    What's the difference between these 2 cases?

    Second, while of course the climate is bounded in many ways that doesn't mean it's not chaotic. The fact that there will always be 4 seasons or the poles always colder than the equator really isn't relevant. Or the fact that mid-latitudes might be between 0'C and 30'C on any given day. I know you didn't put forward these points, but I see them a lot with a kind of "QED climate is not chaotic" and I'm scratching my head..
    In your opinion is this correct? I.e., the fact that the above few points are true doesn't disprove the possibility of chaotic behavior?

    Third, an example. The well-known "Atlantic conveyor belt" of ocean heat is driven by the thermohaline currents. Sufficient melting ice from Greenland/Arctic would disrupt the thermohaline and the conveyor belt stops, northern Europe gets very cold and the Arctic re-freezes. But at what point does this occcur? Prof FW Taylor in his book "Elementary Climate Physics" (2005 OUP) shows the 2-box model of the oceans, apparently originally proposed by Henry Stommel. It's fairly simple but shows an unstable behaviour against peturbations in either direction. He comments that right now (2005) none of the GCMs show the reversal of the circulation, instead they vary between no real change and a 50% drop over the next 100 years.
    However, my point finally arrived at, if this model (extended to a more realistic one) is correct, then surely the thermohaline will provide chaotic behavior at some point. (Perhaps strictly speaking it might not be chaotic, perhaps just unknown and complex at this stage).
    Comments?

    Fourth, without actually knowing the formulae for many important aspects of climate, how can "the climate community" (or subsection thereof?) be so confident that climate is not chaotic?
    E.g. the aerosol effect, a negative feedback, but with error bars stretching between zero and the effect of CO2 at 380ppm (according to IPCC AR4). I could happily theorize about changes in ocean temperature increasing the production of sulphides forming more clouds and providing more negative feedback.. or higher winds from higher temperature differentials picking up more dust from every drought ridden deserts and therefore seeding more clouds.. or not. I have the full extent of the error bars and given that we don't really know the formulae they might exhibit strong negative feedback - or they might exhibit actually positive feedback under some circumstances.

    How to confirm "no chaos" when the equations are somewhere between "cloudy" and "unknown"?

    Sorry for writing such a lengthy set of questions, but a subject that really needs discussion and thanks for posting on the subject.
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  17. I hope you don’t mind me responding to your message stevecarsonr (I happen to be on line, it’s a Friday night, and I’ve drunk a good part of a bottle of wine!).

    I think it comes down to the meaning of “chaos” as I suggested in post #12 just above.

    I'm not sure that the phenomenon you describe (periodic ice-melt-induced cessation of the Atlantic conveyor, which I also used in my post above), is an example of chaotic behaviour. I would say that it is an example of stochastic behaviour.

    Weather has inherent chaotic elements since (i) its evolution is critically dependent on the starting conditions (that’s why weather models, but not climate models, are continuously “reset” to current atmospheric conditions), (ii) it progresses on, and is influenced by events on, a very small spatial scale, and (iii) there are an almost infinite set of influences that determine the temporal evolution of local atmospheric conditions.

    That’s not the case with the example of ice-melt-induced cessation of the Atlantic conveyor. There’s no question that this phenomenon (see e.g. http://en.wikipedia.org/wiki/File:Ice-core-isotope.png ) has a stochastic element. But it is a phenomenon that is bounded within a particular climate regime (glacial period of an ice age with a particular continental arrangement that gives a strong thermohaline heat transport to high Northern latitudes), and is predictable in principle. Presumably if one knew something about the relationship between Arctic ice buildup and its evolution towards instability, then one could make a reasonable prediction (if one was around during the last glacial period say!) of when the next ice-collapse-induced cessation event would occur.

    I think we also have to be careful not to assign the label “chaotic” to behaviour that we happen to lack the knowledge-base to understand predictively. Focussing on the thermohaline circulation (THC) and the effects of melt water, it does seem a possibility (see e.g. ftp://rock.geosociety.org/pub/GSAToday/gt9901.pdf, for an interesting read) that the THC could slow down or stop if sufficient freshwater from Arctic ice melt were to flood the Arctic ocean. But I don’t think this would be an example of chaotic behaviour even if we might not know, at the moment, what specific conditions (how hot does it have to be; how much ice melt and how fast, would be required…). At some point we might well understand this process well enough that it might cease even to be considered “stochastic”.
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  18. This is a great site, because it takes the discussion to a high level, but still not so technical that you can't follow it.

    This particular posting is also quite interesting. I'm wondering about the correct definition and nature of the term "climate". I imagine that "todays weather" is characterized by a large number of parameters, varying in a way which might very well be (weakly) chaotic in the technical sense that the weather of next month is deterministically determined by weather of today, but this dependence is very sensitive to small variations of the initial conditions.

    Climate on the other hand could be defined not as the weather at a particular point in time, but as a subset of the parameter space. The weather can vary wildly, but only inside the bounds prescribed by the "climate".

    Or maybe differently, in analogy to the example of the Lorentz attractor, climate is a subset of the parameter space where "weather" spends "most of its time". This would be a slightly different but possibly more flexible definition of "climate".

    It reminds me of the situation in celestial mechanics. There the laws of motions of the planets in the solar system are quite simple, certainly immensly simpler than the laws governing weather. But you can run into chaos. I believe that at least in some situations you can compute the orbits of the the planets at a time in the far future (climate), but because of chaos you cannot calculate where in the orbit the planet will be at this specific time (weather). In this situation, the orbits of the planets change, so that there is a certain "dynamics of orbits". But it is far easier to predict the development of the orbit of a certain planet at a certain time than to predict exactly where the planet will be at this time.

    One thing which is unclear to me is if the "climate" of meteorological models can vary on its own. Is there is some sort of "dynamics of climate" - long term non-forced natural variation - or is the climate supposed to be completely determined by the various type of "forcing"?
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  19. @stevecarsonr and Marcel Bökstedt

    Thanks for your questions, which I will try to comment on. Please bear with me for trying to answer arguing from the variables of the Rayleigh number.

    Consider two plates (hot and cold) enclosing a liquid convecting fluid:
    http://en.wikipedia.org/wiki/File:Convection-snapshot.gif
    http://en.wikipedia.org/wiki/B%C3%A9nard_cells

    This is the Rayleigh number.
    Ra = gravity * expansion coefficient * system size * temperature gradient / (viscocity * conductivity * diffusivity)
    = g*b*D^3*dT/(v*a*k).

    In the Lorenz attractor Ra must be above the threshold Ra = 13.926 to exhibit any chaos, and below the dynamics is predictable. For instance, my plots are for Lorenz' own choice of Ra = 28.

    The idea that chaos is prevented by boundedness can then be understood: just decrease D or dT sufficiently to end below the threshold. I was using the 'leash'- analogy differently: The mean global temperature is determined as a steady state of huge energy fluxes. It is suspended by the Sun pulling up and the heat loss to space pulling down.

    To exhibit chaos you need to be able to delay heat transport (advection) through fluid dynamics, and with El Niño being the largest phenomenon of relevance we are still far away from fully developed climate chaos. Notice that sea levels increase on the order of centimeters during an El Niño - this is the small expansion coefficient of water. Make b small and you move away from chaos.

    The thermohaline circulation (THC) is a true convection roll resulting from density change. However, the engine of THC is surface cooling in the Arctic which global warming might turn off. If dT cannot drive even laminar currents, then we have smaller dT and lesser probability of chaos.

    I mentioned aerosols in the post, but they are much more transient than CO2. Much like airborne water, aerosols is argued to be fighting a negative feedback: cloud seeding, gravity and precipitation. My understanding is that clouds more or less cancels out in climate models. If aerosols cool they lessen dT for possible oceanic chaos.

    Interestingly, dust depositions on glaciers is hypothesized to be part of the ice age trigger:
    http://forecast.uoa.gr/conferences/iamas/10july/4b/69_smn_dst_dam_iamas_200707.pdf

    I hope you find these comments useful.
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  20. @chris, #15.

    "Surely Ned is basing his prediction on "the system itself".

    Sure, I agree with that, and I see I was a bit unclear with my point, my excuses for that. My point was to try to make a distinction, and to separate, linear and non-linear elements in a system, and to clarify that it is because of the presence of a forcing from linearity in that make Ned able to do the predict as he did. I didn’t meant to say this is not part of the system, but to say it can be seen as a separate from the non-linearity of the system.

    I sometime notice that some people seems to believe that there is proportional linear relation between CO2 levels and global mean temperature. This relation is thou not so trivial as the green house effect from CO2 is said to be a non-linear function of the concentration. In other words, the contributing effect from a linear increase of CO2 will not change as rapid as temperature, therefore, unless we are working with a system that locally can be said to be linear, if both CO2 and temperature increase linear in respect to each other then I would suspect there to be yet another factor in the equation.
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  21. Errata comment #13 (and #20),

    "... less to weather being non-chaotic and more to weather being affected by a ..."

    Above is an editing confusion of mine. I usually edit text a lot before I make a post. I this case I was considering to use the word "chaotic" OR "non-linear", and apparently it all got mixed up in the final edit. :(

    There are also some editing confusion in post #20 as well. For instance:

    "because of the presence of a forcing from linearity in that make Ned able to do the predict as he did."

    Was intended to be:

    "because of the presence of a forcing from linearity in the system that one is able to do a prediction as Ned did"
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  22. Chris, you wrote in #17,

    "the THC could slow down or stop if sufficient freshwater from Arctic ice melt were to flood the Arctic ocean."

    Mostly anything can happen if sufficient conditions are present, but the interesting question is, if this is likely to happen or not. Afaik, current understanding says this is not likely to happen at all - if ever.

    "At some point we might well understand this process well enough that it might cease even to be considered “stochastic”."

    I don't think so. "stochastic" is just another way to label non-linear or "random" system, system we previously not been able to model or control properly. Chaos theory is in principle a theory about non-linear system and how to treat them as non-linear. Before the time we had the mathematical tool to model and understand such system (which wasn’t all that long ago) engineers was busy making sure any non-linear system was modeled with in a certain local region that could be approximated as linear, outside this approximated region the behavior can not be guaranteed.

    The mathematical tools to do this exercises with is know as ‘differential equations’, and one requirement for these tools to be “trivial” to use is linearity – nonlinear differential equations are extremely hard to solve. Stochastic processes lacks the linear properties, thus they are always hard problem to solve with differential equations. Only a few non-linear processes of special interest are understood this way, like fluid dynamics, but even these are solved with numerical methods. However, new mathematical tools and theories theory have help use to better understand system outside the approximated linear boundaries, but such understanding wont make them less stochastic, rather it will help use to even more appreciate the very special behavior of these system.
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  23. Jacob Bock Axelsen:

    First of all, a comment on the thermohaline circulation. You said: "However, the engine of THC is surface cooling in the Arctic which global warming might turn off."

    But there are 2 factors driving the ocean currents. One is temperature and the other is salinity. Both affect density in a non-linear way.

    The reason why it is potentially "a switch" is that as the oceans warm, as they are doing at the moment, there is increased melt from Arctic ice and the Greenland ice sheet.

    If this melt rate increases to a certain point - no one exactly sure what that point is - then the low salinity will outweigh the cold and this water will stop sinking and the "conveyor belt" direction will change.

    Therefore, a very complex situation, and one where increased temperatures will eventually (possibly) lead to a colder northern hemisphere and a refreeze of the Arctic with the consequent (positive feedback) of increasing albedo.

    More on your other points later..
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  24. re #22

    hmmm...this is why I suggested in post #12 that we have to be careful what we mean when we use the term "chaos" in any particular instance. We end up misarguing around the meaning of a word or concept rather than the phenomenon itself.

    Meltwater-induced suppression of the Thermohaline Circulation happened many time is the past. So one can hardly say it can't happen! Of course the boundary conditions are different now (interglacial rather than the many instances identified during glacial periods).

    I (and everyone here, I think) was using it as an example 'though. It's not really chaotic behaviour (it has its chaotic elements on a microscale), but it's really a stochastic phenomenon that is essentially predictable, if not in relation to the precise timing of events, at least as a phenomenon that is a definite and predictable consequent of particular conditions.

    So, for example, it would likely be possible to model N. hemisphere ice sheet dynamics and ocean circulation during the last glacial period to reproduce the Daansgard-Oeschger (D-O) events, within an understanding of the conditions under which these events occurred (not sure if this is yet understood very well).

    Where this differs from chaotic phenomena (as I understand it), is largely the independence with respect to initial conditions. We wouldn't know exactly when a D-O might occur, but we would be able to predict that, independent of inital conditions, once the important factors tended towards threshold values, that a D-O event would have a high probability of occurring..

    Two examples:

    (i) Knocking down a wall with one of those splendid balls on chain swung by a crane. We don't know exactly when the wall will tumble, or exactly the pattern of its disintegration (one might consider the latter to be chaotic). However the event (the wall falling down) is predictable (if not precisely defined temporally speaking), given that we understand the forcings that act in this situation, and is independent of initial conditions.

    (ii) In a warming world we expect coastal flooding events that might have 100 year probability (say) to occur more frequently, as a result of rising sea levels combined with more extreme weather events as sea surface temperatures rise etc. Now, however chaotic the weather is (chaos), the likelihood of an increased frequency of coastal flooding events is predictable. We don't know when any of these events will occur (stochastic), but our prediction of an increase of events in a warming world is likely to be robust, and increasingly so as our knowledge of the climate system increases.
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  25. I think batsvensson's distinction between linear and nonlinear forcings is a red herring here, though I could be wrong (I'm not an expert in this).

    We can predict that (outside the tropics) it will be warmer in summer than in winter because we have a conceptual model of a forcing (the time evolution of the solar zenith angle at a given latitude as determined by the earth's axial tilt) that is large enough to override short-term variability in the weather. There are of course all kinds of feedbacks, positive and negative, that amplify or reduce that radiative forcing. Nonetheless, we can be confident that the magnitude of that forcing is large enough to make the winter-to-summer difference semi-predictable.

    Likewise, we have a good conceptual model of another radiative forcing (absorption of outgoing long-wave radiation by greenhouse gases) that is also becoming large enough to have a detectable influence on climate. We can't predict the weather in 2050 (just like we can't predict the weather next July), but in both cases we know there are predictable radiative forcings that will make it warmer (on average) in summer than in winter, and in a 500 ppmv CO2 atmosphere than in a 300 ppmv CO2 atmosphere.
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  26. Jacob Bock Axelsen:

    Re your follow up comment (06:01 AM on 24 January).

    I think I get the first point. In a well-understood system there are (or maybe) areas of non-chaotic behavior and also areas of chaotic behavior. And this is all non-controversial, chaos 101 perhaps.

    Then your comment which begins: "To exhibit chaos you need to be able to delay heat transport (advection) through fluid dynamics.."

    I might be reading into your comment what others say, so apologies if I am putting words into your mouth.. one way of looking at the earth's climate is almost like the billiard ball model - the basic thermodynamics govern the temperature, the rest is just like the bubbles in the boiling of water - it's extremely well-defined how long it takes to boil a kettle of water - and the fact that we don't know where the bubbles are, although interesting, is irrelevant.
    Ie in that example, chaos probably exists at a micro level, but the key parameters of importance are well-known and simple to calculate.
    So in climate, the heat in due to the sun and the absorption and re-emission of long wave radiation by water vapor and GHGs has to be balanced by the OLR. And nothing can really disturb that because the fluid dynamics of the situation is extremely "non-turbulent".

    If I've not captured the essence of your argument time to step in.

    And if I did get the gist, I would say..

    However, unlike turbulent fluids, the earth's climate is full of coupled positive feedbacks as well as very non-linear and non-understood negative feedbacks.

    The positive feedbacks include water vapor (increasing with temperature), CO2 outgassed from the ocean (increasing with temp all other things being equal), ice albedo reducing with temp (therefore increasing solar radiation received).
    Negative feedbacks include the T^4 increase in outgoing radiation with temp. Large unknowns that are probably negative include clouds and aerosols.

    Any system with positive feedbacks is likely to be unstable. Start a movement in one direction and your positive feedback re-inforces it. Now that does not prove "chaos". But it certainly does create instability.

    That's why the THC is an interesting one because heating up the arctic can lead to cooling down the arctic, even though we don't have "turbulent fluid flow".

    Let's delve into the uncertainty here for a minute - heating up the arctic might instead lead to large releases of methane gases from permafrost thus a large further warming. (Not sure where the research is on that at the moment). Heating up the arctic might reverse the THC, thus a cooling.

    Which one happens first? We have a system with large positive and negative feedbacks. That is just one part of the climate.

    So I believe that to claim as your original post did that the system is "not chaotic" needs a lot more evidence.
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  27. Chris:

    Re your comments from 10:31 AM on 24 January

    You said: "I (and everyone here, I think) was using it as an example 'though. It's not really chaotic behaviour (it has its chaotic elements on a microscale), but it's really a stochastic phenomenon that is essentially predictable, if not in relation to the precise timing of events, at least as a phenomenon that is a definite and predictable consequent of particular conditions.

    "So, for example, it would likely be possible to model N. hemisphere ice sheet dynamics and ocean circulation during the last glacial period to reproduce the Daansgard-Oeschger (D-O) events, within an understanding of the conditions under which these events occurred (not sure if this is yet understood very well). "


    How do you know this is "essentially predictable"?

    The thermohaline currents are complex with difficult boundary conditions - the shape of the oceans along with the starting point of exactly the salinity, temperature and momentum vectors on your modeling day zero. Then you have the complexity of the interaction with the atmosphere, where dependant upon those conditions you might have more or less heat transferred, more or less momentum transferred, and depending on the cloudiness, more or less solar radiation received. And then you have the equations governing the melt rate of the Greenland ice.

    A nice analogy - the wall being knocked down - perhaps relevant, perhaps not. Analogies can be useful illustrations, but first of all, is it a correct analogy?

    I'm actually amazed that you are so confident that the THC shutdown can be modeled accurately. The dual pendulum is much simpler. But it's chaotic. THC might be non-chaotic and just very complex.

    But surely the starting point in determining chaotic or not is actually to know what equations we are dealing with?
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  28. stevecarsonr,
    although i'm not Jacob Bock Axelsen (and not even expert on chaos), let me comment on a couple of points you made.

    First, it's not true that "Any system with positive feedbacks is likely to be unstable". It's just a possibility, they _might_ be unstable. It's definitely true if they have only positive feedbacks, but the climate system has at least on strong negative feedback, thermal radiation. Indeed, earth climate proved to be quite resilient to warming much stronger than projected for the near future.

    The example of slowing or shutting down THC is misleading in what you consider regional changes in temperature but using global heat fluxes. The correct heat balance for the northern hemisphere alone would include also ocean and atmospheric circulation, i.e. all the sources of heat fluxes. So, that an initial warming may lead to cooling should not come as a surprise, nor should the resuming of warming afterward.
    Indeed, the seesaw behaviour is hardly considered chaotic.
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  29. As we while away our Amateur Hours here, lest we get too lost discussing various subsystems it's a good thing to remember that-- chaos at or near the surface or not-- energy has to escape from the Earth if it is not to become warmer. So maybe it would be good to focus on effects that can actually change the planetary radiation equation.

    It seems to me that it's easy to pay too much attention to surface temperatures, ocean and land. Surface temperatures can be misleading; absent some means for improved radiation from the planet, there is not going to be any net "cooling" in absolute terms going on regardless of what surface dynamics and temperatures indicate. The elephant in the room here is the oceans; energy hidden in the ocean does not disappear.

    For example, can thermohaline currents affect global temperature? How? If energy is released at a unusually high rate from the oceans in one region it does not follow that it will automatically escape back into space.

    Focusing on chaotic processes is a bit of a distraction, unless one can envision a chaotic process that removes heat from the planet. What might that be?
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  30. doug_bostrom: "how can thermohaline currents affect global temperature?"

    By refreezing the arctic and northern europe and therefore increasing the albedo of the planet.
    If the albedo of the planet increases by 1% that's equivalent to about 3W/m^2, or a little more than the current impact of CO2.

    The current albedo of the northern mid to high latitudes is around 45-55%.
    If we took an area that was grassland, soil or forest and covered it with snow or ice it would go from around 20% to 60%-80% albedo.
    If we took ocean and covered it with ice it would go from 6% to 60% albedo.

    So an extra 7M km^2 of water with ice and an extra 3M km^2 of land with snow/ice would make that change.

    That's about 50% more sea ice in the arctic at the moment and a pretty big increase in northern europe/n.russia ice & snow.

    Also, if as a result of the THC shutdown the energy in the climate system is simply "re-arranged" then the tropical/sub-tropical regions will be warmer as a consequence of the northern high latitudes being colder.

    As energy is radiated at T^4, this will also result in a higher overall radiation. Back of envelope..
    20% of the world's surface decreases from 0 to -5'C and 20% of the world's surface increases from 15-20'C then there will also be a 1.1W/m^2 increase in radiation from the earth's surface (averaged across the whole planet).
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  31. doug_bostrom: "..unless one can envision a chaotic process that removes heat from the planet. What might that be?"

    Other effects (apart from THC as already outlined) include aerosols in reflecting incoming radiation and creating more clouds. As the IPCC AR4 (2007) shows, the error bars for aerosols match the size of the CO2 effect.

    The error bars are there because as yet the effects of aerosols are not really understood. "Medium low level of scientific understanding" is the description by the IPCC.
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  32. stevecarsonr at 13:18 PM on 25 January, 2010

    Good points, and thank you for the illustrations.

    So in sum insufficient help from turning the subarctic into an ice cube and refreezing the Arctic, unless forcing estimates are wildly inaccurate.

    I guess that illustrates what I've been thinking, which is that any unidentified issues w/the current crop of models are going to have to be huge and thus likely quite difficult to overlook.
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  33. Sorry, I did a poor job w/the last sentence of my last post.

    The point I was trying to make is that the central effects being predicted by climate models are so large as to require a rather enormous and therefore unlikely unidentified lacuna in order to make those central findings significantly less threatening.

    Steve did a nice thought experiment w/some rough numbers describing a scenario resulting in a discontinuity in the predicted mode of forcing of major scope, yet that change was not enough to fully offset the net impact of our activities.

    Another more fundamental problem comes along with relying for comfort on model errors of the type envisioned as accompanying failure to identify and include various dynamic features of the climate. For instance, if as Steve hypothesizes we should upset the accustomed thermohaline circulation of the North Atlantic as an upshot of our activities, though it might reduce warming roughly as he suggests we'd still have a major climate disaster on our hands.

    Errors in models unaccounted for do not change the basic physics underpinning AGW. The energy budget of the planet is being altered and though we may make errors in predicting the effects of that change we're very unlikely to find net benefit as a result. We and the rest of the biological systems here are adjusted for a particular energy budget and if that budget changes too fast we'll have a hard time coping.
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  34. @stevecarsonr

    (I here use 'deterministic chaos' and 'chaos' interchangeably.)

    If you want to establish chaos you need to prove extreme sensitivity to initial conditions. It is virtually impossible without computer modelling and non-linear mathematics.

    I understand that you could imagine chaos 'in the error bars' of clouds and aerosol, but you need more advanced arguments than feedbacks. I could be my own devil's advocate by pointing to the surprising fact that all-negative feedbacks can be chaotic:

    http://arxiv.org/abs/0905.3672
    - just to emphasize the necessity of proper mathematical modelling.

    Personally, I am skeptical that your proposed THC-mechanism is chaotic because I see no possibility of heat being trapped in a truly fluid fashion. Would THC shut down lead to Arctic freezing to start with?

    I have still not touched upon how to actually detect chaos, but one way is the Lorenz map: take the sequential period-maxima of your data (m1,m2,m3,m4,...) and plot the pairs ((m1,m2),(m2,m3),(m3,m4),...). If you have a predictable cycle, as non-linear as you like, you will get a cloud of points mostly tracing the diagonal (like for ice ages). Do it for a strange attractor and you could get some off-diagonal contour. Lorenz produced a 'teepee' and proved how to extract order from chaos(!):

    http://ams.allenpress.com/archive/1520-0469/37/8/pdf/i1520-0469-37-8-1685.pdf

    I note that you use turbulence and deterministic chaos somewhat interchangeably, which is a clear misunderstanding. In your boiling kettle any possible chaos in heat advection is destroyed by turbulence and vapour bubbles. Please visit the links I provided before for a much more precise picture of convection.

    In searching for origins of minor chaos one obvious candidate is El Niño's trapped equatorial surface waters. In 1998 roughly 10^21 J was transiently trapped in the atmosphere before it could leak to space. It is now understood to be almost entirely due to deterministic chaos in weather spilling over to the global energy budget. Again the Rayleigh number is useful: it is the wind that initially is chaotic because it has a high Rayleigh number.

    The best way to predict large-scale chaos is then to have El Niño and the Lorenz attractor in mind when hypothesizing about a much stronger mechanism.

    Finally, the ice core record shows no signs of CO2 and CH4 leading to major chaos despite huge outgassings etc. There were also varying aerosols, clouds and all other feedbacks throughout the period, so the record is a strong indication that we cannot expect spontaneous wild chaotic behavior of the future climate.

    Again, it is just my highly informal comments - not science.
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  35. Jacob:"Personally, I am skeptical that your proposed THC-mechanism is chaotic because I see no possibility of heat being trapped in a truly fluid fashion. Would THC shut down lead to Arctic freezing to start with?"

    I think you may have missed the point of steve's hypothetical. It is not necessary to "trap" heat at any point in order to get chaotic behavior at least in theory.

    Steve Carson is suggesting a situation where albedo has the potential to vary more in different places than in others. The polar regions for instance have the potential for wide variation in albedo while the tropical regions have fairly small potential for variation. Thus, it follows that simply moving more heat to polar regions away from tropical areas will have a greater warming effect on the globe as a whole than something that does the opposite.

    Does the climate system have the capacity to move sufficient heat to make its overall behavior chaotic? Beats me, but there is no need for heat to be "trapped" anywhere for it to be present.

    Cheers, :)
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  36. I think chaos doesnt really exist , its just that we are unable to see the reasons for events because they are either so complicated or spread over such long time scales that we cant see the patterns or get enough information to so what going to happen next .
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  37. Jacob Bock Axelsen:

    I'm definitely a novice on chaos so I need to keep asking questions to make sure I've understood your points correctly. So please don't think I'm trying to be argumentative as I press on again - it's the way I learn...
    (And I'll take a look at the Lorenz paper you provided but it will take a while and everyone will have moved on by the time I have digested it)

    1.
    You said: "If you want to establish chaos you need to prove extreme sensitivity to initial conditions. It is virtually impossible without computer modelling and non-linear mathematics.."

    I'm throwing in the idea that climate *may* be chaotic to find out how well *your* original claim stands up.

    Of course I'm very glad that you posted the article because it is a subject that needs discussion, and therefore refreshing to find it here. But you claimed "climate is not chaotic".

    I'm asking you to really demonstrate it, or justify how your article demonstrated it.

    I'm certain that I can't *prove* that climate is chaotic.

    So to turn it around, to you the poster, can you actually establish that climate is not chaotic without the same burden?

    2.
    Back to one of my first questions because I am very interested in knowing the answer.. My second question from 08:35 AM on 23 January about the fact that the poles will be colder than the equator, that there will still be seasons etc - is it true that this *doesn't* demonstrate that climate is NOT chaotic?
    It's just that I see arguments along these lines quite often (they seem so flawed as a demonstration of non-chaotic behavior that I think maybe there's something I don't understand).

    3.
    You said: "Finally, the ice core record shows no signs of CO2 and CH4 leading to major chaos despite huge outgassings etc."
    Again maybe I just don't get chaos..
    What is "major chaos" and what would one see?
    To me - if climate was chaotic I would expect to see ice ages, interglacials etc, like we have seen, but that their appearance, timing, coldness/hotness etc was "sensitive to initial conditions" - and therefore unpredictable.
    I noticed with your original 20th century temperature graph that you said "The climate is definitely non-linear, but also not chaotic in this plot."
    How can you tell?
    What is it that I don't understand about chaos?
    Can you pick it up by eye or did you apply a mathematical formula to it?
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  38. Steve

    It is not an easily understood subject, and in this discussion forum I will only expand on what I wrote. Anybody who is interested in real learning needs to read a good textbook, like "Nonlinear dynamics and Chaos" by S.H. Strogatz or perhaps this online book:
    http://staff.science.nus.edu.sg/~parwani/c1/book.html

    Seasons are probably a very minor global temperature regularity: just plot every 6th global temperature of the monthly averages - all you get is the well-known global warming curve. Seasons are not a proof that global climate is not chaotic.

    The 'not-chaotic' statement should have been after I mention Stefan-Boltzmann's law. The trained eye really requires a record of periods of unpredictable oscillations to say anything. It could be a small fraction of a chaotic oscillation - but before anyone makes that leap they find it is strongly prevented by the records (even just the Hockey Stick) and the radiation physics.

    There are two overall possibilities of chaos:

    1. Either heat is trapped or moving around on global scale - and the best illustration is the autonomous fluidity of air in the weather. For the climate El Niño is the incredible Hulk of such phenomena - but it has its origin in the weather.

    2. The external forcings vary chaotically. A chaotic Sun variation of 10% over 10 years will undoubtedly lead to chaotic global temperatures. Note that turbulence is not the predecessor of deterministic chaos. Turbulence is not a deterministic system i.e. the trajectories cannot be plotted with confidence and no simple equations can describe them.

    Consider the stunning power of radiation physics: It can explain the Earths surface temperature, greenhouse effect included, to about 288K -+ 10K or an error of 3%, perhaps much lower depending mostly on the variations in the general albedo term.

    BTW: I can easily see how the T^4 term for the radiated power can make one think that the radiation can fluctuate wildly for a small temperature change. However, T^4 is almost linear for the temperature interval from 280K to 300K.

    The surface albedo, greenhouse gases, aerosols and clouds are mostly described by the radiation physics, and they almost explain the 4K variation in the ice cores. Also, the error bars in the IPCC report are about the error in the understanding of their effect on radiation.

    The expected stochasticity/chaos is then bootstrapped by the records in combination with the level of understanding of the present state.

    Or less concisely:
    What is unknown is perhaps only -+1K or less, including chaotic external forcings as well. That is where chaos has been confined historically and that is why I mention the oceanic climate indices in my post. The most reasonable expectation must then be, to first order, that this is what we can expect in the future.

    Runaway feedback effects in the global temperature of e.g. crossing the bifurcation point at 450 ppm CO2 are not ruled out. It is just not chaos.
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  39. It's funny. We may be on the verge of a serious glaciation, deeper than any of the recent ice ages and much more permanent. I would not like that, not even in ten thousand years. The last time most of Europe was under ice, it was damn cold and dry where I live. Biodiversity has definitely improved since then.

    Also, I don't think carbon dioxide would be a remedy. The albedo discontinuity at the edge of the ice sheet they are talking about is very real. Ice is white in the visible and pitch dark in IR.

    It is the same for the snow which is all over the place right now, making the nights chilly. I have no choice but to generate carbon dioxide by destroying some methane, otherwise the family, including the kids would get frozen.

    "For the best-fit run, transition to the large Eurasian ice sheet occurs shortly after the present. Our results therefore suggest that the actual climate system may have been geologically close (10^4­-10^5 yr) to the final phase of a 50-Myr evolution from bipolar warm climates to permanent bipolar glaciation. (Presumably, future society could prevent this transition indefinitely with very modest adjustments to the atmospheric CO2 level.)"

    Vol 456 | 13 November 2008 | doi:10.1038/nature07365
    LETTERS
    Transient nature of late Pleistocene climate
    variability
    Thomas J. Crowley & William T. Hyde
    http://www.geos.ed.ac.uk/homes/tcrowley/crowley_Nature08_iceages.pdf
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    Response: I'm not sure I'd call between 10,000 to 100,000 years as being "on the verge of a serious glaciation". The forcing from orbitally forced albedo changes is quite small compared to the forcing from CO2 and operates over geological time scales as opposed to the dramatic response of climate to CO2 that happens over decades.
  40. Not much chaos in Crowley et al paper. Insted, it would be appropiate as a positive impact of global warming:
    "(Presumably, future society could prevent this transition indefinitely with very modest adjustments to the atmospheric CO2 level.)"
    ;)
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  41. Jacob, thanks for taking the time to respond to questions. Plenty for me to think about.

    I have a chaos book out of the uni library: "Non-linear ordinary differential equations" (Jordan & Smith), but the library also has the Strogatz book so I might do a swap.
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  42. I think some illustrative movies are called for.

    Experiments.

    1. Non-chaotic convection (from above):
    http://www.youtube.com/watch?v=UhImCA5DsQ0&NR=1

    2. Mildly chaotic:
    http://www.youtube.com/watch?v=lbRzeXPuJIo&feature=related

    3. Jupiter's weather is both chaotic (f.ex. the red spot) and turbulent (eddies of all sizes):
    http://www.youtube.com/watch?v=yjU_CJWzAd4&feature=related

    -----------

    Simulations.

    1. Non-chaotic Rayleigh-Bènard convection:
    http://www.youtube.com/watch?v=yRjSXBLvIFo

    2. Chaotic:
    http://www.youtube.com/watch?v=1yBz4U7Xbmc&feature=related

    3. Strong chaos/soft turbulence (but still nice):
    http://www.youtube.com/watch?v=fGvXK_qb5fY


    Enjoy!
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