Shifts, phase-locked state and chaos in climate data

by David Douglass

Recent studies of observational climate data have shown that Earth’s climate system:  has many abrupt climate shifts; is phase locked to an annual cycle of Solar origin; and is chaotic. These phenomena are related and are summarized below.

The climate shift of the mid-1970s is well known. Among the first to characterize this particular Climate Shift was Trenberth [1990], who in a study of the Pacific mean sea level pressure reported a “different regime after 1976”. Numerous studies have confirmed this and many other Climate Shifts. A new innovative way to identify Climate Shifts using networks of climate indices was introduced by Swanson and Tsonis [2009]. Their method, which can be thought of as a different quantitative “teleconnections” scheme,  consisted of two parts. The first part was a definition of a “distance” d between two indices involving the Pearson correlation function. Second, the correlation among more than two indices was defined by “synchronization”, which is a particular function of the individual distances. In a study of a network of four northern hemispheric climate indices (Nino; Pacific Decadal Oscillation, PDO; North Atlantic Oscillation, NAO; and  the North Pacific Index, NPI) they reported “synchronization peaking” showing five Climate Shifts since 1900 including the mid-1970s Climate Shift.

Douglass [2010] (here) improved the Swanson and Tsonis scheme in a number of ways. A different distance “d” was defined so that it satisfied the 3 triangle conditions to be in a mathematical metric space. This then allowed the correlation among 3 or more indices to be easily measured using the diameter D from Topology. Minima in D corresponds to high correlation.  These new definitions were used to study of a different “more global” set (Nino3.4, north and south Pacific indices and Atlantic Multidecadal Oscillation, AMO) of climate indices that included both north and south hemispheres . The more sensitive D plots showed 18 strong minima (Climate Shifts) since 1870, which included the 5 of Swanson and Tsonis. The 3 most recent Climate Shifts occurred during 1976–77, 1986–87 and 2002–03. A further improvement in the scheme was made by Towsley, Pakianathan  and Douglass [2011] (here). They showed that the topological area A defined also from the distances d was even more sensitive.

Figure 1 below from Towesly et al. sums the results. Both the diameter D and the area A since 1940 are plotted. The minima which corresponds to the Climate Shifts are indicated by arrows.  The three most recent Climate Shifts  are: 1976-1977 (the mid-1970s shift); 1986-1987 (missed by Swanson and Tsonis); and 2001-2002 (this Climate Shift is seen to be particularly strong.

Figure 1 from Towsley et al.[2011] showing seven minima in D or A since 1950.

Phase-Locked Climate States

What is the nature of the climate state between Climate Shifts?  In a later paper Douglass [2011a] (here) studied the Pacific sea surface temperatures in greater detail and showed that the climate system is frequently  phase locked to an annual cycle. The abstract of that paper gives a summary.

The Pacific sea surface temperature data contains two components: NL, a signal that exhibits the familiar El Niño/La Niña phenomenon and NH, a signal of one-year period. Analysis reveals: (1) The existence of an annual solar forcing FS; (2) NH is phase locked directly to FS while NL is frequently phase locked to the 2nd or 3rd subharmonic of FS. At least ten distinct subharmonic time segments of NL since 1870 are found. The beginning or end dates of these segments have a near one-to-one correspondence with the abrupt climate changes reported by Douglass [2010].

The plot below is a Figure from Douglass [2011a] of NL from 1990 to 2012. The plot shows two complete phase-locked segments. The first is a segment of 3 cycles of period 3-years from about Dec 1991 to Dec 1999. Note that the El Niño of 1997-98 is the third oscillation in this segment. The second segment shows 3 cycles of period 2-years from about June 2002 to Mar 2008. A new phase-locked segment  began about April 2009; the period is not yet determined [as of Nov 2011] but may be 2-years. The explanation for the abrupt beginnings or endings is not known.

Figure 2 from Douglass [2011a]

Chaos of the climate system

Douglass [2011b] (here) also studied various ENSO time series and showed from a determination of the Lyapunov exponents that the underlying climate system is chaotic . See table below. In particular:

  • dimensionality  d = 3
  • one exponent is positive, which by definition means that the system is chaotic
  • the El Niño index NL has one exponent ≈ 0, which mean that the underlying dynamics are described by d (=3) first order differential equations

Since the time series are bounded and the phase-locked states are predominately 2nd and 3rd subharmonic of the annual forcing, then the chaos of the climate system is probably deterministic (bounded) of low order.

Table of Lyapunov exponents of ENSO indices. From Dougass [2011b]

Conclusion

Studies of various climate indices suggest that the global climate system is chaotic of low order.  In particular, the ENSO indices show many time segment that are phase locked to an annual cycle probable of Solar origin. These phase locked states have abrupt beginnings and ending, which have occurred at least 18 times since 1870.

Predicting future climate phenomenon depends upon knowledge of the past and an extrapolation to the future that assumes continuity of the relevant climate variables. Continuity of climate indices of interest across an abrupt Climate Shift cannot be assumed and any analysis that does so may be meaningless.

References

Douglass D.H. (2010) Topology of Earth’s climate indices and phase-locked states. Physics   Letters A 374  p4164-4168. doi:10.1016/j.physica.2010.08.025.  (pdf)

Douglass D.H. (2011a)The Pacific sea surface temperature  Physics Letters A 376 p128-135.    doi:10.1016/j.physica.2011.11.10.042  (pdf)

Douglass D. H. (2011b) Separation of a Signal of Interest from a Seasonal Effect in    Geophysical Data: I. El Niño/La Niña Phenomenon David H. Douglass International   Journal of Geosciences, 2011, 2, **-** Published Online November 2011 (pdf)

Towsley A., J. Jonathan Pakianathan and D.H. Douglass.(2011) Correlation Angles and    Inner   Products: Application to a Problem from Physics  International  Scholarly Research   Network, ISRN Applied Mathematics, Volume 2011, Article ID  323864,   doi:10.5402/2011/323864 (pdf)

Swanson K. L. and A. A. Tsonis (2009) Has the climate recently shifted? Geophys Res. Letters 36  L06711, doi:10.1029/2008GL03022

Trenberth K. E. (1990) Recent observed interdecadal climate changes in the northern  hemisphere. Bull. Amer. Meteorol. Soc. 71 p988–993

JC comment.  This guest post arose from an email that David Douglass sent to me pursuant to the Santa Fe Conference about some of his recent papers.  I invited him to do a guest post, since this has been a topic of interest at Climate Etc.   Previous Climate Etc. posts on this topic include:

I would like to thank David Douglass for his post, and remind you that guests posts implies no particular endorsement by myself.

180 responses to “Shifts, phase-locked state and chaos in climate data

  1. Consensus Climate Theory allows Ice Volume to grow because earth is cold. Consensus Climate Theory grows the ice volume in the north while earth has no liquid water in the north.
    Consensus Climate Theory allows Ice Volume to diminish because earth is warm.

    Pope’s Climate Theory puts the Ice Volume in place when earth is warm with lots of snow. The ice advances south, as far and it can and then piles up while the oceans drop and get earth gets cold because of the Ice Volume and the high Albedo. Pope’s Climate Theory stops the snowfall when earth is cold, because the water is frozen and cannot deliver moisture for snow. The ice melts and the ice sheets thin and keep earth cold for a long time. That is what ice does when it melts. It keeps a constant, cold, temperature. Once the ice sheets thin enough and start to retreat, earth warms as the Albedo reduces.

    For Major Ice Ages, the oceans drop enough to stop the flow of warm water into the Arctic and the long cold is on the order of a hundred thousand years.
    For the current, very stable, narrow, cycle the warm and cool periods are a few hundred years that can vary due to other forcings. The temperature limits are well defined and cannot exceed plus or minus 2 degrees C from the mean. They have not exceeded this plus or minus two degrees for the past ten thousand years, in spite of all forcings.

    • Herman,

      I look at it differently.
      Energy is the ONLY reason water is not solid ice.
      Lack of solar energy, circulation, friction, etc. is the lack of the ability to change ice to water.

      • That is not different. In my Theory, the sun does provide the Energy and does melt the ice. Earth uses the ice to regulate the thermostat. When it gets too warm it snows. When it gets too cold, the snow stops.
        It is that simple. Why do so many of you try to make a simple Thermostat so complicated.

      • Herman,

        The interactions of this planet and sun are far more complex than what you have included.
        Does this include the positioning of moving ocean heat or cold currents?
        No, these are part of planetary circulation and does have an effect on precipitation patterns.

      • Joe,
        The total climate is very complicated.
        The Thermostat is very simple.

      • Joe,
        No. I am not trying to model the climate.
        I am just explaining that when earth is warm it snows and when earth is cold, the snow stops. All the other forcings do force the cycles, but cannot force the cycles outside of their stable range. When extra people come in my house in summer, the AC runs more. When the sun shines hotter, the AC runs more. When a solar cycle or wind cycle warms earth more, it will warm the oceans and that will melt more Arctic Sea Ice and the snow will compensate. Sometimes this takes a few hundred years. During the major ice age and warming cycles this took thousands of years but the cycle was the same. When earth was too warm, it snowed like crazy. When earth was cold, the snow stopped.

      • Herman, you are right. Earth’s climate is complicated because Earth’s heat source is very different from the stable heat source “in equilibrium” that world leaders and their consensus scientists perpetuated for the last four decades by hiding or ignoring experimental data and observations:

        “Deep roots of the global climate scandal (1971-2011)”

        http://dl.dropbox.com/u/10640850/20110722_Climategate_Roots.pdf

        David Douglass is to be congratulated for having the courage to address reality in the current atmosphere of fear and uncertainty over implications of the National Defense Authorization Act bill that the US Senate is set to vote on today or later this week:

        http://www.infowars.com/yes-americans-will-be-targeted-as-terrorists-under-the-ndaa/

        With kind regards,
        Oliver K. Manuel
        Former NASA Principal
        Investigator for Apollo

      • “Joe,
        The total climate is very complicated.
        The Thermostat is very simple.”

        Perhaps, what you describing is mechanism of climate.
        Do you think a world would behave differently if continents were in different location. Or completely different mix of ocean to land on a planet?

      • gbaikie,

        We do have 2 hemispheres that show that they ARE slightly different.
        But this is the point science misses…the slight differences have absolutely no meaning to current scientists.

        Meanwhile, the different velocities at the different latitudes have differing densities.
        Our planet looses 1.25mm of water to space every 10,000 years. |
        So, calculating back…this planet had vastly more water.
        This would make a great deal of sense when the pressure exerting down suppresses the toxic gases that should have been present if we had no water.

      • The location of the continents and the Arctic Ocean is very important.

  2. David,

    If you do NOT look at ALL parameters, how can you generate a good conclusion?

    • Do you mean…lack of knowledge then generates the conclusion of chaos?
      But most scientists are perfectly happy with the current parameters and do NOT want any new knowledge that may interfere with the research they have been funded to study.

  3. Excellent post! I have been wondering about that squiggle right before 1994. I was thinking that was a precursor to a shift interrupted by the ’97/’98 El Nino.

    • Dallas, thanks for restoring some sanity to this thread.

      Judith, everything above the Captain’s remarks should be deleted as off topic and, franky, nuts!

      • RobB –

        I’m not averse to the occasional OT diversion but in this case I agree. There should be a designated corner of the blog where people with ‘interesting’ ideas can go and talk amongst themselves. And yes Oliver, that includes you! :)

      • the should get a room with the dragons: the Moshpit

  4. So, it looks like it was the thermostat in the water pump after all. Here is the bill for you engine overhaul. Have a nice day.

  5. But seriously folks:

    “Studies of various climate indices suggest that the global climate system is chaotic of low order.”

    Predicting future climate phenomenon depends upon knowledge of the past and an extrapolation to the future that assumes continuity of the relevant climate variables. Continuity of climate indices of interest across an abrupt Climate Shift cannot be assumed and any analysis that does so may be meaningless”

    I don’t know much about chaos. I have been told it is impossible to model a chaotic system deterministically. However, if the chaos is “of low order” is it possible to approximate it with a deterministic model? Note I am not suggesting this has been done.

    • Choatic systems have deterministic solutions. The problem is that an arbitrarily small change in the initial condition can produce an arbitrarily large difference in the outcome.

      What I believe David is suggesting is that we may be able to bound the system with a lower order, non-chaotic model, but I would be surprised if you can get a particularly good result using this technique as the bound is going to be conservative.

    • Well yes and no. A dripping tap (faucet) is a chaotic system; the timing between each drop and the size of each drop fall within quite narrow bounds and there is quite a good correlation between droplet size an the interval until the next drop. However, you cannot predict the beat of the drops, nor the size of the drops, starting at time = t. You can state that the volume delivered, when t is large, very well, but when t is small, you cannot.

    • “However, if the chaos is “of low order” is it possible to approximate it with a deterministic model?”

      Yes and no. First of all. chaotic systems are deterministic. If you run the same equations starting with the same initial conditions on the same computer, you get the same answer every time. But if you use different computers that implement floating point arithmetic in different ways, the solutions will diverge. Lkewise if you compare single vs. double precision on the same computer, or, in the case of differential equations (DEs), different DE solvers.

      To visualize what goes on, see the animation at http://bill.srnr.arizona.edu/demos/mixing/mix.lorenz.html. Just click where indicated and watch 100,000 solutions initialized within the yellow dot separate. Eventually, the only thing you can say is that there is a probability distribution that specifies where the system will be at a given time.

      Note that unpredictability is consequent not to randomness but to measurment error. Being mortal, we humans cannot specify the a system’s position to infinite precision. Sensitivity to initial conditions, as the unpredictability of chaotic systems is called, is a consequence of measurement error which is unavoidable.

      Suppose you can specify the magnitude of the error. How long is long enough? Answer is that it depends on the system and, for the same system, on the parameter values. That’s where the LCEs come in. Each one of them specifies the rate of divergence / convergence in a different direction. These directions are with respect to the system’s position, which means that as the trajectory evolves, the orientation of the directions changes.

      David tells us that the climate data are consistent with there are three LCEs, which means that trajectories evolve in a three dimensional space. Strictly speaking, this cannot be true. The climate system is spatially extensive, which means you’re dealing with partial differential equaltions, which means that the equivalent set of ordinary differential equations is of infinite orde, which means that the space in which trajectories evolve is infinte dimentional. This does not preclude the possibility that within this space is a finite dimensional subspace to which trajectories are confined.

      If you’re dealing with models, you can figure all this out, either by extensive simulation or, in some cases, by mathematical analysis. If you’re dealing with data, the problem is more difficult. This is especially true if the system gets batted about by random inputs. As noted in the post below, the late 1970s / early 80’s witnessed a number of attempts to devise methods to determine the correct “embedding dimension” of experimental / observational time series. Eventually, people moved on, essentially because none of the methods proved robust.

      • In the case of climate shifts, is not the ‘Butterfly’s wings’, really the random injection of smoke/ash/SO2 into the atmosphere from varied sized and located volcanoes?

      • No. Lorenz’ equations have no random inputs. Just three ordinary differential equations. See the commentary at http://bill.srnr.arizona.edu/demos/mixing/mix.lorenz.html. On the other hand, you could certainly add random inputs designed to mimic volcanoes or anything else. Impt point is magnitude of such inputs. If large enough / sufficiently frequent, you obliterate the underlying deterministic behavior.

  6. Okay folks, let’s be serious. Just what are the design specifications for a planet in a chaotic solar system, such as ours? We don’t want to throw a rod now, do we?

    • After that rebuild, how can we help but?
      =============

      • Kim as noted in 2002:

        Section II: “Life is fragile. Mankind lives in fear of calamity on the surface of a tiny, iron-rich planet that comprises about 0.0003% of the mass of the solar system. To calm these fears, public funds are channeled to the scientific community to explain the occurrence of natural events. The results are not always reassuring, e.g., witness the current debate over global warming.”

        Abstract: “efforts to understand unusual weather or abrupt changes in climate have been plagued by deficiencies of the standard solar model (SSM). While it assumes that our primary source of energy began as a homogeneous ball of hydrogen (H) with a steady, well-behaved H-fusion reactor at its core, observations instead reveal a very heterogeneous, dynamic Sun.”

        http://arxiv.org/ftp/astro-ph/papers/0501/0501441.pdf

  7. Hmmm. Phase locked to the sun. Wonder if some manifestation of the sun might indicate the sex of the Abruptio Climata?
    =================

  8. David Douglass

    Thank you for this analysis.

    I believe more work in this direction will lead to interesting and productive results, and is vastly superior to much of what has come before.

    Conversely, I observe there are unacknowledged issues in the paper, in particular the high reliance on Swanson K. L. and A. A. Tsonis (2009), or rather moving forward so ambitiously with limited means to confirm what sounds somewhat speculative, and while leaving out some strong alternative explanations.

    Anyone with much experience in graphical analysis knows to be suspicious of the suggestions to the eye made by bumps in the curve, and to tend to require far more evidence before agreeing they reflect real effects than are required of linear hypotheses.

    For instance, I can readily accept the odds are 1000:3 that there is a dramatically warming global trend since the last third of the past century compared to the prior trend. I can’t accept that there’s a 60ish-year regular alternating tendency in the same trend for another century at least, however, there being too few periods available to confirm this speculation and a well-established alternate explanation of convolved regional and other trends. (http://www.woodfortrees.org/plot/sidc-ssn/mean:7/mean:11/from:1800/normalise/scale:-0.1/offset:-0.5/plot/best/mean:41/mean:61/isolate:123/normalise/scale:0.15/offset:-0.5/plot/jisao-pdo/from:1908.5/to:1935/trend/normalise/plot/jisao-pdo/from:1935/to:1961.5/trend/normalise/plot/jisao-pdo/from:1961.5/to:1988/trend/normalise/plot/jisao-pdo/from:1988/trend/normalise/plot/esrl-amo/to:1882/trend/normalise/scale:0.5/plot/esrl-amo/from:1882/to:1914/trend/normalise/scale:0.5/plot/esrl-amo/from:1978/to:2010/trend/normalise/scale:0.5/plot/esrl-amo/from:1946/to:1978/trend/normalise/scale:0.5/plot/esrl-amo/from:1914/to:1946/trend/normalise/scale:0.5)

    We have perhaps two centuries of instrumental record, and it’s questionable if so much as one third of that could be called global (though land only — oceans being much much worse) at sufficient resolution and confidence to support robust topological analysis. Is this really enough for the sort of analysis being done?

    And that’s the instrumental record, with almost 40,000 sites and 1.5 billion distinct data points. The paleo record, although it reaches much farther back, has only a fraction of the sites, of the data points, of the effective resolution (for temperature) due its proxy nature, and the confidence, so is little help to us without better metrics of what is a meaningful trend.

    Can you demonstrate what certainty we ought vest in this type of analysis, and speculate on what uncertainties may pertain?

    Are all five identified Climate Shifts real, or is it possible one or more of these shifts is an artifact of technique?

    We know there are strong recurring trends in several ocean basins; is it really a “Shift” if what we see is mere superposition of the extreme peak or trough of a pair or trio of regular trends, absent a change to a new and dominant mode that either disrupts one of these contributors or leads to future global climate measures to no longer show the influence of these components?

    I look forward to more good research in this vein.

    • B, your certainty of the ‘dramatically warming global trend since the last third of the past century’ (sic), is betrayed by your uncertainty as to the rest.
      ===================

      • kim [sic]

        It’s called math. It lets me do things like determine confidence intervals all by myself. (Don’t be turned off by the long words. They’re not that hard.)

        You should try it sometime.

        It’s a lot of fun.

      • The big words are easy Bart, It’s the little words that are v. tricky. They don’t use hand-waving.

      • Heh, Phil Jones heself told me that there are three episodes in the last century and a half of temperature rise at the same rate as the last quarter of the last century. But tell me, why should I believe him?
        ========================

      • RoyFOMR

        Then use them without handwaving, if you can.

        kim [sic]

        Yeah, I still prefer math to Phil Jones.

        That you prefer Phil Jones to math ought surprise one of us.

      • Unless one is confident in the probability distribution, you cannot be confident in the confidence intervals.

        Climate is a fractal distribution, in that it shows scale invariance. This has lead Climatology to underestimate natural variability, by using the wrong statistical assumptions – the assumption that climate is a “normal distribution”.

        The central limit theorem and law of large numbers break down for fractal distributions. You cannot assume that average temperatures have a long term average. Confidence Intervals that rely on a long term average are an illusion of the assumptions.

      • ferd berple | December 1, 2011 at 11:05 am |

        Yes, and no.

        If you see and use the temperature curve on the timescale in question as having a long term average, you must produce a CI for that timespan to identify the statistical reliability of the curve for such durations.

        If you see and use the temperature curve as fractal, then you would wish to determine its fractal dimension over multiple timescales. Which again, you would be hard pressed to do on observational data without examining assumptions and also knowing the CI, error bars, and so forth characterizing the curve of the observed data.

  9. Curious Canuck

    I have watched this several times and would suggest it to all.


    (It’s split into 5 parts – above is 1/5)

    It’s the BBC-Learning’s “Freak Waves” broadcast, of which (legally or not) portions of are available on youtube.

    It’s an interesting broadcast that is relevant to elements of the discussions here on models and their limits, tunnel vision and the development of better methods to observe and predict frequency of a wave type that previous (linear) models said could only occur once every 10,000 years.

    Waves, being so basic within the climate context, seem to represent a relatively tiny component of the entire climate system. It seems that certain ways of approaching them have been discovered since the frist scientific observation of a rogue wave, the Draupner Wave (en.wikipedia.org/wiki/Draupner_wave) in Janauary of 1995.

    At this point it was discovered by a Dr. Al Osborne (Universit of Turin) and his recognition of how a deep-water wave version of the Schrödinger equation could explain the hitherto ‘impossible’ proportions and profile of the Draupner Wave given the overall sea-state it occured in.

    It seems it took until the late 90s for us to realise where we stood with ocean wave science in regards to quantum theory in order to design models that better reflect reality. Just where we are with knowing when to take this beyond this most basic aspect(s) of the climate system to the larger system-wide models and just how succesfully we are doing so?

    I’m not asking so much for an answer, as sharing the sorts of questions raised when watching this with the larger climate debate and modelling attempts in mind. I hope anyone who has some time goes through enjoys it and walks away with questions, or in the cases of those with professional understanding of the matter some answers to the logical questions it could raise among climate debate observers and belligerents.

    Seems relevent to the discussion of chaos.

    • That is a good one. Peter Davies had a link to paper on the typical frequencies of non-ergodic systems, which climate may be one, I do get flack on that, but it appears non-ergodic to me. With synchronization, the 5% confidence levels used in Climate science, could allow for 250 year periods of “climate” outside of the not as tight as normal confidence interval use in climate science. The one percent level may be a glacial lasting about 90K years. So most of the black swans would appear to be in the wrong direction of those predicted based on global warming theory.

      A climate rouge wave can last a long time. The system is a bit more complex than most theories seem to indicate.

  10. Curious Canuck

    Cool thoughts Capt. Large scale climate rogues, never thought about it on that level as such but yeah.

    And look how simple the models were, even in the updated approach. Simply focussing on the for and after wave, in the case of the energy draw of the rogue and also in the discussion of of tide/current effect opposing a given sea. Both only showed ‘head-on’ opposition. You get into cross-seas and, like the Antarctic mariners saw, the freaks (relative to average wave height) usually come in at wild angles. You’d think in reality there’d be some potential to draw energy from waves of one or more other wave trains interacting with your aspiring rogue wave at varying angles and distances and durations, this in addition to the effects of tides and currents.

    Multiply the example out to the number of wave/variables involved with a climate system. Seems to me that these extrapolation models we see in use don’t really seem anywhere sophisticated enough to do what they claim predictively (the C, in CAGW) based on a relatively small change of a single variable in addition to natural variablitiy (ie.Vostok’s suggestion that civilization has only ever existed on the dangerously brief-looking, crest of an extreme climate wave, MWP, LIA, etc).

    It’s reassuring that the climate isn’t as ‘solved’ as some of would have us believe. The real discoveries are going to be so informative and exciting taking place outside of this umbrella of ‘messaging’, castigation and secrecy.

    Tight lines and thanks for the engage.

  11. I am interested… Enough so that I am downloading the references. I believe the following one has an incorrect doi: though…

    Swanson K. L. and A. A. Tsonis (2009) Has the climate recently shifted? Geophys Res. Letters 36 L06711, doi:10.1029/2008GL03022

    It should be
    doi:10.1029/2008GL037022

    I may have more comments after I have digested the the reference material.

    Thanks again…

  12. David, Judy –

    Agree with David’s bottom line conclusion; more generally that you don’t understand the climate system til you’ve got a model that predicts observed cyclicity. Calling the latter “natural variability” gives you a name, nothing more. In addition:

    1. Periodic forcing of a nonlinear system one way of generating chaos. Interestingly, high frequency forcing can generate cycles or arbitraily long period Impt question is whether climate chaos is a) intrinisic, b) a response to annual forcing, c) response to solar forcing, itself chaotic, d) some / all of the preceding.

    2. Estimating Lyapunov exponents (LCEs), fractal dimensions, etc. from short, noisy time series fraught with difficulty. Became a cottage industry in the late 70’s / 80s that produced mutliple methods; none robust w.r.t. distinguishing chaos (deterministic) from non-chaotic dynamics plus noise.

    3. Observation of multiple cycles alternative to estimating LCEs because the former ties directly to the topology of chaos. Specifically, periodic orbits (cycles) “dense” on chaotic sets – i.e., every point on such sets arbitrarily close to such an orbit.

    5. These cycles not attractors, but rather have the stability of saddles. That is why chaotic systems “drift” from vicinity one cycle to the next. In real world systems, transitions from vicinity of one cyle to another effectively impossible to predict. See any text on nonlinear dynamics that treats subharmonic resonance for details; alternatively Schaffer, W. M. 2009. A surfeit of cycles. Energy and Environment. 20: 985-996 (http://bill.srnr.arizona.edu/mss/Surfeit.pdf) for discussion with reference to climate.

  13. It would have been helpful in a paper introducing a new metric if its properties ,significance, and skill had been established with a rigorous
    examination of synthetic systems. Otherwise it tends to look somewhat like
    those mannian methods invented and applied to data in one swell foop

    • steve,

      you mean:

      “Douglass [2010] (here) improved the Swanson and Tsonis scheme in a number of ways. A different distance “d” was defined so that it satisfied the 3 triangle conditions to be in a mathematical metric space. This then allowed the correlation among 3 or more indices to be easily measured using the diameter D from Topology. Minima in D corresponds to high correlation.”

      as the new method?

      as far as Mannian methods I though the problem was the follow up, not the original application. follow up i.e. can’t get data, overhyping etc. etc.

      • Bill –

        Although it’s probably very elementary for you – I thought you might find this interview interesting (I just re-listened to it the other day and remembered that you wrote about multiverses).

        http://www.npr.org/player/v2/mediaPlayer.html?action=1&t=1&islist=false&id=132932268&m=133120238

      • Joshua –

        Interesting link.

        ‘Multiple Universes’ is one of the few phrases that turns me into a pure pedant. What, then, can one mean by Universe?

        For everything else I can be (and usually am) a semantic relativist, but this IMO is wrong :)

      • Multi-verse? Cool! Now we can get into thermal/non-thermal flux interaction and its impact on climate!!! I knew we would get around to relativity sooner or later :)

      • Anteros –

        I’m descriptive grammar guy (not prescriptive) – and I also have my limits, but he does talk about the term.

        He says that if you define the universe as absolutely everything, then you can’t say that there are multiple universes, but that they’ve come to theorize that what they previously thought of as absolutely everything is only a small fraction of what actually exists.

        Simple enough explanation, right?

      • All the possible jokes from the vast field of the Theory of Branes, and we can’t do better?

      • Thanks Joshua, but you do me undeserved honor. My only understanding of multiverses from Stargate and the like. String theory above my pay grade.

      • Speaking of, this is probably right up Lubos Motl’s alley.

      • Which is just where Lubos likes it

      • Are you BillC?

      • No Joshua. I’m Bill S.

      • No Joshua. I’m Bill S.

        That’s logical!

        My original comment in this nest was directed at BIllC (not that I wouldn’t have assumed you to be knowledgeable about multiverses as well – but he had mentioned an interest in that topic in a previous exchange with me) and so it seems that the honor you thought was undeserved wasn’t directed at you to begin with.

        So I guess in a sense it wasn’t really an undeserved honor after all.

      • Joshua,

        Actually I don’t find this stuff elementary. I am actually listening to it right now but probably won’t finish it. I listened all the way through a similar segment with Saul Perlmuter while driving last week, and it was very interesting.

        But my evolving general conclusion is that my “contributions” whatever they are, do not require exploring things like the origins of the universe or the workings of subatomic particles. Too far from the systems of my primary interest. Until for instance someone demonstrates that the quantum behavior of atmospheric gases is different in the atmosphere than in the lab.

        This post (and more so Bill S’s commentary on it) and WHT’s recent post have some common themes, most strikingly the possibility of “forcings” to swamp “natural variability” in some cases.

        If I was Ben Santer, I’d use this stuff to backpedal off the 17 year thingy, and move the goalposts again.

      • BillC –

        If I was Ben Santer, I’d use this stuff to backpedal off the 17 year thingy, and move the goalposts again.

        What is the dividing line between modifying analysis on the basis of new information and moving goalposts?

        I suppose that answering that question is at some level important, but it is also inherently fraught with subjective datapoints, and in the long run more than likely irrelevant except for those whose primary interest is tribal.

      • “If I was Ben Santer, I’d use this stuff to backpedal off the 17 year thingy, and move the goalposts again.

        What is the dividing line between modifying analysis on the basis of new information and moving goalposts? ”

        When the information isn’t new.

      • Note that I don’t mean personal insult to Santer and his 17 coauthors, who are probably related to someone I may try to work with someday :).

        But seriously, why write a paper that suggests a 17-year criterion, which turns out to be just slightly greater than the current downturn, necessitating confirmation in terms of a warming by 2015, when internal chaotic oscillations MIGHT cause it to stay relatively cool until 2030 and then warm like a mother.

      • I agree, they are setting themselves up for an unnecessary “fall” with this one. One motivation seems to be to make sure that the cool period in the 1940’s – 1960’s can’t be explained by natural internal variability. Personally, I suspect that much of what was going on mid-century was indeed natural internal variability.

      • But seriously, why write a paper that suggests a 17-year criterion, which turns out to be just slightly greater than the current downturn, necessitating confirmation in terms of a warming by 2015, when internal chaotic oscillations MIGHT cause it to stay relatively cool until 2030 and then warm like a mother.

        I would suppose to possible explanations. The first is political CYA. The second is that 17 years is what his analysis returned.

        Ironically, the first explanation may be less likely for exactly the reason you alluded to; a merely political motivation is likely to be counterproductive. Of course, it may just be that he’s a bad politician.

      • err….two.

      • Yes I mean that passage.
        as for mannian methods see jeffids work

      • steve,

        jeff id does a lot of work and i think most of it is good, but i don’t know where to start with your suggestion. i don’t think it matters, because my point was that haven’t we more or less agreed that the original mbh paper was “just a paper” and the problem is all the stuff that occurred afterwards.

    • Mosher,

      I don’t think it is introducing any new metrics. It is just analyzing probability states. The mechanisms are not defined, just that the probability that solar is linked is greater etc. Verifying the mechanisms with thermo, fluid dynamics or another field is still required, there are just better indications of the strength of some connections or the weakness of other connections, which appears more likely.

      So if solar has a definite impact under certain conditions, radiant CO2 forcing under those same conditions could not be greater that a certain magnitude. Just more confirmation that sensitivity is likely less than 3C and more likely on the order of 1-1.5C.

      • They call it a new metric.
        You jump to fantastical conclusions. yawn

      • First two sentences, no problem. Your yawn just let you down.

      • At least I get exercise :) They may call it a new metric, it may be, but they had a post on the phase locked approach a while back, if memory serves.

        Sorry I bore you, but Leif Svaalgard has made it pretty clear that TSI change even with the UV is not enough to drive climate more than 0.1C or so. When I look at the ratio of surface to solar absorption, it looks like twice that is possible. With the Monckton/Lucia blow up, my take was sensitivity was closer to 0.8. 0.2 is not insignificant when compared to 0.8 or less.

        Time will tell :)

    • Steve –

      I think you’re right but also wrong.

      Right: As noted above / below, attempts to calculate dynamical quantities from observational / experimental time series eventually petered out. It all got started well enough. FlorisTakens proved that you could calculate such quantities for an n-dimensional system from a (2n+1) embedding of a univariate time series, and he explained how to construct the embedding. Harry Swinney applied the technique brilliantly to experimental time series obtained from the Belousov-Zhabotinskii (BZ) reaction. But the BZ data were unrepresentative of real world data. In the first place, the experiments were well controlled. Second, the experiments generated large amounts of data, where by “large” I mean extensive relative to the longest time scale of the system. The latter is critical. Every point on chaotic sets is recurrent. Start out on such a point; draw an epsilon ball around it. If you wait long enough, you come back inside the ball. If your data’s not recurrent, and most real world time series are not, there’s no hope. Likewise, you’re also out of luck if the environment changes on time scales commensurate with the longest natural time scale of the system. Then you’re stuck looking at transients.

      Wrong. Nonlinear dynamics is not Mannian foppery. The underlying theory was worked out by serious / talented scientists and mathematicians (Poincare, Arnol’d, Lorenz, etc.) in the days when science was done for its own sake. Climate cycles, ye olde natural variability, are undoubtedly real. Cycles are also the essence of chaos and the reason chaotic systems manifest sensitivity to initial conditions. Job 1 is not, I submit, predicting the quantitative time evolution of a system we don’t understand with models that can’t be understood. That’s job n >> 2. Job 1 is understanding the system’s qualitative behavior. That means dispensing with all-but-the kitchen sink simulations and going back to what models are all about, i.e., the simplest caricatures of reality that iterate to whatever data we happen to be interested in.

      Getting back to David’s post, with a properly formulated model, one could study study the following: 1) The relative frequency with which one observes statistically periodic behavior of different frequencies; 2) the durations for with which said behaviors persist; 3) the transition probabilities of going from cycles of period i to j. In each case, one could compare prediction with observation to the extent that the data are available. Do this with a model simple enough to accumulate the necessary simulated data and to have some understanding of what’s going on mathematically, and you’re in a position to begin determining whether or not you have a clue with regard to the real world.

      To the extent that climatology is driven by a desire to make quantitative predictions, it’s putting the cart before the horse. A model parametrized on the basis of motion in one region of phase can have maximum skill for that region, and be worthless when the system jumps to another region.

      • Bill, I would be most interested in hosting a guest post by you on the issues that you have raised here. If you are interested, pls send me an email curryja at eas.gatech.edu

      • by mannian foppery I was referring to the practice of introducing a new method and simultaneously using it on real data, rather than studying the metric first with synthetic data where the underlying truth is prescribed.
        Then the metric is tested for its ability to capture that aspect of the system.

        in this case what is the ability of the metric to capture actual synchronization.

      • “By mannian foppery I was referring to the practice of introducing a new method and simultaneously using it on real data, rather than studying the metric first with synthetic data where the underlying truth is prescribed.”

        Oh, I absolutely agree. And that was what happened. Someone would propose a new way to compute something or other, test it out on a model system, maybe with noise added, and then put it into the literature. Then it would fail when tested on another model. Eventually people got sick of estimating dynamical invariants and moved on. At least, that’s my reading. However, in fairness, I should note that I recently met a very bright / enthusiastic graduate student at a meeting who thinks my pessimism not justified.

      • yes I would l like to see a guest post on this stuff by Bill S (not Bill C).

        I feel like the paper you linked to (I read it) was better than the journal it was published in.

        and I thought some of us at least agreed that “mannian foppery” was ok within limits, like publishing an original paper and then getting the sh*t kicked out of it in the ensuing literature, minus the part about amplifying echo chambers.

      • Good post. Mainstream climate science is badly in need of a fresh approach as it seems now to be running around in circles and not advancing. I am of the belief that climate shifts do occur due to external forcings that cannot be predicted due to the chaotic nature of the earth systems in play.

      • w. m. schaffer

        In What Is A True Model? What Makes A Good One?: Part VI Climate Model Focus, Statistician William M. Briggs observes:

        Climate models don’t have strong skill (as far as I’ve seen) at predicting yearly global average temperature. They do not (again, as far as I’ve seen) beat persistence. Thus, their predictions cannot yet be trusted.

        A number of researchers have modeled global temperatures as a trend since the Little Ice Age with a superimposed PDO type oscillation. e.g.
        Syun-Ichi Akasofu,

        On the recovery from the Little Ice Age, Natural Science Vol.2, No.11, 1211-1224 (2010); or
        Natural Components of Climate Change

        Appreciate your comments on the “skill” of persistence, or “simple” models such as Akasofu’s oscillation on a warming trend, compared to Douglass’ versus the Global Climate Models and waiting 17 years, in light of Briggs

  14. The ‘phase’ operating system in the North Atlantic is certain:

    http://www.vukcevic.talktalk.net/CET-NV.htm

    and possibly globally

    http://www.vukcevic.talktalk.net/CET-NVa.htm

    Solar influence is fundamental but not predictable and certainly not via the TSI

    http://www.vukcevic.talktalk.net/HMF-T.htm

  15. Dr. Douglass,
    Thank you for this post. You bring a fresh perspective and present it well.

  16. Minima in D corresponds to high correlation. These new definitions were used to study of a different “more global” set (Nino3.4, north and south Pacific indices and Atlantic Multidecadal Oscillation, AMO) of climate indices that included both north and south hemispheres.

    This is all a bit irrelevant, if you whish to find out quantity of timber in a tree, you do not need to microscopically study leaves structure.
    As far as the AMO is concerned, there is a lot nonsense written (starting with M. Mann), making it a mysterious climatic process which may explain this or another. It is nothing of a sort it is a simple consequence of another more direct and better data documented process. Anyone interested in the AMO, scientist or amateur, AGW supporter or sceptic can find more new info here:

    http://www.vukcevic.talktalk.net/theAMO.htm

  17. re. Mosher’s comments. I think D. Koutsoyannis has done work in this area from a hydrological point of view. DK’s work seems to be ignored by the Climate science franternity yet hydrology is arguably is the most important impact of climate. Why introduce new measures when there is years of work behind things like the Hurst coefficient? At the very least Douglass should explain why Hurst is irrelevant to his work.

  18. By inspection, the earth’s climate is extremely stable. When it is warm, it always cools. When it is cold, it always warms.

    • Sometimes it takes a few hundred or a few thousands years, but it always warms when it is cold and cools when it is warm. That is a stable system! Does anyone out there see that?

      • Sometimes it takes a few hundred or a few thousands years, but it always warms when it is cold and cools when it is warm. That is a stable system! Does anyone out there see that?

        Then why hasn’t the temperature gone down by 33 degrees Celsius and warmed and cooled about that point?
        By your reasoning it warms when it warms, and then it warms only slightly less when it is less than maximally warm.
        If that doesn’t make sense, then you should understand why we have so much trouble with your logic.

      • The current bounds are plus or minus two degrees with strong tendency to be plus or minus one. Before the current warm period, before the most recent major warming, before 20 thousand years ago, the bounds were much wider. Then, earth could and did warm and cool with much wider bounds. That does not happen now.

    • Herman – You’re confusing boundedness with stability. By definition, a chaotic attractor exists within a region of phase space wherein initial conditions tend to the attractor. Once on the attractor, you bounce around, but remain thereon. If one of the state variables is temperature, you get warming followed by cooling followed by warming, etc. But that’s not an equilibrium.

      If you’re looking at data, equilibrium plus random perturbations is an alternative hypothesis. But it doesn’t get you cycles. Of course, if the range of temperatures on the chaotic attractor is small enough, you might just as well consider it the system equilibrial and be done.

      To illustrate these points, try this: Iterate x’ = rx(1-x), where x is the present value of x, and x’, the next value, with r = 3.944 and the first value of x = .01. That is, you compute the 1st point, then take the result, plot it, then plug it into the formula and compute again. Do this many times. The system converges to an attractor that ranges (roughly from x =.12 to .97). Within that range, it hops around. In otherwords, the dynamics are bounded, but not equilibrial. You get the same qualitative result for all x0 (initial point) between 0 and 1.

      Try it again with r = 3.864. Same overall result, but you should see a lot of period three behavior. Now try x0 > 1 or x0 negative infinity, i.e., these initial conditions are outside what they call the “basin of attraction.” Can you see why? Hint. Plot x(i+1) vs x(i) – not by interating the equation – but by plotting the function.

    • Many of the things that drive earth temperatures may very be random or chaotic. But when earth gets too warm, Arctic Sea Ice Melts and it snows and cools earth. When earth gets too cool Arctic Sea Ice Freezes, the snow stops and earth warms. Ice and Water have a set point.

      • Herman Alexander Pope | December 3, 2011 at 9:36 am |

        I don’t entirely disagree with the observation that when the Earth has in the past gotten warm, it has cooled and the obverse — when it has cooled, it warms. We may quibble about specific details like mechanism in this apparent homeostasic tendency, but we agree it appears to be there to some degree.

        We diverge on two elements:
        1.) Whether the Earth “warms or cools” as its homeostasis, or “tends to balance its inputs by warming or cooling” as its homeostasis;
        2.) Whether Earth’s mechanisms for homeostasis may be crippled by large scale changes.

        1.) The homeostasis of warming, you predict, would directly cool the Earth. The homeostasis of new inputs would work to absorb the new inputs and integrate them into the overall system. That integration, that overall new system state, is a warming one and will remain so as long as the CO2 perturbation continues and then once the extra source emissions stop thereafter until the CO2 level falls to its convergent range of 230+/-50 ppmv. Which will be hundreds or thousands of years, given WHT’s estimation of a 44 year halflife of anthropogenic CO2 in the atmosphere.

        2.) Land use changes, albedo changes due anthropogenic particulates, other atmosphere changes like anthropogenic aerosols, sea life changes due chemical dumping and fisheries, UHI’s, deforestation, all of which may tend to limit or upset the ordinary function of homeostasis of the Earth system, are of large enough scale to consider in this question.

        So while your hope is optimistic, your Gaian faith in this magickal homeostasis is strong, it isn’t rational to just say it must always be because in the past it has been.

      • Which will be hundreds or thousands of years, given WHT’s estimation of a 44 year halflife of anthropogenic CO2 in the atmosphere.

        In fat-tail terms a “half-life” for CO2 adjustment time doesn’t really exist. The 44 years does not turn into an exponential decrease, but merely indicates a characteristic diffusion time. What this means is that dozens of the 44 year times strung together will not turn into an infinitesimally small level, but instead will show considerable excess CO2 concentration. That is why the hundreds or thousands of years is the correct yardstick, as you indicate.

        I just wanted to clarify this because skeptics tend to twist the physics to fit their purpose, since the term half-life has a specific meaning and will eventually get misinterpreted.

      • WHT

        I stand corrected, and thank you for the clarification.

      • If something happens for a whole bunch of times, over a period of ten thousand years, it is most likely that that same thing will happen again. It is most unlikely that some totally different event will happen on the very next cycle.

    • w.m. You would have me say that temperature is bounded rather than say that it is in a stable cycle. I can see some logic to that. According to Arctic and Antarctic Ice Core Data, for the past ten thousand years temperature has always been within plus or minus two degrees of the average. For most of the time temperature has been within plus or minus one degree of the average. It is currently inside the plus one level. Temperature is bounded by limits it does not exceed.

      • Herman Alexander Pope | December 3, 2011 at 10:48 am |

        I’m driving along the freeway. Sometimes there’s a slight rise in the road, and because my foot is steady on the accelerator pedal, I slow down to a slight degree; when there’s a dip in the road, I speed up to a slight degree.

        Speed is bounded by limits it does not exceed while my foot is steady on the accelerator pedal.

        That was up until a quarter millennium ago.

        Since then, I’ve been flooring the pedal.

      • Go ahead, turn the heat up, more snow will fall to correct your mistake.

  19. David, I assume you know that the chaotic nature of El Nino is not a new claim. See http://www.sciencemag.org/content/264/5155/70.short from 1994.

    The issue is the chaotic nature of climate per se.

  20. Speaking of climate shifts I recently published an article discussing the change in stratospheric and tropospheric temperature trends that occurred in the late 90s.

    http://climaterealists.com/index.php?id=8723&linkbox=true&position=6

    “CO2 or Sun?”

    I think the issue highlighted is important and relevant and deserves attention.

    The observations are clearly incompatible with AGW theory but compatible with a solar connection.

  21. I am an engineer not a scientist so much of this is beyond my depth. I do understand some things about coupled nonlinear dynamics though, e.g. saddle points. Can we say anything about what future possibilities are based on what past climate has been during this and past inter-glacial periods? If we did warm by 3C wouldn’t that be the highest temperature during this interglacial? Can we say anything about the possibility of temperatures above any that have occurred during the current glaciation?

    • To cut to the chase………..Aren’t deterministically chaotic systems bounded?

      • “Aren’t deterministically chaotic systems bounded?”
        I guess the clue lies in the deterministically bit.
        But, apart from well-bounded cases where determinism is a given, how can we comfortably apply that term to less certain cases?

      • Indeed Doug, chaos is actually a form of stability. In simple cases once the system gets into the attractor it will stay there, so it is highly bounded. Climate certainly looks like that, which might well mean that the kind of positive feedback excursions postulated by the models simply cannot happen. The price of this stability is intrinsic unpredictability within the allowable range.

  22. Dr. Douglass’ stimulating material is appreciated. It deserves careful consideration. Although it’s the busy season at work, I’ll see if I can find time to share some cautionary notes a few days from now. I hope Tomas Milanovic will also have time to share cautionary notes from his perspective. The communication needed to achieve common understanding will be tedious, but nature’s certainly fascinating enough to keep us dedicated.

    • I have to admit that I read the first paragraph of Dr Douglass and at once did a search for Tomas Milanovic, to see his reaction, which is how I ended up here.

      Nature’s certainly fascinating enough to keep us dedicated.

      What a lovely way to put it. Happy Christmas :)

  23. Dr. Douglass’ contribution is for sure interesting, and I hope to profit from reading the originals. I also liked the comments by w.m. schaffer.

    I may comment, but only after a few days.

  24. I don’t think anyone has yet pointed out the obvious:

    “What is the nature of the climate state between Climate Shifts? In a later paper Douglass [2011a] (here) studied the Pacific sea surface temperatures in greater detail and showed that the climate system is frequently phase locked to an annual cycle.”

    The key is “locked to an annual cycle”. Well, “annual” is just seasonal variations in climate, with the asymmetry between northern and southern hemispheres obviously leading to a yearly annual variation. More water in the southern hemisphere will lead to some sort of wobble, which is definitely observable, even in CO2 readings.

    Not a big deal, but thought I would mention it. The harmonics beyond the annual cycle are really the more interesting feature I would think.

    • my terminologically ill mind assumed “phase locked to annual cycle” sort of included harmonics.

      i am confused about the meaning of “annual cycle of solar origin”.

      this means the sun does something every year besides move back and forth between the tropics while rising in the east and setting in the west every day?

      • this means the sun does something every year besides move back and forth between the tropics while rising in the east and setting in the west every day?

        This is enough to cause a stimulus which can reinforce a harmonic. Say an effect happens every other year at a certain time, then the seasonal effect could reinforce that harmonic (and it doesn’t cancel the other year). It’s actually a subharmonic, which don’t typically occur naturally although one can run into them, for example, a loudspeaker can produce subharmonics when you hear it starting to buzz.

  25. Idle thoughts of an idle fella. If we haved a series of dates of discontinuities, is it possible to electronically search all the data bases in cyberspace, to see if this series coincides with ANYTHING.

    • That way madness lays. The problem is ‘spurious correlations’. If you have 1,000,000 data sets, looking at completely unrelated phenomena, there is a very good chance that one will observe a lovely correlation between the 1,000,001th and one of the the others. Indeed, the correlation between skirt lengths and the trend in the stock market has been shown to have a strong inverse relationship. Ice cream sales and the number of shark attacks on swimmers are correlated as is the number of cavities in the teeth of elementary school children and the size of their vocabulary .

      • DocMartyn writes “That way madness lays`

        I very strongly disagree. If there are spurious correlations, than it should not be too difficult to show that they are spurious. The classic example I know of is from the military. The more you use artillery, the slower you advance. I am sure I dont need to explain. However, if just one correlation was found that could not easily be shown to be spurious, who knows where this might lead.

      • Jim, spurious signals are very often anything but obvious. It would make signal processing very much easier if they were.

    • The correct application of scientific method would be to develop theories and suitable sets of hypotheses (based on these theories) for testing on the available datasets PRIOR to any further study of the datasets themselves.

  26. The basic problem is that climate science is not taking nonlinear dynamics seriously. As I pointed out above, there was a major paper arguing that ENSO was chaotic in 1994. About the same time there was a paper in JGR demonstrating that a simple nonlinear model of ocean upwelling could explain the entire 20th century global temperature (estimated) record. No one seems to have noticed.

    Nonlinear dynamics is a fundamental problem for science itself, because it brings in intrinsic unpredicatabilty, and therefore also intrinsic unexplainability of observed phenomena. In climate science it is pushed off as noise, when it should be the first thing looked at.

    No one that I know of is asking how much of climate change can be explained by chaotic oscillation (perhaps all of it). This should be the fundamental question, the central focus of research. Instead they are starting with the nonsense of zero feedback sensitivity and working out from that. If climate is chaotic, which I strongly suspect, it explains why the science continues to thrash about looking for Newtonian solutions. They do not exist.

    Chaos is a mathematical property, not a physical one. The physical question (as always) is to which phenomena the math applies? It is a strong possibility wherever we find nonlinear negative feedbacks, and climate certainly qualifies. Where then is the research? Alas, there is no money in intrinsic unpredictability.

    • Historical note: twenty years ago I was involved with an effort at the Naval Research Lab, called the Special Project in Non Linear Science, or SPINLS. The SPINLS goal was to get the NRL scientists to at least think about the implications of nonlinear dynamics in their respective fields. Science as a whole still has to come to terms with the math of nonlinear dynamics, although some fields have done well, especially quantum dynamics. Climate science has done almost nothing, beyond the sub-decadal chaos evident in the models. It is the dec-cen chaos that we need to be looking at.

    • Re David’s (imo perceptive) observations:

      “In climate science it [chaos] is pushed off as noise, when it should be the first thing looked at.”

      A while ago, Schmidt posted a (somewhat condescending) note, on Climate Audit, I believe it was. Used the word chaos but, as David says, equated it with noise. Then claimed that with enough data, the anthropogenic signal would be detectable. Maybe yes, maybe no. Can’t say til you do the proper numerical expts – see previous posts. These haven’t been done, because, as David says, people are asking the wrong questions.

      “Instead they are starting with the nonsense of zero feedback sensitivity and working out from that. If climate is chaotic, which I strongly suspect, it explains why the science continues to thrash about looking for Newtonian solutions. They do not exist.”

      Amen. But please substitute “linear” for “Newtonian”. No conflict between Newton and chaos.

      ” … there was a major paper arguing that ENSO was chaotic in 1994. About the same time there was a paper in JGR demonstrating that a simple nonlinear model of ocean upwelling could explain the entire 20th century global temperature (estimated) record.”

      If ENSO chaotic, then entire climate system chaotic with probability one – consequence of Takens’ theorem. That doesn’t mean that it may not be more apparent in ENSO.

      Thanks for the refs.

      • +1 to both Bill’s and David’s comment on this. Nonlinear dynamics as it relates to the climate problem is something I want to spend more time on here.

      • Dr. Curry, will you please consider e-mailing Tomas Milanovic to let him know some of us would appreciate his comments on the notion (apparently) embraced by many here that this can be treated as a time-only problem?

      • … there was a major paper arguing that ENSO was chaotic in 1994

        The Jin paper ie the Devils Staircase introduces a number of important problems such as structual stability( instability ) in the Andronov and Pontryagin criterium eg J. Guckenheimer and Yu. S. Ilyashenko, The duck and the devil: canards on the staircase. Moscow Math. J. 1 (2001).

        Ghil 2008 ( arXiv:1006.2864v1 ) eg Theorem B.3

        Before applying this result, let us explain heuristically how a Devil’s staircase step that corresponds to a rational rotation number can be “destroyed” by a sufficiently intense noise. Consider the period-1 locked state in the deterministic setting. At the beginning of this step, a pair of fixed points is created, one stable and the other unstable. As the bifurcation parameter is increased, these two points move away from each other, until they are pi radians apart. Increasing the parameter further causes the fi xed points to continue moving along, until they fi nally meet again and are annihilated in a saddle-node bifurcation, thus signaling the end of the locking interval.

      • Sounds interesting Maksimovich, elegant even. Fixed point annihilation! (I love this stuff, but I have the impression that the Russians are way ahead of us.) What does it mean?

      • Andronov and Pontryagin (1937) that was used by Ghil 2001. 2008 and Zaliapin and Ghil 2010 interesting paper (that was poorly understood by many) .The underlying theory is indeed very elegant as described by Arnold.

        Poincare’s bifurcation theory was elaborated by the Russian mathematicians Pontryagin and Andronov already in the 20’s and in the 30’s (due to the need to apply these bifurcations to radiophysics).

        Andronov published (with all the proofs) the theory of the birth of a periodic motion of a dynamical system under the generic loss of stability of an equilibrium position, in the case when two eigenvalues of the linearised system cross the imaginary axis, moving from the stable to the unstable complex half-plane.

        Andronov’s theorem claims that (depending on the sign of some higher term of the Taylor series) exactly two generic cases may occur: Either the stability of the equilibrium position is inherited by the new-born limit cycle (whose radius grows like the square root of the difference between the new value of the parameter and the value at the stability loss), or else the radius of the attraction domain, diminishing like the square root of the difference between the growing parameter value and the future value, at which the stability will be destroyed, disappears at the stability loss moment.

        The first case is called the mild stability loss, the new-born periodic motion-attractor describes a small oscillation near the old stationary regime. The second case is called the hard stability loss, the behaviour of the system after this stability loss being very far from the equilibrium, loosing its stability.

        The proofs of these results of Andronov on the phase portraits bifurcations were based on the Pontryagin’s extension of Poincare’s results in the holomorphic case to that of the smooth systems of differential equations.

        Ghil 2008 now uses this criteria with the underlying geometric theory
        ( circle maps arnold 1967) to make the underlying analytic information more amenable to a prior description as opposed to PDF per se. eg

        Changes in this double-gyre circulation occur from year to year and decade to decade. We study this low-frequency variability of the wind-driven, double-gyre circulation in mid-latitude ocean
        basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular
        ows documented in the observations. This sequence involves local,
        pitchfork and Hopf bifurcations, as well as global, homoclinic ones.

        The natural climate variability induced by the low-frequency variability of the ocean circulation is but one of the causes of uncertainties in climate projections. The range of these uncertainties has barely decreased, or even increased, over the last three decades. Another major cause of such uncertainties could reside in the structural instability in the classical, topological sense of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics.

        This is a standard approach ( in the dynamical sense) and the underlying information that is being sought is the evolution laws and differentiating from the random and the necessary.

        Dynamics and thermodynamics are different species and the approaches are differing eg the former is used to ascertain the dissipative structures, the later the forcing components a clash of doctrines. Eaddy(1951) suggested a form of Darwinan evolution and selection in flows (baroclinc and baratrophic) and states.

        Thermodynamics dominate in the hadley cells ,dynamics in the ferrel and polar cells.

        The relevance to this post on phase evolution is indeed well described in the literature eg Cvitanovic)

        The physical significance of circle maps is connected with their ability to model the two–frequencies mode–locking route to chaos for dissipative systems. In the context of dissipative dynamical systems one of the most common and experimentally well explored routes to chaos is the two-frequency mode-locking route. Interaction of pairs of frequencies is of deep theoretical interest due to the generality of this phenomenon; as the energy input into a dissipative dynamical
        system (for example, a Couette flow) is increased, typically first one and then two of intrinsic modes of the system are excited. After two Hopf bifurcations (a fixed point with inward spiralling stability has become unstable and outward spirals to a limit cycle) a system lives on a two-torus. Such systems tend to mode-lock: the system adjusts its internal frequencies slightly so that they fall in step and minimize the internal dissipation. In such case the ratio of the two frequencies
        is a rational number. An irrational frequency ratio corresponds to a quasiperiodic motion – a curve that never quite repeats itself. If the mode-locked states overlap, chaos sets in. The likelihood that a mode-locking occurs depends on the strength of the coupling of the two frequencies.

        The implications for the mechanisms of enso complex become more clearly by reframing the problem ie obvious.

    • David, I appreciate your comments on chaos. My instincts tells me that the advancement of climate science depends on further development of discretisation techniques for the study of non-linear systems.

      The work of Mary Selvam that Dallas has referenced suggests that the current dichotomy of quantum and classical science for the study of the smallest phenomena to the universally large phenomena, respectively, is false, since all phenomena surely belongs to the same spatial continuum.

  27. More to say in days ahead, but for now brief notes to get simplest matters aside:

    1. Stating explicitly what we all know: Annual terrestrial spatiotemporal insolation tides – not to be confused with solar irradiance – are due to Earth’s tilt & orbit. Rather than calling the 1 year signal “solar forcing” (which will upset at least some solar scientists), let’s consider calling it “orbital” or “thermal tide” (to avoid protracted straw-man argumentation that does nothing to advance discussion).

    2. The “nonlinear effects” have SPATIAL components. Please see the article I referenced here: http://wattsupwiththat.com/2011/10/15/shifting-sun-earth-moon-harmonies-beats-biases/#comment-771684 . Time travels in only one direction, but spatial gradients turn in time. Cognizance of the potential to unwarily interpret spatial phase reversals as temporal evolution must be vigilantly maintained at all times.

    More meaty issues later…

    Regards.

  28. Is the climate chaotic?

    http://bit.ly/qGcD9M

    No!

    For 160 years, it has a single pattern:

    A long-term global warming rate of 0.06 deg C per decade with an oscillation of about 0.5 deg C every 30 years.

    Why does the “climate establishment” deny this obvious result?

    • Girma, there is nothing in you graphic to suggest that climate is not chaotic. The short term oscillations (less than a decade) are very likely chaotic. The longer term oscillations, including the overall trend which may be part of a multi-century oscillation, may well be chaotic. The fact that these oscillations occur across many time scales, including many smaller scales not shown, is strongly suggestive of chaos.

      Chaos is a form of non-periodic oscillation, due to nonlinear feedback. Your graph looks like chaos to me, as does virtually all climate data. Random looking non-periodic oscillation within seemingly fixed bounds (at a given scale) is the footprint of chaos, especially when the oscillations are bipolar, such that the average is a rare event. Most climate data looks this way.

      • David

        How can you say something that is clearly predictive is chaotic?

        http://bit.ly/ocY95R

      • Girma, how is this “predictive”? Are you claiming this trend will continue? What do you mean? The past is always predictable in hindsight.


      • In the above animation graph, don’t the pink and blue balls meet every 30 years since 1880?

        What does the above fact tell you about climate?

        My interpretation of the global mean temperature data is that there has not been any shift in the climate since 1880. Until the next shift occurs, it reasonable to assume the last 160 years pattern will continue.

      • Shift is a relative term and there are several in the data, as Douglas notes. But chaos is not about shifts, it is about oscillations. I see no reason to believe the warming trend will continue, but this is beside the point. Your diagram proves nothing, it is just data. What we need is science.

      • Girma, let me put it another way. The trend you have blocked out (ignoring the fact that this data is suspect) may simply be the system emerging from the LIA. But the MWP-LIA-present may simply be a single chaotic oscillation, the latest of many on the millennial scale.

        Even the ice ages and interglacials may be nothing more than chaotic oscillations. It may all be due simply to constant solar input plus feedbacks. The math says this is possible, but it seems that nobody is willing to look this way.

        A funny thing happened to the IPCC TAR. When it first came out online, there were a series of graphics showing the primary feedback loops. The caption of each graphic said, quite correctly, that such a system was capable of self induced oscillation. I could not find these graphics referenced in the TAR text, and they eventually disappeared. Go figure. Somebody was trying to get the message out.

    • Girma,

      A planet’s motion around the sun is chaotic, including earth. The issue is one of predictive time horizon. A simple two body system is predictable for all time, but once a third body is added the system is deterministically chaotic. There is no real controversy about this fact. For the solar system the prediction horizon is millions of years. Alternatively, a forced pendulum may have a prediction horizon of only a few seconds.

  29. There’s lots of good info in this thread. I hope the discussions will continue and hope especially that my attempts at contributions don’t shut it down.

    The following are a few comments that are based on my understanding of temporal chaotic response as exhibited by numerical solutions of small systems of autonomous ODEs having constant parameters. The original Lorenz system of ODEs from 1963 is very likely the best known example. There are several others, and my comments apply to those systems as well.

    I think the 1963 system, and others developed by Lorenz, have been described to be models of the mathematical models of the Earth’s climate systems. They are simplifications, and generally extreme simplifications, of already idealized mathematical models of the Earth’s climate systems. The original Lorenz system does not correspond to any actual thermo-hydrodynamic system.

    The primary characteristic of chaotic response from these ODEs is the bounded aperiodic temporal distributions for all the dependent variables. The response is bounded because there is an extremely delicate balance between the energy input into the systems and the dissipation of that energy internal to the systems. The original Lorenz system, for example, does not exhibit chaotic response for all combinations of the numerical values of the parameters. Some combinations will attain a steady equilibrium state with no variations in time for the dependent variables. Other combinations will lead to divergence of the numerical results. An important characteristic of temporal chaotic response is that steady equilibrium states are not ever attained.

    As noted above in this thread, chaos is basically a mathematical property. And more nearly complete, in my opinion, a numerical mathematical property. One can not know a prior if a system of ODEs will exhibit chaotic response: numerical solutions, and analysis of these solutions, is the only procedure to determine chaotic response.

    While a kind-of convergence can be attained in the phase-space trajectory of the dependent variables, the numerical solutions of the ODEs is known to be a function of the form of (1) discrete approximation applied, (2) the discrete step size used, (3) the order of the discrete approximations, and (4) the number of digits used to represent the numerical values in a computer. To accurately numerically integrate the ODEs out to long times requires both very high order discrete approximations plus very large numbers of digitals.

    What I have not yet seen is a mapping of (1) the continuous equations used in NWP and GCMs to the mathematical requirements for chaotic response, or (2) a rigorous mapping of the physical system to one that exhibits chaotic response. The latter is very likely not possible, but noting that the physical system being ‘non-linear’ and ‘complex’, in undefined ways, is nothing more than arm waving. The ODEs that are the foundation of temporal chaotic response, while being non-linear, are very simple: not complex. Some are in fact extreme simplifications of ‘complex’ systems. The universe of real-world applications of mathematical models is dominated by non-linearity and most might be designated to be complex.

    And more importantly, I have not yet seen the calculated numerical values from NWP or GCMs shown to be solely the accurate results of the discrete approximations to the continuous equations and not some kind of artifact that simply and roughly ‘looks like’ chaotic response. There are examples of applications of unstable numerical methods to equations that cannot exhibit chaotic response in fact giving chaotic response. Has this possibility been eliminated in the case of GCM applications? None of the requirements for accurate integration of the model equations as required for temporal ODEs is even remotely approached in the case of NWP and GCMs.

    Corrections for incorrectos will be appreciated.

    • Dan, what is an ODE?

      • ODE = Ordinary Differential Equation

      • Many thanks gents. I think I agree with Dan. However it is important that if a simple component of a system is chaotic, then nothing added will change that, so far as I know.

      • On reflection, I agree with Dan to the extent that he outlines part of the needed research program. But at one point he seems to be claiming that chaos may be merely an artifact of numerical solutions to ODEs. If so then the rejoinder is that ODEs are themselves artifacts, because nature is nowhere continuous. Calculus itself is thus an approximation technique. I am quite sure that chaos is physically real, probably even pervasive.

        But I may have misunderstood Dan’s points. They are far from clear to me, albeit quite interesting. Language is like that.

      • David,

        What I think Dan is saying is that Chaos “can” be an artifact, which I totally agree with. I try to follow the energy to avoid the artifacts. Like when you have a normal distribution of waves, twice the energy anomalies is a good indication of what is real and not an artifact. More than twice is more likely an artifact or at least an indication that something is up.

        Like for if a doubling of CO2 does cause 3.7Wm-2 of forcing, twice that value at a surface at 288K would produce 1.34K of temperature increase. That is not an artifact that is just a fact.

        Co2 doesn’t have to be felt at the surface. if the forcing is felt at 600mb, then twice that impact at that temperature could produce greater temperature change because of the S-B relationship.

        For the rogue (or rouge :) ) wave video above, a perfect rogue would be twice the peak to peak value when waves synchronize. To me, that is the difference between deterministic and not, the two step boogie :)

    • Thanks Dan, I think you provided an answer to my question above.

    • Dan,

      Interesting comments. To me, just about anything that requires statistics can produce whatever you are looking for, the physical connections need to be made. This author, http://cdsweb.cern.ch/search?ln=en&p=Selvam%2C+A+M&jrec=1&f=author has quite a few nonlinear dynamics papers on climate. I have been messing around looking at various physical boundary layers and estimating time constants and nonlinear feed backs. I could take a year or two just to set the problem up. A lot more of what I have seen jives better with self organizing criticality than plain vanilla global warming theory. Even CO2 forcing appears to just shift responses not dominate.

      it really would be nice to have the temperature data divided into a Bucky ball to get a better handle on things. Global averages are not what it will take IMO.

  30. What the heck does “low order” mean?

    a) Disorderly
    b) Less than (i.e. x 1)
    c) Order of the lower court

    I really don’t know. Anybody?

    • Crap… b) Less than (i.e. x is less than x squared)… I think my comment was “interpreted” by the webware.

  31. Dr. Douglass,

    Your paper misses a shift in 1993/4 :) Sorry, but I think I just found the 94 squiggle, http://policlimate.com/tropical/global_running_ace.jpg
    The Ace peaks and troughs would make sense as precursors.

  32. Tomas Milanovic

    I wouldn’t like to spoil the party but a word of caution is imperatively needed for those who take the Tsonis and Douglass notion of “metrics” literally.
    I have already written here a rather long and comprehensive analysis of the Tsonis paper so will not come back to that.

    The word of warning is related to Paul Vaughn’s comment about the strict temporality of the data considered in these papers.

    Historically a seminal paper about chaos in climatic systems is Fraedrich 1986 : http://www.mi.unihamburg.de/fileadmin/files/forschung/theomet/docs/pdf/frae86.pdf
    The paper uses rigorous mathematical procedures based on Taken’s embedding theorem. Both the theory and the application are well established and do not stand to discussion.
    The conclusion is that the time series considered (data from Berlin) is a low dimensional chaotic system with an attractor whose dimension is estimated.

    So where is the problem?
    Well I have been living in Berlin too and know as well as Fraedrichs does that the weather and climate in Berlin is strongly influenced by weather and climate in Danmark because most pressure waves travel preferentially from Danmark to Berlin. But he has no data about Danmark in his analysis which implicitely postulates that the time series in Berlin is independent of any other time series at any other place.
    How does he solve this difficulty?
    Well in a single esoterical phrase which a casual reader would simply fly over without really reading and understanding it.
    He writes at the beginning of §2 :“The dynamics of the weather and climate system is simulated by partial differential equations describing the underlying physical processes. These equations can be transformed in a set of n time dependent ordinary differential equations, IF (!) the space variability is expanded to a set of n orthogonal functions.”
    Even if it looks innocently, this is the one paramount hypothesis on which the whole paper relies.
    This hypothesis is trivially wrong in the general case (the Hilbert space of the climatic fields is infinite dimensional). It should then be proven that this is an acceptable approximation for the considered system. We are very far from being able to bring such a proof and I am pretty convinced that it is actually wrong.
    To finish with this paper, even if one accepts the hypothesis, the use of Taken’s embedding theorem needs a very large number of data that we have not. That’s why it would not really be sure whether the results are not just artefacts of a too short time series.

    In a sense Swanson&Tsonis are doing the same thing with another justification. Like Fraedrichs they also need a finite dimensional phase space.
    But instead of taking physical variables (temperatures, pressures etc), they take 4 indexes.
    By definition their phase space is now 4 dimensional and they look at trajectories of points with those 4 coordinates which are, again by definition, only functions of time. What does such a 4 dimensional point mean physically? Well nothing. It is implied that it somehow represents the dynamics of the whole system but this implicite hypothesis fails for the same reason as Fraedrich.
    The space variability can’t be expressed by a finite n of orthogonal functions and certainly not by a very low number (for Tsonis n=4!).

    So we are back to what is really the most fundamental questions by which should begin every dynamical study of this system.
    1) What is the phase space and its dimension?
    2) What is the metrics?
    What is strange for me is that we actually know the answers on those questions.
    1) The phase space is the infinite dimensional Hilbert space of climatic fields (some vectorial, some scalar).
    2) The metrics is the Hilbert scalar product [f(x),g(x)]= ∫ f(x).g(x).dx
    Why is this fundamental?
    Because as the system is chaotic, if one wants to find attractors and I remind that an attractor is a subset of the phase space one needs to be able to say whether one point of the phase space is near or far from another one.
    What does it mean that an observed temperature field T(x,y,z) is near to a point of the attractor Tattractor(x,y,z)?
    It means that ∫ [T(x,y,z)-Tattractor(x,y,z)]²dV smaller than epsilon.

    This shows clearly among others that averages are not and cannot be a correct metric because that would be even worse than Tsonis – the phase space would be only ONE dimensional.
    Btw in that case the system couldn’t be chaotic because there is a theorem that any dynamical system can only be chaotic if the phase space dimension is more than 3.

    I realize that the word of warning became largely more than 1 word :)
    But the gist of it is that while all those approaches(Fraedrichs,Tsonis,Douglas) are quite rigorous , based on established temporal chaos theory and proven methematical theorems, they all rely on a very heavy assumption which is the finite (sometimes very low) dimensionality of the phase space.
    As this assumption is likely wrong, the results can’t be extrapolated to the dynamics of the whole system and even their validity in a restricted, finite dimensional, part of the phase space may be just an artefact.

    • “This shows clearly among others that averages are not and cannot be a correct metric because that would be even worse than Tsonis – the phase space would be only ONE dimensional.”

      Without a doubt averages are not the best approach. Most of the information is lost in the smoothing. At some point though, averages are all you have got, so they can give you information on what you need and where. Comparing global averages with regional averages with Bucky sphere averages allows you to hone in on the physical relationships. Then the analysis can move forward.

      And I still think the Earth is non-ergodic :)

      • Please be careful to not misunderstand. It would be grossly incorrect to interpret this as suggesting integrals & averages have no utility & no meaning in specific highly informative contexts.

        Judicious diagnostic interpretation requires both great awareness & great care.

        Math is an infinitely big abstract world. If one opts to spend one’s time exploring every abstract branch of math rather than staying faithful to nature as observed, one will easily consume many lifetimes.

        Under the influence of abstract seduction, sober human judgement is rare. If a new math or stats idea or method is “cute”, it will gain epidemic-scale traction & application even if it hinges on patently untenable assumptions.

        For example, human attraction to the seductive notion of temporal chaos has completely derailed this thread, which could have gone somewhere a whole lot more interesting & worthwhile.

        Great patience is clearly required for discussion participants who are serious about carefully advancing the discussion in a manner that is consistent with nature as observed. Please make an effort to notice assumptions underpinning arguments, particularly patently untenable ones.

        Study of climate is multidisciplinary and we are very far from common understanding at this point.

        – –
        I hope to find time to offer some cautionary notes on Douglass’ subharmonic groupings sometime over the next few days…

        Regards.

      • Paul, That was exactly the point I was trying to make. Making the best use of the data you have available. The Douglass paper points out shifts which generally agree with other methods. The 1993/94 precursor I call it, is identical to solar precusors, which is an indication that the solar and internal climate system are similar per A.M.Selvam. So there would be a solar relationship, because of the commonality of the systems. When the two systems are in sequence, there would be amplification, perfect synchronization would be double impact, like a rogue wave.

        I know this is an over simplification, but twice expectations is an indication of synchronization, the question is which systems are synchronizing and how well?

      • Hi Dallas,

        The coherence is in the **rate of change** due to differential transmission. Ask any mechanic if the drive wheels rotate at the same rate as the crank shaft.

        sun = crank shaft

        equator-pole temperature gradients
        = differential transmissions

        westerlies = drive wheels

        Until the general public secures a better handle on interannual geometry, it’s best to look at interannual inter-index relations as a single unit of complex correlation (as in complex numbers, due to the spatial dimension). The set of indices as a whole is globally constrained, but local seasonality, geometry, & circulation differ.

        I’ll share more details at a later date when the pool of online climate discussion participants with a handle on EOP is larger. (There’s no point squandering more effort on an audience that’s unreceptive due to an innocent lack of preparation.)

        Best Regards.

      • Paul,

        I am not dismissing anything you say out of hand. You mention the westerlies, with increased conductivity at the ocean/atmosphere surface boundary there can be more heat transferred. There is quite a bit to the problem. I am only finding parts of interests for those looking for mechanisms to go with theories.

      • There are many roles to be played in this multidisciplinary effort that involves many thousand contributors. My role is to help people stop misinterpreting stats and looking at the wrong metrics & pattern markers, so they can start doing better than theorizing about physical mechanisms abstractly – i.e. without due respect for nature as it actually exists, as recorded in observational multivariate geophysical data. Regards.

      • Well shucks Paul, I thought I was on to something :( I guess I could start over with the assumption that man has no impact on nature.

      • Must be some misunderstanding Dallas. I was not criticizing you. I appreciate your appreciation for nature. Best Regards.

    • Danka Tomas, I think you answered my question above, whereas:

      Per Douglass:
      “…the global climate system is chaotic of low order…”

      Per you:
      “…(data from Berlin) is a low dimensional chaotic system …”

      I wasn’t quite sure what they meant by that. Perhaps it’s obvious to one who has been chaoticly educated :)

  33. Tomas Milanovic

    I also think that the system is not ergodic because spatio temporal chaos generally isn’t.
    The comment about averages looks to me quite like the famous comment that if one has a hammer, every problem looks like a nail.
    And I only remark that I see no nails in the problem :)
    The difference between regional averages and global average is just a difference between n= 1 and n = something.
    For an infinite dimensional phase space it is an irrelevant distinction – both give wrong results.

  34. I have been asked to expand on my views of the numerical mathematics aspects of chaotic response.

    Because there are no analytical solutions available for any equation system that exhibits chaotic response, the numerical aspects are a very important focus. Critically important in my opinion. And, attention to this aspect is totally ignored when NWP and GCMs are the subject.

    Consider the large amount of work that has gone into getting a handle on accurate numerical integration of the original Lorenz ODE system. It was only in the last couple of years that the requirements for achieving step-size independent solutions has been determined. As I mentioned in my comment, in ( X, Y, Z ) space the trajectory seems to approach a more-or-less converged visual picture, but the time values of the dependent variables clearly have not. One of the books has plotted this for decreasing step sizes, but it would take a while for me to find that book and section.

    I think that even the famous 3- ( or n- ) body equations have been used to demonstrate that step-size independent solutions can be achieved for longer times as the number of significant digits is increased. Special computational engines have been built for application to this problem.

    The ODE systems that exhibit chaotic response, so far as I am aware, are simple, pure, autonomous systems having constant coefficients. This is an exceedingly simple numerical problem, yet consider the numerical difficulties that have been uncovered. Each solution method, and each step size, represents a different mapping of the continuous equations and calculated values of the dependent variables. The effects of the methods and step sizes are of the same importance as the dependence on small changes in ICs.

    Here’s some recent information on this problem:

    Teixeira J, Reynolds CA, Judd K (2007), Time step sensitivity of nonlinear atmospheric models: numerical convergence, truncation error growth, and ensemble design. Journal of the Atmospheric Sciences, 64:175–18. DOI: 10.1175/JAS3824.1.

    The abstract is here.

    There were several discussions of this paper and resolution of an issue seems to be summarized in this paper, Long-Time Computability of the Lorenz System.

    Now, when we get to PDEs the numerical issues are even more critical. As I mentioned, there are examples of calculated chaotic response from simple PDEs, PDEs that cannot exhibit chaotic response. A single equation in two independent variables can be used to demonstrate this possibility. Again, a pure simple PDE having constant coefficients. See Cloutman and H. Yee, among others. I have a short bibliography: many URL links might be broken there. I think you’ll agree that under these conditions one should dig deeply into the numerical-methods aspects of the problem. The usual goal is to demonstrate independence of the calculated solutions from all aspects of the discrete approximations. Such an investigation would lead to determination that the apparent visual chaotic response is false and solely an artifact of the numerical methods.

    Now we come to the GCMs, and other models of real-world problems. These models are not pure ODEs or PDEs, they are far far from simple systems, or even simple equations. They are instead a complicated system of complex equations. They are a mixture of PDEs, ODEs, algebraic equations, plus a very large number of instantaneous switches comparing calculated quantities with critical, or threshold values, and a multitude of other confounding issues. And these characteristics refer to only the continuous models. I think it’s important to acknowledge that none of the continuous equations correspond rigorously to any that might have been investigated relative to chaotic response.

    I think that even at the continuous-equation stage that there are un-asked, and thus unanswered, questions. For example, have any equation systems with these characteristics been shown to obtain the properties necessary for chaotic response? More specifically, has the equation system used in any GCM been shown to meet theoretical requirements for chaotic response?

    And finally we come to the numerical solution methods for the NWP and GCM models. At this stage there are an almost un-countable number of issues relative to calculated chaotic response. It is well known that the discrete approximations for the fluid flow includes terms that are not contained in the continuous equations. It is also already well known that the GCMs are not employed in applications with discrete approximations that are in the asymptotic range of the numerical solution methods. Probably never will be. The algebraic switches, and sub-grid parameterizations which are functions of the discrete approximations, make this a difficult goal even if the spatial and temporal ranges of interest were not so enormous.

    So far as I am aware, there have been no investigations that attempt to demonstrate that the calculated numerical values from GCMs are in fact an accurate representation of chaotic response, in contrast to being solely due to numerical-solution methods.

    In summary. I think that there are important issue relative to chaotic response that have not yet been addressed at the continuous-equation level for the GCMs. I know that the critically important numerical-methods issues have not yet been addressed: they have been completely ignored. Sometimes it seems to me that concepts from purely temporal chaotic behavior associated with simple ODEs have been assumed by osmosis into the GCM statio-temporal world of PDEs plus ODEs plus algebraic parameterizations plus switches plus numerical methods plus unresolved solutions of the discrete approximations.

    Corrections for all incorrectos will be appreciated.

    • Any climate model that can’t reproduce EOP (Earth Orientation Parameters) can be dismissed.

    • Dan, once again, I think you have outlined an important climate research program, far more important than, say, the carbon cycle, which is today’s biggest program. But I am still having a problem with one point. I have no problem with the notion that the form of the chaos one gets is extremely dependent on the method and step size one chooses. On reflection that makes a lot of sense.

      But I am troubled by the apparent claim that one can get a chaotic response simply by using a numerical method to describe a system, and that chaos is somehow false thereby. You seem to be taking the ODE’s as the reality, but nature is nowhere continuous. I am inclined to think that if you can get a chaotic response then that is a fact about the system, not just a fact about the method.

      Moreover, if a simple aspect of a system is chaotic, atmospheric turbulence for example, I do not see how adding complexity can make that go away. But I am certainly prepared to be wrong, if one can demonstrate how this happens.

      • David said, “Moreover, if a simple aspect of a system is chaotic, atmospheric turbulence for example, I do not see how adding complexity can make that go away. But I am certainly prepared to be wrong, if one can demonstrate how this happens.”

        I totally agree. All systems, including the computers used for modeling, are chaotic to some degree. Finding the right level of complexity for the “systems” being analyzed is the trick. Multiple approaches that tend to converge make more sense to me.

      • CD: Speaking as an editor, there is nothing in your article related to your title, not that I can see. You do not even mention chaos, much less defining a chaotic system.

      • The question mark in the title is because if there is a physical explanation, is it chaotic? Selvam’s work relates to a lot of systems, even the stock market. So I am beginning to think it is our perception that is chaotic not nature. This Global Warming debate tends reinforce that thought. :)

      • The turbulence is constrained globally. (Earth’s shells wrap on themselves.) Tip: Stop ignoring EOP.

      • Paul,
        I am not ignoring EOP. http://redneckphysics.blogspot.com/2011/11/thermodynamic-layer-convergence-and.html I believe that is an example of wrapping on themselves. Actually, the Antarctic seems to have a number of convergences which require a more sophisticated approach.

        I am just starting from a fluid dynamics point and working from there. Different strokes for different folks :)

      • The fluid loops (including the Southern Ocean one) vary in speed with vertical distance (positive or negative) from the surface.

        Air moves fast.
        10m: http://i44.tinypic.com/28rgyzo.png
        (Note: These animations work in Mozilla Firefox, but not Internet Explorer.)
        850hPa:

        Wind-driven water: slower.

        Something floating in the water?
        in between – varies with buoyancy & sailing profile.

        Way up above the surface friction layer:
        whipping along – e.g. 200hPa:

        550K: http://i56.tinypic.com/14t0kns.png

        Recognizing opportunities for generalization simplifies conception.

        What do speeds of different layers have in common?

        CHANGES in terrestrial equator-pole temperature differentials are coherent with CHANGES in solar cycle length – i.e. equator-pole-temperature-gradient-driven west-east fluid acceleration on Earth is proportional to pole-equator acceleration on the Sun.

        2m temperature: http://i55.tinypic.com/dr75s7.png
        column integrated water vapor flux with their convergence: http://i51.tinypic.com/126fc77.png

        The solar cycle’s the crank shaft. Terrestrial equator-pole temperature gradients are the differential transmissions. The westerlies are the drive wheels.

        Very Important:
        Crank shafts are NOT phase-locked to drive wheels. Coherence is in the RATE OF CHANGE due to DIFFERENTIAL transmission.

        Perhaps Douglass & Milanovic will spend some time carefully considering global constraints: http://wattsupwiththat.files.wordpress.com/2011/10/vaughn-sun-earth-moon-harmonies-beats-biases.pdf

        When the time is right, I’ll bring geomagnetic aa index into the picture…

        I support your attention to fluids Dallas.

        Best Regards.

      • Paul, When you are ready for the geomagnetic field, Antarctic thermal flux measurements appear to have some magnetic noise. I am not sure how to approach the problem yet, but around 65 to 70 Wm-2 low energy thermal photons appear to be interacting with non thermal photons or the magnetic field.

        It may be science fantasy, but this indicates a magnetic influence on the performance of the tropopause that strengthens as the energy of the photons decreases. I had expected something like this at 100K or less as a result of the form of the S-B equation, ~185K was a bit of a surprise.

        Even if it isn’t real, it could make for a good sci-fi plot line :)

      • Paul, I have no idea what you are talking about, much less how it relates to what I am talking about. Sorry, but if you can’t relate what you are saying specifically to what I have said then you are wasting your breath, and my time. Try starting with something I have said.

      • Must be some misunderstanding caused by the dreadful comment-tree format.

  35. Towsley A.; Pakianathan, J.; & Douglass, D.H. (2011). Correlation angles and inner products: application to a problem from physics. International Scholarly Research Network (ISRN) Applied Mathematics. doi:10.5402/2011/323864.

    http://www.pas.rochester.edu/~douglass/papers/Towsley_Pak_Douglass_published_.pdf

    Key hinge worth noting:

    “We must put one mild hypothesis upon V in order for it to have the desired properties. The hypothesis is that the vectors must be “probabilistically independent.” […] It should be noted that this independence is in no way related to the linear independence of the random variables.”

    Fundamentally important omission in the paper: mention of the 84-month window.

    Spatiotemporal evolution of sign isn’t irrelevant. I would suggest using a multivariate 20-dimensional hypercomplex correlation (integral, 0th, 1st, & 2nd derivatives for radius, longitude, latitude, time, & scale across a range of geophysical variables). (I have developed a prototype I call multiscale complex correlation.)

    Important: Scale canNOT be omitted.

    Douglass alluded to this in the earlier paper:

    Douglass, D.H. (2010). Topology of Earth’s climate indices and phase-locked states. Physics Letters A 374, 4164-4168. doi:10.1016/j.physleta.2010.08.025.

    http://www.pas.rochester.edu/~douglass/papers/Topology_PLA20011.pdf

    “The 84-month window in the computation of the correlation coefficients should also be varied”

    This is a serious understatement.

    Spatiotemporal aggregation criteria FUNDAMENTALLY affect statistical summaries (going so far as REVERSING phase, for example). Scale, shape, & orientation of windows not only deserve but additionally DEMAND full, unblindered attention.

    The simplest first step is to vary the timescale (easy to do). (This is routine in some branches of advanced physical geography.) Multivariate coherence can be plotted with color (ice-blue, blue, black, red, yellow-flame) on a z-axis with the y-axis as timescale and the x-axis as time (i.e. a color-contour plot). See some bivariate examples here:
    http://wattsupwiththat.files.wordpress.com/2011/10/vaughn-sun-earth-moon-harmonies-beats-biases.pdf (Watts always spells my name wrong — no offense taken.)

    It’s valuable to have experts from other fields helping with the development of metrics. Towsley, Pakianathan, & Douglass: Thanks for joining the discussion.

    It will take LOTS of back-&-forth communication to raise mutual awareness to a level where cross-disciplinary awareness converges. My background spans the skeletons of 7 disciplines and is supported by plenty of meat in specific areas, so I can efficiently play the role of hybrid “go between” messenger. Based on what I know from strong influences in other fields:

    Climate index developers NEED to study EOP (Earth Orientation Parameters) [ http://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html ] WITHOUT any more delay. Due to the persistent delays (I acknowledge & respect that people are legitimately busy), the whole conversation is stuck at a major bottleneck. When the timing is right, we’ll cooperatively break the logjam of misinterpretation on the river of Statistical Paradox.

    Comments regarding Douglass’ subharmonic groupings later (an opportunity to tell part of “The Tale of SAM & SOI”)…

    Best Regards.

  36. Thanks to Dr. Douglass for this interesting post (and to our host for posting it).

    It’s a lot to go through, but the concept that ENSO cycles are possibly of Solar origin is very interesting, and this may be a missing link in the puzzle.

    Max

  37. Christopher Game

    A small suggestion for Dr Douglass. A symmetric season-effect-removing filter that is centred on nominal years rather than on nominal half-years has weights 1/2,1,1,1,1,1,1,1,1,1,1,1,1/2. It must be very nearly the same as a 12-month evenly weighted box-car filter 1,1,1,1,1,1,1,1,1,1,1,1. The removed (sifted-out) quantity is seasonal. The residual (sifted-in) quantity is non-seasonal. Christopher Game

    • Your suggestion is well-placed; you neglected to mention, however, that the weights need to be divided by 12. The filter thus obtained is a convolution of the “boxcar” weights of the standard 12-month moving average with the binomial weights 1/2, 1/2. This reduces the sidelobes of the amplitude response of the MA, while retaining the zer0-amplitude response for 12-month components and all their harmonics.

      Decades of experience indicates that this refinement is nowhere near as critical in obtaining truly non-seasonal series as the act of filtering itself. It’s the customary practice of subtracting estimated climatic “norms” to produce “anomalies” that leaves substantial seasonal components in the record, which act as a confounding factors in refined signal analysis. Sadly, climate science is largely unaware of such analytic basics.

  38. david h. douglass

    I appreciate the many comments and would like to respond and to also clarify some of the issues.

    1. General comment
    My approach has been to focus on understanding the observational data first. What good is a theory based upon data that may be what you thought it was? One of the most important discoveries that I made was that the commonly used climate indices which were constructed to be “season free” were NOT “season free” Douglass [2011a]. I showed that the method of construction actually made the data set worse and, in fact, would never work. I also showed that a running box filter could be used to produce a “season free” index. The analysis in all my papers uses indices produced by this box filter.

    2. Models
    Although the results in my paperst come from analysis of data and not from models I am not unaware of them. In regard to the comment by David Woljick. I know the Jin et al. “Devils Staircase” paper very well. It is reference 25 in Douglass [2011a]. This paper and three other papers (references 26, 27 and 28), published nearly simultaneous proposed very interesting models of El Nino showing chaos and and phase-locking phenomena similar to results from data that I have discussed in this post. However, these were models only. The data available to these investigators were not good enough to use. They stated that the data sets were not long enough.

    What I have shown is that the data sets available to them were actually long enough but were “contaminated” with the seasonal effect. They should revisit their model calculation using the indices I have defined keeping the results that I have found to constrain the many model parameters.

    David Douglass