Guest Post by Willis Eschenbach
After I’d published my previous post on the Hurst Exponent entitled A Way To Calculate Effective N, I got an email from Dan Hughes which contained a most interesting idea. He thought that it would be productive to compare the Hurst analysis of the records of weather phenomena such as temperatures and the like, with the Hurst analysis of the corresponding climate model outputs. He proposed that we take a look at the question and discuss and share our findings. Hey, what’s not to like?
Along the way, as such things happen, the topic of our discussion and investigation turned to a larger and more interesting question—`which of the various natural datasets (temperatures, rainfall, eruptions, pressures, etc) and/or the corresponding global climate model outputs are chaotic?
I must, of course, start with the obvious question … what is meant by “chaotic”? A chaotic system is a system wherein nearby initial states either converge or diverge exponentially. An example is the ocean surface. If you drop two sealed empty bottles overboard in mid-ocean, one on each side of a boat, they will drift apart over time. This separation will be slow at first, and then faster and faster as the two bottles encounter different winds and currents in different areas.
Whether a dataset is chaotic or not is generally assessed by looking at the Lyapunov exponent. This is a measure of the “stretching factor”. In our ocean example above, the stretching factor measures how fast the two points are moving apart over time. In a chaotic dataset, the stretching factor generally increases or decreases with time. In non-chaotic datasets, on the other hand, the stretching factor doesn’t vary with time. Figure 1 shows the “Lyapunov curves” of the evolution of the stretching factor with time for a variety of natural and calculated datasets.
Figure 1. Lyapunov curves for a variety of datasets. All datasets have been detrended and standardized before analysis.
Now, I’ve graphed four types of datasets above, indicated by the four colors. The first type, shown in red and mostly obscured by the blue lines, shows four different varieties of random numbers—normal, uniform, poisson, and high Hurst exponent fractional Gaussian random numbers. Basically the Lyapunov curves of the random number datasets are all plotting right on top of each other. Starting from time = 0, they climb rapidly to their maximum value and then just stay there. As we would expect from random data, there’s no trend in the stretching factor over time.
The next group, in blue, shows the Lyapunov curves for a half-dozen climate-related datasets, viz:
• HadCRUT4 Monthly Mean Surface Air Temperatures 1850-2015
• Annual Nilometer Minimum River Height 622-1284
• Stockholm Monthly Tides 1801-2001
• Central England Temperature Record Daily Maximum Temperatures 1878-2015
• Armagh Ireland Daily Mean Temperatures 1865-2001
• Annual Average Nile Flow, Cubic Metres/Sec 1870-1944
As you can see, in terms of the Lyapunov analysis, all six of these climate-related datasets (blue lines) are indistinguishable from the four random datasets (red lines), which in turn are indistinguishable from each other. None of them show any trace of chaotic behavior.
Another group of datasets, those at the bottom in gold colors, are quite different from the random and the observational datasets. They are a variety of chaotic datasets. Note that they all share a common factor—as mentioned above, over time the rate of separation (as measured by the “stretching factor”) increases. The rate of separation doesn’t just go to a certain high point and stay there like the random or climate-related datasets did. The rate of separation in chaotic datasets continues to rise over time.
Finally, there are a couple of other datasets in purple. These show observations of phenomena that are usually thought of as “forcings”. One is changes in the solar activity, with daily sunspots as a proxy for the activity, from 1880 to 2015. The other is annual aerosol optical depth from 800 to 2000, which is generally a function of volcanic action and is calculated from ice core data. Curiously, these two datasets plot somewhere in between the random observations at the top, and the chaotic datasets at the bottom. In addition, both of them show significant variation in stretching factor over time. The sunspots have a slight but significant increase. The aerosol optical depth goes down and then looks like it’s starting back up. So it seems that these two datasets are weakly chaotic.
Now, these results were a great surprise to me. I’ve long believed, without ever checking it, that the climate was chaotic … however, this analysis shows that at least those six observational datasets that I analyzed above are not chaotic in the slightest. Hey, what do I know … I was born yesterday.
Are there any climate datasets which are chaotic, even weakly chaotic? I think so. It appears that tropical ocean temperatures are weakly chaotic … but that’s a question for the next post, which will look into Dan Hughes’s idea regarding the use of the Hurst analysis to distinguish between chaotic and non-chaotic datasets.
Regards to everyone,
w.
MY USUAL REQUEST: If you disagree with someone, please quote the exact words you disagree with. That way, we can all understand the exact nature of your objection.
CODE: To calculate the Lyapunov exponent I’ve used the lyap_k function from the R package tseriesChaos. Here are the functions I used to make Figure 1:
lineout=function(col="black",testdata=testdata,line0=F,lwd=3,s=110){
output <-lyap_k(scale(dtrendlin(testdata)), m=3, d=2, s=s, t=40, ref=100, k=4, eps=4)
lines(output,main="",cex.main=.95,col=col,lwd=lwd)
invisible(output)
}
plotout=function(col="black",testdata=testdata,line0=T,lwd=3,s=110){
output <-lyap_k(scale(dtrendlin(testdata)), m=3, d=2, s=s, t=40, ref=100, k=4, eps=4)
plot(output,main="",cex.main=.95,new=F,ylim=c(-6,1),col=col,lwd=lwd,
ylab="Log(Stretching Factor)")
if (line0) abline(h=c(0,1),lty="dotted")
invisible(output)
}
dtrendlin=function(timeseries,doplot=FALSE){
outseries=timeseries
thelm=lm(timeseries~time(timeseries))
thegood=which(is.finite(timeseries))
outseries[thegood]=outseries[thegood]-thelm$fit
if (doplot){
plot(outseries)
}
outseries
}
Natural laws give limitations of variations and even within those limitations. It is easy to predict by carefully observing and calculating.
Beyond the difficulty in merely agreeing upon a definition of “chaos” I believe you are making a critical mistake by trying to find chaos in data sets of single variables (like daily temperature at a single location or lowest annual Nile-o-Meter reading). I would expect a chaotic system to produce data sets exactly as you found–highly random particularly when time frame of the data sets is either very short relative to what appear to be profound chaotic events in the system (like ice ages) or very long relative to the sublime chaos over a very short period of time (say a single day or night of temperature at a specific location sampled by the millisecond).
The inclusion of the Nile-o-Meter in your piece is a curious one. While I have no idea if there is anything like a continuing record, there is still at least one functioning ancient Nile-O-Meter in Egypt at the temple of Dendera. (I saw it this year.) Yes, I am fully aware that the construction and operation of dams on the Nile have profoundly affected the natural fluctuation. Even more profound effects are likely to occur in the near future when the construction of another dam on the Nile (this time in Sudan) is complete. If a reasonably continuous record is available that begins in ancient times and continues to the present day, I suspect that your analysis of the data set would be quite different.
Willis please add a blow-up version of the range -1 to 1. Now there are too many wiggles overlapping.
Summary of definitions of chaos above:
” a system wherein nearby initial states either converge or diverge exponentially.
It is time dependent feedbacks that define chaotic systems.
Systems are chaotic if very small difference in initial conditions yield widely different outcomes.
If the climate wasn’t chaotic, the models would have it projected already,
to be sensible to initial conditions, to have dense periodic orbits and to be topologically mixing.
If Man doesn’t really know what’s going on, call it “Chaotic””
Me, I’m partial to the last one. Possibly the only verifiable initial condition is the big bang?
I’m surprised that any thinking person would assume that the climate is chaotic when it is trivially obvious that it is not- at least for the last 6 -700 million years. During this time the temperature and the oceanic and atmospheric chemistry has stayed within the narrow limits required for carbon based organic life.
We can also e.g. back cast the ephemerides for about 60 million years before the differences in how different computers round off decimal points causes divergence.
There is evidence in the geological record for the 405,000 year eccentricity cycle more than 400 million years ago.
The millennial solar activity cycle which has probably just peaked in 1971 has been identified in the Holocene and as far back as the Miocene.
http://3.bp.blogspot.com/-NuOJUXIC050/U9A9WuN2thI/AAAAAAAAAUg/o2nWMaYMBlY/s1600/KernMioHolo.png
This does not mean, however, that climate can be forecast using reductionist climate models.The modelling approach is inherently of no value for predicting future temperature with any calculable certainty because of the difficulty of specifying the initial conditions of a sufficiently fine grained spatio-temporal grid of a large number of variables with sufficient precision prior to multiple iterations. For a complete discussion of this see
Essex: https://www.youtube.com/watch?v=hvhipLNeda4
Earth’s climate is the result of resonances and beats between various quasi-cyclic processes of varying wavelengths combined with endogenous secular earth processes such as, for example the changing geomagnetic field.. It is not possible to forecast the future unless we have a good understanding of where we are at present time in relation to the current phases of these different interacting natural quasi-periodicities which fall into two main categories.
a) The orbital long wave Milankovitch eccentricity,obliquity and precessional cycles which are modulated by
b) Solar “activity” cycles with possibly multi-millennial, millennial, centennial and decadal time scales.
The convolution of the a and b drivers is mediated through the great oceanic current and atmospheric pressure systems to produce the earth’s climate and weather.
After establishing where we are relative to the long wave periodicities , especially the solar millennial cycle, to help forecast decadal and annual changes, we can then look at where earth is in time relative to the periodicities of the PDO, AMO and NAO indices and based on past patterns make reasonable forecasts for future decadal periods.
For the latest forecasts and complete discussion see
http://climatesense-norpag.blogspot.com/2014/07/climate-forecasting-methods-and-cooling.html
The reference for the Figure is
Fig.6 Kern et al http://www.sciencedirect.com/science/article/pii/S003101821200096X
Sorry -typo – the solar peak is 1991.
http://4.bp.blogspot.com/-WXTGA3eyHeY/U9VG8LpW1FI/AAAAAAAAAVA/2vyuOLUU7kA/s1600/oulu20147.gif
What are the units of the vertical axis in the figures above and what measurements were they derived from?
I read the paper, and as far as I can tell, you won’t be able to tell us the units because there are none. The data is derived from a total of 3 proxies (e.g. bore drill of clay in Austria), and then munged through numerous processes such as arcsin transform. By the time you are done, it’s unrecognizable and the peaks could mean anything as they might just be aliasing or the result of some nonlinear transformation on random noise. I was unable to figure out the final units after all that munging. Please clarify, I might have missed something in the paper.
I do appreciate the use of confidence level bars, but the paper doesn’t say how they were created. Monte Carlo against appropriate spectra of noise is my assumption from your references, but it’s unclear.
I like the analysis approach of looking at spectra and wavelets, but the proxy munging makes me cringe a bit…
Peter
Sorry, I read too fast. I understand the vertical units now (gamma ray CPS from your radiameter instrument). I believe the standard is to also measure for the null hypothesis by running one that is shielded from the experiment but otherwise exposed to the outside world. I didn’t see where that was done. Something that tripped up the cold fusion folks I believe…
Still think you need more proxies and a reproduction…
Peter
I think we can all agree that the Earth’s climate is a complex system with many poorly understood positive and negative feedback loops. However, one can deduce a very important characteristic from its behavior. Dynamic systems are either stable or unstable. Dynamically stable systems have negative feedback loops that return the system to equilibrium (picture a ball inside a globe, it can oscillate but it returns to the bottom). In an unstable system, a change in one variable can cause the system to obtain a new equilibrium point (picture a ball on top of a globe, a small shove and it falls off and obtains a new equilibrium). The theory of man-made global warming relies on the assumption that the earth’s climate is an unstable system. Global warming advocates argue that a 3% change in one minor system parameter (CO2 concentrations) causes the system to achieve a new equilibrium. However, the earth has experienced changes in solar radiation, volcanic eruptions, great meteor strikes, and even significant changes in CO2 concentrations, yet the earth’s average temperature has fluctuated just 18 degrees F over the last 600 hundred million years. The earth’s climate is clearly a stable system and if it were not, the earth would have fried or frozen eons ago and climate change advocates along with the rest of the human race would never exist.
Thanks, Willis. Very interesting essay.
You write:
“Now, these results were a great surprise to me. I’ve long believed, without ever checking it, that the climate was chaotic … however, this analysis shows that at least those six observational datasets that I analyzed above are not chaotic in the slightest.”
W.M. Briggs and Roy Spencer seem to believe otherwise and I agree with them. But I’m looking forward to your developing understanding of this matter.
Willis’s analysis does NOT show “not chaotic in the slightest”. Willis’s analysis shows that he doesn’t know the difference between random and chaotic. He then conflates the two, sees only randomness then concludes that there is no chaos.
This is in direct contradiction to Judith Curry’s work that DOES show a stadium wave effect affecting climate. Judith knows what she’s talking about. Willis does not and he misleads people.
Give it a rest.
Willis,
The standard definition of climate implicitly assumes that climate is not chaotic. Otherwise you could not define climate properties in terms of means and standard deviations. To have any hope of creating a climate model requires that climate not be chaotic, otherwise you could not have any hope of modelling it.
Weather is chaotic. But if a chaotic system is bounded, then the Lyapunov plot must eventually approach a constant value, since the distance between trajectories can not increase forever and stay within the bounds. So if you take a Lyapunov plot for such a system and plot it on a suitable timescale (one that is long compared to the time needed to approach the constant value), it will look like a plot for a random number (rapid rise to a constant). For temperature, the rise takes place over weeks, so it is not surprising that a Lyapunov plot for monthly and globally averaged data looks random. During the initial rise, the initial value is important, for the constant part of the plot, only the boundaries matter. That is what climate modellers mean when they say that weather is an initial value problem but climate is a boundary value problem. If right, that makes the statistical properties of climate predictable (at least in principle) even though the weather is not predictable beyond maybe a week. Lorentz called that predictability of the second kind.
The problem, I think, is that the models focus on the atmosphere; but one of the boundaries for the atmosphere is the surface of the ocean. This following is speculation on my part.As long as that surface stays reasonably constant (or periodic), the climate should be constant. But the ocean has its own internal chaotic dynamics. Those processes occur on a much longer time scale (decades to millennia) than atmospheric chaos (weeks or less) so on sufficiently long time scales the climate is indeed chaotic. But your analysis does not pick that up since the instrumental record is not really long enough. Then when you get to millennia, the ice caps get in the picture and push the chaotic behavior out to even longer times scales.
The modellers say that since they include the ocean in the models, the only boundary that matters is the top of the atmosphere. But that only works if they get the internal dynamics of the ocean right. I think that the failure of the models to get the sort of variability indicated by the Medieval Warm Period or the Little Ice Age (or even the pause) is an indication that the models underestimate the chaotic variability in the oceans.
The climate models on which the entire Catastrophic Global Warming delusion rests are built without regard to the natural 60 and more importantly 1000 year cycles so obvious in the temperature record. The modelers approach is simply a scientific disaster and lacks even average commonsense .It is exactly like taking the temperature trend from say Feb – July and projecting it ahead linearly for 20 years or so. They back tune their models for less than 100 years when the relevant time scale is millennial. This is scientific malfeasance on a grand scale.
The temperature projections of the IPCC – UK Met office models and all the impact studies which derive from them have no solid foundation in empirical science being derived from inherently useless and specifically structurally flawed models. They provide no basis for the discussion of future climate trends and represent an enormous waste of time and money. As a foundation for Governmental climate and energy policy their forecasts are already seen to be grossly in error and are therefore worse than useless.
The Fig below shows the sort of irrational thing the modelers do when they project a cyclic trend in a straight line..
http://4.bp.blogspot.com/–pAcyHk9Mcg/VdzO4SEtHBI/AAAAAAAAAZw/EvF2J1bt5T0/s1600/straightlineproj.jpg
For a better way see
http://climatesense-norpag.blogspot.com/2014/07/climate-forecasting-methods-and-cooling.html
Dr. Page,
I have never seen a convincing demonstration of 60 or 1000 year cycles. Even if there, such cycles account for no more than a small fraction of the variance in the temperature record, so they are no good for forecasting.
I am skeptical of this definition of chaotic (and therefore the conclusions drawn). It fails to account for non-linearity, complexity, and feedback mechanisms, to name just a few factors. I need to learn more about this, but even textbooks on chaos theory and non-linearity avoid defining the term. See: http://mathworld.wolfram.com/Chaos.html
Hi Willis,
A couple of minor comments. First I very recently decided that I just didn’t know enough about dynamics (where nearly all nonlinear dynamical systems are at least potentially chaotic and where, as it has been said, studying nonlinear dynamics should be like a biologist studying non-elephant mammals — the exception rather than the rule). So I’ve bought three fairly serious books on the subject and am working my way through them. Because I like to watch, I’ve set up a directory in which I am building a sequence of projects in octave to exhibit e.g. logistic chaos (which is a bone-simple iterated map with a very clear chaotic regime, perfect Feigenbaum tree, etc) and other chaotic systems. Following notes and comments by Sprott (who has a superb website devoted to chaos and fractals that accompanies his equally excellent book Chaos and Time-Series Analysis) one of the first things I did was to take simple ODEs known to have NON-chaotic solutions and integrate them with a large fixed stepsize using Euler integration, verifying that for a sufficiently large stepsize they develop strange attractors and become chaotic (something I’d pointed out to Nick some time ago, but I wanted to see it for myself).
By simple ODEs, I mean absolutely trivial ODEs that move a point around in a circle, a Van der Pol oscillator, things like that. The point being that even well-behaved systems of ODEs with perfect closed analytic solutions can become chaotic iterated maps if one integrates with too large a stepsize. As Sprott points out, this is likely to be true for many, many systems of differential equations, including linear ones.
I spent today integrating the rigid driven pendulum, a second order nonlinear ODE with a well-known chaotic regime. Or, perhaps I should say, with a near infinity of not-so-well-known chaotic regimes. My purpose there was to investigate just how sensitive results were to stepsize using a good ODE solver with tight control of tolerance — Octave’s built in lsode with absolute and relative tolerances set to 10^{-9} and full use of adaptive stepsize (but with a user-selectable grid that can force a moderately finer granularity). Note that this is a much, much simpler system of only two coupled ODEs (to solve a simple nonlinear second order equation). Since even laptop computers are at this point supercomputers, even using octave instead of C to do the solutions (probably a factor of 10 loss of speed) let me play with millions of timesteps (plus adaptive solution between timesteps) across a transient and out to where the solution becomes either quasi-periodic or at the very least non-repetitive if not chaotic. Much to my interest and surprise I learned that this enormously simple system has multiple chaotic regimes when one varies a single parameter, e.g. the driving torque amplitude, interspersed with regimes of perfectly repetitive cyclic behavior (rarely pure oscillations but some N-fold cyclic oscillation that is apparently stable). The cyclic regimes display nothing like period doubling — one can run the system with a torque amplitude of 0.778 and get one behavior with a clear double cycle and run it for 0.779 and get chaos and run it for 0.780 and get a different periodic cycle. The solutions are equally sensitive to butterfly effect perturbation of initial conditions, the number of timesteps in the integration with fixed time intervals, pretty much everything.
From this I conclude that it is not at all implausible to assert that the fond hope of the GCM modellers that using a stepsize 30 orders of magnitude larger than the Kolmogorov scale will even qualitatively reproduce the distribution of possible future climate trajectories may be unreasonable, as I think has been pointed out before by several people but without quantitative support. I’m hoping to set up a simple open system with some of the features of a “climate” and run it as an iterated map to see if the system develops a chaotic behavior for certain operating regimes. From what I’ve already learned from my reading, it seems very likely that it will.
To get back to your observations above — I wouldn’t take them too seriously as evidence that the climate is or is not chaotic. This is where having some models (like the rigid oscillator) to play with can help a lot. If I just plot the phase angle of the oscillator against time, I can get a dazzling array of possible behaviors well out beyond the usual “transient” expected for the damping rate. In fact, I often run for times of 100,000 or 200,000 ‘units’ (on the scale of the natural frequency of the oscillator) because sometimes after 150,000 timesteps the system “discovers” an apparently stable quasi-period oscillation. Other times it finds one after 50,000, it persists for 30,000, and then turns back into chaotic oscillation. It isn’t even possible to do scaling of this stuff — change one digit in the sixth column in the inputs and you get completely different behavior.
The chaotic parts can easily look just like the timeseries we have for temperature over a fairly renormalizable set of time scales. The problem is that those time scales may not be large enough to display any sort of divergent or phase space covering behavior. It is also difficult to map out a strange attractor with just one variable and time — the “chaos” is probably evident only on a torus in some (probably large) dimensionality so that when you look at global average temperature in particular, you are seeing only a projective view of the torus into a single dimension, nothing much to see and (for a bounded torus) no divergences per se.
So you are looking at the wrong thing, in the wrong dimensionality. My recommendation is — invest in Sprott. It’s worth it. Learn more about dynamical chaotic models up front. I still don’t know enough to feel comfortable with chaos and statistics and dynamical maps, and I’ve spent many a fond hour integrating ODE systems that exhibit collectively nonlinear behavior and catastrophes in an earlier life. We both probably need to study — a lot — before asserting any sort of conclusion based on a simple glance at a short timeseries.
rgb
Humility in the face of actual dynamical systems is always laudable.
Thanks, Robert, always good to hear from you. I learn more from your comments than just about any.
I thought I was being cautious with my conclusions. What I can say is that I used a method that very clearly separated the datasets into three groups—one chaotic, one containing random data and also observations, and one containing a couple of forcing datasets (sunspots and volcanoes).
Not only that, but that analysis agrees completely with a separate analysis I’ve done but not posted yet, using the Hurst exponent to differentiate between chaotic and non-chaotic datasets.
So do I think that this is the final word? By no means. I just find it quite interesting, in that I had expected at least some of the six datasets I investigated to show some sign of chaoticity.
How accurate is the method I used? As you say:
Mmm … you say that I’m looking at “only a projective view of the torus into a single dimension”. Actually, that’s not true. The single dimension is “embedded” into a higher dimensional space. All of those were embedded into a 3-dimensional space. So I do need to look at using larger embedding dimensions. However, the simplest analysis has done well at identifying known chaotic datasets using just an embedding dimension of 3 … hang on …
OK, I just looked at the results using embedding dimensions up to 8 … very little difference in the results.
Also, I just looked at determining the optimum embedding dimension using the method of false nearest neighbors, and the answer is an embedding dimension of either 3 or 4 for all of the datasets, natural and chaotic. By that, I mean that after 3 or 4 dimensions, adding further dimensions only marginally improves the analysis. (Of course, for the random datasets the embedding dimension is meaningless.)
So my question would be … since this method I’m using clearly and correctly identifies the various chaotic datasets, including Lorenz’s dataset which was based on atmospheric dynamics, and it clearly and correctly identifies the random datasets, then why does it identify the natural datasets as not being chaotic?
It may be, as you say, that the datasets are too short. But chaotic is chaotic, whether it is short or not. I just re-ran the analysis using only N=663 data points for each of the chaotic datasets … and the results were basically unchanged.
Again regarding time, that’s one reason why I used different datasets. The longest in time was a 1,200 year annual dataset (AOD) The smallest N was the annual Nilometer data, N=663. I also showed daily data, both average daily data for a small area (CET) and single-station data (Armagh). Both of these had N ≈ 50000. I had thought perhaps one or the other would be chaotic.
Please don’t get me wrong. I’m feeling my way here. All I’m saying is that by the test I showed above, the six natural datasets are not looking chaotic. And I’m well aware that there are more types of chaos in Heaven and Earth, Horatio, than are dreamed of in my philosophy …
In any case, you raise further interesting points here:
Now, that makes perfect sense to me. I learned from Bejan and the constructal law that natural heat engines speed up until they run slightly into the turbulent regime. It seems like what you are talking about amounts to the same thing, that nature is running at the edge of chaos.
What’s new for me in what you’re saying is the idea of the cyclical venturing into the turbulent, chaotic realm and then returning to laminar flow. I like that.
Without conscious consideration, I’d always envisioned that a natural heat engine would run a bit into the turbulent realm, and then somehow just stay there, up against the stops as it were. And that is certainly possible
But in fact, your description of “porpoising” into and out of chaos seems equally likely in any natural system as is the alternate condition of a stable position slightly into the turbulent realm.
Egads … so much to learn, so much to consider. Thanks so much for your contributions, always interesting and thought-provoking.
w.
You got that one right, good buddy.
Consider the humble “random number generator”. There is no such thing, of course, but many of the ones that exist are pulled from chaotic maps. This parentage is not mathematically distinguishable by any simple statistical test, because if it were, the sequence produced would not be random! Or if you prefer, the whole point is to make the sequence look random in the specific sense that it will produce a uniform distribution of p-values in an ensemble of statistical tests against the null hypothesis “this is a perfect random number generator”, where success makes it indistinguishable from a truly random sequence, if such a thing has any meaning in a deterministic Universe, if the Universe is truly deterministic. Lots of ifs in there, some of them pretty fundamentally unresolvable and hence with answers on the edge of religion.
I should also likely have qualified my remarks. It is my opinion that the Earth is a complex system — a system with a very, very high dynamic dimensionality and with “truly chaotic” dynamics visible in abundance throughout. For example, turbulent flow is chaotic. Does it make sense to assert that the climate is not chaotic in some average variable when one can observe chaotic turbulent flow in everything from the whistling of the wind to the crashing of the wave and water in the local dynamics? To put it another way, suppose one takes a limit cycle — any limit cycle, one with a normal or strange attractor — and averages it. Do you expect the average to be chaotic?
This again is a particularly apropos question with regard to temperature in particular, since temperature is formally defined in terms of an equilibrium average in the first place, and has meaning in a non-equilibrium open system like the sun-driven coupled rotating earth-ocean-atmosphere only by generating a coarse grained local definition of temperature that is itself subject to many assumptions that are often untrue.
To take this coarse-grained average definition of local temperature in a large system and then to superaverage it over the entire system into an “average temperature” that literally has no microscopic meaning and is not connected to any sort of sensible assertion concerning the distribution of energy within the system is itself a bit sketchy. But why one would expect to see chaotic internal dynamics within the system reflected in this average is beyond me.
To understand my point, take (say) 100 simple iterated maps in the chaotic regime — the logistic map with R=4, for example — and start them with distinct x and/or a distinct R. Imagine that each x is a sample of local dynamics on a 10×10 grid. Average the resulting series. Is the result still distinguishably chaotic? Now try a 1000×1000 grid. Still chaotic? Now try a grid like that of a climate model, or a grid like the piss-poor examples of sampling that are used to generate an average global temperature anomaly (where we’ve added yet another step to the processing by subtracting an arbitrary number from the average itself so that we are no longer even looking at the original quantity). You are still thirty spatiotemporal orders of magnitude short of the Kolmogorov scale of local atmospheric turbulence.
This is why I constantly bitch about the idiocy of the IPCC et al averaging over the output of many perturbed parameter ensemble results from the individual GCMs, and then superaveraging the results and calling it a “prediction” (or projection, or guess, or whatever you like) of the future climate. O. M. G. There are only a few dozen unproven or known false assumptions in that process. One reason I’ve started to look into chaos more carefully is to get some idea of just how false they are. I’ve already been able to show to my own satisfaction that systems with nonlinearities capable of producing chaos are not only potentially infinitely sensitive to initial conditions, they are infinitely sensitive to things like stepsize and integration algorithm (which is kind of obvious, but it is nice to watch it happen). Changing stepsize for any algorithm can change the character of the system in dramatic ways. Averaging can completely destroy the appearance of chaos (or make it look “merely random”), and do it quickly as you observe above.
There is a temptation to say that climate is some sort of average over weather and hence is not chaotic. Perhaps that is true. But that does not mean that climate is predictable when the underlying weather is chaotic, because our climate today is the result of integrating over the chaotic weather over the entire interval in between, and we do not end up in a “state” that is the average of many such states, we end up in one specific state.
rgb
“It is also difficult to map out a strange attractor with just one variable and time”
Plot on one axis, “y”, on the next, “dy/dt” on the third, “d2y/dt2”, etc. It jumps right out. It’s how I make my Arctic Ice predictions a few months before the minimum.
Robert –
Your comments about the oscillator simulation reminded me of the case where one is attempting a fixed-point (integer) digital sinewave oscillator where something very much like an “attractor” saves the day.
First – an aside: A popular DSP lecture “War Story” is that at Bell Labs, the “theoreticians” sent instructions to the technicians to construct such an oscillator, but then thought better of it, and belayed the order. Of course, when the halt message arrived, the oscillator was already running. Great story of course – Details lacking. It was generally supposed that it was simply a matter of recognizing that for the oscillator to work, we seemed to need poles EXACTLY on a unit circle, almost always impossible with Cartesian coordinates. In hearing and retelling the story for dozens of years, I became curious about the details, and my colleagues told me it was Jim Kaiser’s group. So I asked Jim (then down with you at Duke!) and he told me that he had never even HEARD the story!
The figure below shows the basics.
http://electronotes.netfirms.com/FIXEDPOINTOSC33.jpg
I will post the full 50 page of this 1995 report on my Electronotes site if there is any interest (email me). The two examples selected here show a NEAR nominal result (A) where the major non-ideal effect is a small amount of harmonic distortion due sample roundoff. Example E is more typical. It is a (NON-CHAOTIC) length 154 repeating sequence (originally termed a “limit cycle”) that closely resembles a sinusoidal oscillation, except the 13 cycles should have been 13x(360/30) = 156 samples. The frequency is thus a tad high (and there is a lot of harmonic distortion). A third case (not shown here) is where the sequence never returns to the initial state, but nonetheless becomes periodic by returning to SOME OTHER two states (often with quite different amplitude).
Here the structure “finds” a sequence. Why does this non-linear time-varying deterministic device seem to behave reasonably well (although LOOK temporarily chaotic)? It seems unlikely. The simplest explanation is that eventually, it has to find two states it has previously visited because there are only a finite number of consecutive pairs of states (fixed-point arithmetic). Moreover, because of the sinusoidal “model” of the second-order network, it is primed to find a solution sooner, rather than later.
rgb See comments
http://wattsupwiththat.com/2015/10/22/is-the-climate-chaotic/#comment-2054886
and
http://wattsupwiththat.com/2015/10/22/is-the-climate-chaotic/#comment-2054960
above.
Don’t you think that ,for purposes of discussing the climate system as a whole over geological time scales, the system is obviously stable.
Within that framework the rate of change of particular parameters may vary significantly as the combination of the absolute values of certain individual parameters crosses particular absolute thresholds.
As an example see the rapid rise in temperature at the end of individual ice ages.
Those living during such times might well get the impression the climate is unstable – or even chaotic !!.
Oh dear, some others that do not bother to find out what chaotic means. A double pendulum has also a bounded trajectory, it does not mean it is not chaotic. There are plenty of very simple systems with bounded trajectories, which are chaotic.
One post even talks about some sort of ‘equilibrium’. Sorry, there is no equilibrium. The Earth system is not at equilibrium and it cannot be. It’s a rotating body which receives energy from the Sun while sending energy into space. That cannot reach equilibrium. Not even a dynamical one.
When discussing the behavior of the climate, temperature is the most useful parameter.
MathWorld says
“The boundary between regular and chaotic behavior is often characterized by period doubling, followed by quadrupling, etc., although other routes to chaos are also possible (Abarbanel et al. 1993; Hilborn 1994; Strogatz 1994, pp. 363-365). ”
Using this practical criteria there is no evidence of chaotic climate behavior for the last 6 -700 million years.
When discussing the behavior of the climate, temperature is the most useful parameter
////////////////////
Why? It would appear that you have been suckered by the warmists who have been forced to change the theory from global warming to climate change, and when they do this they only advance evidence that temperature has increased, since there appears to be little evidence that other climatic parameters have changed at all. You are playing their game.
What about floods or drought which could be much more significant.
What would happen to places like India if there were say 6 or 7 years without monsoon?
We know what happened to Egypt when the Nile didn’t flood. The problem was not temperature.
The reason we are fixated on temperature is that the radiative properties of certain gasses can absorb and re-emit photons and it is claimed that this property leads to a temperature increase. It is not a laboratory characteristic of CO2 that it causes climate change. Temperature is important for considering AGW, but not necessarily climate change. As I said in my post (October 22, 2015 at 6:22 pm0 above:
“Far too many people equate temperature with climate. Temperature is merely one of the variables that make up climate (others include humidity, precipitation (amount, nature and cycles thereof), air pressure, wind, windiness, influenced by El Nino/La Nina conditions etc etc). Each of these parameters is variable, and constantly varying within bounds, and each may interact with each other.”
There has been some modest warming in the 20th century, but there has not been any climate change. No region has changed its Koppen (or equivalent) characterisation.
richard verney,
Correct. And I would even quibble with this:
When discussing the behavior of the climate, temperature is the most useful parameter
I think the temperature trend is the most important and useful parameter. Temperature is only a snapshot in time.
Another climate parameter: is the trend accelerating? Some folks still claim that the rise in sea levels is accelerating. If so, that would be a serious cause for concern. Fortunately, they are wrong.
And Arctic ice: is the reduction around 2006 – 2012 a trend? Or is it just a natural fluctuation?
The ‘climate’ consists of many parameters. Those parameters have been wide ranging over the past century or so. But the current climate parameters are not unusual, and they are certainly not unprecedented. They are normal and natural; every climate parameter observed now has been exceeded in the past, repeatedly, and to a much greater degree than now.
Therefore, what we are observing now is just the normal, natural, and cyclical changes that always occur. When Occam’s Razor is applied, we see that there is nothing to be concerned about.
The climate Null Hypothesis has never been falsified. That means that the Alternative Hypothesis — that human CO2 emissions cause global warming and other calamities — has been falsified, because the Null Hypothesis and the Alternative Hypothesis cannot both be right at the same time. One of them must be false.
That’s the rational, scientific view. Not many skeptics say that CO2 has zero effect. Most skeptics say that the effect of human CO2 emissions on global T is so minuscule that it can be completely disregarded for all practical and policy reasons.
But the climate alarmists’ goal is not scientific knowledge. If it was, the question would be settled by now. The real, unstated goal of the alarmist brigade is the passage of a carbon tax, and handing over national enforcement authority to a world body.
A carbon tax would greatly increase the cost of all goods and services, without a commensurate rise in incomes. And of course, the UN is always angling for more power. But no one in their right mind would hand the UN or anyone like it that kind of power, so they play the game of climate alarmism. That requires that they lie about it.
That’s the elevator speech explaining what’s going on. It isn’t about science; it never was. Nor consensus. It is greed for money and power. No more and no less.
Richard. You say – “There has been some modest warming in the 20th century, but there has not been any climate change. ”
Complete nonsense -the climate changes continually and the changes in global temperature are the most useful metric as a general measure of that change – Especially when the Northern and Southern hemispheres are considered separately.
If you think I have been suckered by the warmists you have never read any of my blog-posts at
http://climatesense-norpag.blogspot.com
where you will find my forecasts of the coming cooling.
=============
“The boundary between regular and chaotic behavior is often characterized by period doubling, followed by quadrupling, etc., although other routes to chaos are also possible (Abarbanel et al. 1993; Hilborn 1994; Strogatz 1994, pp. 363-365). ”
Using this practical criteria there is no evidence of chaotic climate behavior for the last 6 -700 million years. – Dr. Page.
=============
1 polar vortex x 2
= 2 polar vortices:
http://eoimages.gsfc.nasa.gov/images/imagerecords/2000/2832/antarctica_ept_2002267.jpg
That was only 13 years ago.
January last year:
http://climatechangenationalforum.org/wp-content/uploads/2014/01/GoHomeArcticYoureDrunk.png
The flat “average global temperature” trend since ~97 hasn’t been useful for predicting climate on the ground. Richard Verney makes a good point.
http://www.sea-way.org/blog/From_space02.jpg
I am well aware of your posts, and your predictions, and we have exchanged comments upon some of your posts. Time will tell whether your predictions come to pass, and perhaps, if this is accompanied by better knowledge and understanding, time will also enable us to ascertain whether you were right as to cause.
But with respect, Dr Page, on this particular issue (ie., whether temperature is climate or the most important proxy for climate), it is you who are speaking nonsense, and this is because of a fundamental failure to grasp what Climate is.
Climate consists of a range of parameters, temperature being just one of many parameters that make up Climate. Materially, each of these parameters is never in stasis, and each of these parameters is constantly varying within bounds (some of which can be quite wide). This means that change of any one parameter is not Climate change, nor even evidence of Climate change. Change is just what Climate is. It is only when a parameter or a number of parameters varies beyond the bounds of natural variation AND stays there for a protracted period of time, is there evidence of Climate change, and depending upon the extent to which the bound has been broken and the length of time that that variable has exceeded its natural bound will there be Climate change.
If I look at the UK weather forecasts over the next week or so, I will find temperature variations of up to 7 or so degrees. Consistently, I will find areas of Scotland being 1 to 2 degrees lower than other areas of Scotland, Scotland generally being 1 to 2 degrees lower than the boarders, the boarders being 1 to 2 degrees lower than Wales, Wales being 1 to 2 degrees lower than the Midlands, the Midlands being 1 to 2 degrees lower than the South West, the South West being 1 to 2 degrees lower than the South East, the South East being 1 to 2 degrees lower than the Channel Islands, Ireland and the Scottish Isles will also have different temperatures. One could divide the UK into at least 7 may be 10 different temperature zones, but no one has ever claimed that the UK has 7 to 10 different Climates.
The fact that there are variations in temperature of this order over what is a very small country just confirms that temperature does not alone dictate Climate. Climate is regional (not global), and even on a regional basis there is much variation in it, even to the extent that you get micro climates for example in the shadow of mountains, on the coast, particularly if warm currents pool, or if you have mountains and coast in close proximity etc. etc
In the UK, it is thought that the 1530s/1540s were the warmest decade on record. Did the UK have a different climate then? I would say no, it was simply that it was slightly warmer.
Climate cannot be measured over 30 years, or even a 100 years. If one goes back through the Climate history of a country say for a 1000 years then you get a good insight into what the Climate of that country is. and the extent of the upper and lower bounds of temperature, rainfall, snow fall, floods, droughts, storminess etc etc.
There is no evidence of actual Climate change, albeit that there is some evidence that there has been some warming since the depths of the little ice age with that warming being variable in fits and starts, and not correlating with the rise in CO2. Whether it correlates with anything else is moot, and presently we do not understand why there was the Holocene Optimum, the Minoan Warm Period, the Roman Warm Period, the Medieval Warm Period, the Little Ice Age, the late 20th century warming, the current pause. All we can say is that CO2 does not correlate with such events, and that it appears that the planet is on a downward trend which will be interrupted in fits and starts, heading back into the throws of the current ice age. ..
I know that you are not a warmist, and I shall continue to read all your posts and continue to watch to see whether your predictions come to pass.
Mr. Page,
The Webster says;
Chaos;
complete confusion and disorder : a state in which behavior and events are not controlled by anything.
You win this debate, I say. No contest.
(Yikes, Chaos apparently stole the P in your name ; ).
Temperature would be the last metric I would look at for this issue. It is the end result of a mix of factors, of which, different mixes can result in the same temperature outcome. These other factors are themselves a result of topography and regional climate made up of temperature range, humidity, sun days, cloudiness and cloud patterns, etc.). Wickedly complex question. Wickedly complex investigation.
Pamela
The question is so wickedly complex that it is inherently insoluble by numerical modeling methods so that another forecasting approach is necessary. I will repeat part of an earlier comment.8:41 AM above,
“This does not mean, however, that climate can be forecast using reductionist climate models.The modelling approach is inherently of no value for predicting future temperature with any calculable certainty because of the difficulty of specifying the initial conditions of a sufficiently fine grained spatio-temporal grid of a large number of variables with sufficient precision prior to multiple iterations. For a complete discussion of this see
Essex: https://www.youtube.com/watch?v=hvhipLNeda4
Earth’s climate is the result of resonances and beats between various quasi-cyclic processes of varying wavelengths combined with endogenous secular earth processes such as, for example the changing geomagnetic field.. It is not possible to forecast the future unless we have a good understanding of where we are at present time in relation to the current phases of these different interacting natural quasi-periodicities which fall into two main categories.
a) The orbital long wave Milankovitch eccentricity,obliquity and precessional cycles which are modulated by
b) Solar “activity” cycles with possibly multi-millennial, millennial, centennial and decadal time scales.
The convolution of the a and b drivers is mediated through the great oceanic current and atmospheric pressure systems to produce the earth’s climate and weather.
After establishing where we are relative to the long wave periodicities , especially the solar millennial cycle, to help forecast decadal and annual changes, we can then look at where earth is in time relative to the periodicities of the PDO, AMO and NAO indices and based on past patterns make reasonable forecasts for future decadal periods.
For the latest forecasts and complete discussion see
http://climatesense-norpag.blogspot.com/2014/07/climate-forecasting-methods-and-cooling.html
Surely, enthalpy is more important a metric, but it seems to be harder to get a hold of. If the earth was a glass of water full of ice cubes a lot “warming” would need to go on before temperature began to rise. Temperature is a kind of ‘remainder’ of all the other forms that the sun’s energy contributes to the earth and is therefore very incomplete a measure of how much ‘heat’ is retained.
I am convinced that we best study the polar regions for our indications of a warming planet – polar amplification serves as a kind of vernier for examining essential climate change. If the Arctic ice recovers, – it certainly doesn’t seem to be getting worse- then all alarm is empty noise. To bad we didn’t do more study in the polar regions during the 1930s. We probably would be able to say that today’s ice extent is nothing to worry about. Regarding the temperature in the tropics – I recall the same temperatures in Lagos in the mid 1960s to what I see now. Let’s stop worrying about temperature change in the tropics – it aint happening there and even CAGW theory says it shouldn’t
“A chaotic system is a system wherein nearby initial states either converge or diverge exponentially.”
Um. No. You’ve linked to a post on the lorenz attractor. Ever put in the initial state (0,0,0)? It neither converges nor diverges. A chaotic system, by definition, is a system with a non-integral dimension, a fractional dimensionality. The Lorenz attractor, for instance, has a Hausdorff dimension of 2.06, according to wiki https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension . It covers more than an area. It fills less than a volume. It has a fractional dimension, ergo it is chaotic.
“Whether a dataset is chaotic or not is generally assessed by looking at the Lyapunov exponent.”
No, it’s not. One assesses it by looking at the dimensionality. The Lyapunov exponent (singular, when did it become singular?) is not generally used. That’s why wiki has half a hundred Hausdorff dimensions listed for various attractors…. and no list of Lyapunov exponents for various attractors, AFAIK. Mandelbrot originally used a Hausdorff dimension to describe his fractal. He did not use a Lyapunov exponent or even more than one Lyapunov exponent (I love the fact that you stated it as a singular. It makes me chuckle.). Professionals use Hausdorff dimensions. Authors like James Gleick use Lyapunov exponent(s).
(Aside #1: I just read this in wiki, “Thus, there is a spectrum of Lyapunov exponents— equal in number to the dimensionality of the phase space.” Didn’t you at least read wiki? Didn’t that clue you into the idea that there was more going on than a single exponent? Didn’t the phrase “dimensionality of the phase space” stop you short? Didn’t you wonder what wiki meant by “dimentionality” or “phase space”? Do you even know what “phase space” might be?
“An example is the ocean surface. If you drop two sealed empty bottles overboard in mid-ocean, one on each side of a boat, they will drift apart over time. This separation will be slow at first, and then faster and faster as the two bottles encounter different winds and currents in different areas.”
This is an assertion. There is no evidence that “then faster and faster” will happen. That’s an assertion. You have not differentiated between chaos and two bottles on a random walk. Hint, if the two bottles can hit each other then there is randomness involved. They can hit each other. Again, this would be impossible if they were in a system dominated by chaos.
(Aside #2: I should turn this aside into another post… Didn’t the sentence from wiki “A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness)” bring you up short? Did a word like “compactness” prompt you to think why two bottles in the ocean might not be a good example of chaos and a better example of randomness….. No? NO?)
“Now, these results were a great surprise to me.”
Oh we know. Physics and math is generally a surprise to you. You began your post stating as a fact that the ocean is chaotic and not random. You then do some plots and find that the systems you are interested in are dominated by randomness. It never occurs to you to go back and figure out that the two bottles you used to explain chaos are sitting there hitting each other over and over which would be impossible if the ocean were dominated by chaos. Once again you’ve made a hash of things.
That result not a great surprise to me.
Once again Willis I ask that you stop doing this. You are consistently misinforming people.
1) Of what relevance is fractal geometry to the question of chaos?
2) How much oil from the Deepwater Horizon blowout randomly walked ashore to Florida?
http://www.scientificamerican.com/article/walls-of-water-make-chaotic-currents-more-predictable/
1) Of what relevance is fractal geometry to the question of chaos?
Think of it as Willis's bottle.
Not geometry per se but a fractional dimension has everything to do with chaos. That’s why everyone gets so interested in Mandelbrot’s work. Because he showed a real-world thingy that had a fractional dimension <2, i.e. more than a line and less than an area. That is why everyone gets so excited but the Lorenz attractor. Because it has a fractional dimention <3, i.e. more than an area and less than a volume.
Take Lorenz's equations for instance (btw- I'm about to create a conceptual example from equations…. I hate doing that.) as the state moves around, in a 2-D (area), it seems to have the same 2-D state but goes off in a different direction. So 2-D isn't enough. As the state moves around in a 3-D volume one sees missing holes between the states that are never, ever filled. 3-D is too many dimensions.
A random variable would end up filling that 3-D volume. Eventually Willis's bottle in the ocean passes through the previous location of the other bottle. Eventually, they share exactly the same spot. Then even though the two bottles had the exact same location, they randomly walk away from each other. The bottles do not move chaotically but randomly.
Here is a nice picture from wiki showing the random walk of a large object being pushed around by other smaller objects.
Here are a few questions I don’t know the answers to but are always of interest?
1. If the ‘global temperature index’ (whichever one you choose) is NOT chaotic, is it still possible that useful predictions cannot be made in any better way than applying the equivalent of quantum mechanical frameworks to say: ‘there are a number of allowable states (aka future temperature evolutions) which can emerge from this situation, and all we can say is that there are statistical likelihoods assigned to those potential allowable states (whose error bars may nonetheless be rather large)?
2. Do climate indicators like ‘global temperature’ show features similar to what you see on trading exchanges, where ‘resistance points’, ‘break-out events’, ‘singularities’ occur etc etc? Singularities would, in this case, be analogous to e.g. transitions between inter-glacials and ice ages. Resistance points might be natural boundaries of variability within an inter-glacial, within a centennial or inter-centennial set up or even within PDO cycles, Hale Cycles or Sunspot cycles. I”m not sure the Quants on Wall Street will wish to be poorly paid to transfer their knowledge from algorithms to predict gold futures to algorithms to predict climate regimes, however……..
3. If temperature indices are not ‘chaotic’, does that imply that apparently stochastic input variables like e.g. major volcanic eruptions, solar events like major flares etc etc are not, in fact, stochastic but, with sufficient scientific research and data collection/analysis, will become amenable to reasonably accurate predictions?
4. Are ‘global temperature indices’ actually much use in predicting local/regional weather patterns (since actually what is important on an annual basis are things like rainfall patterns, temperature patterns etc not across the globe but in e.g. the Mid West Corn Belt, the SE of the UK, the Corn Belts of Ukraine and southern Russia etc etc)?
The big questions are, I think, whether useful predictions on a 9 – 12 month scale are intrinsically possible (my hunch is they are, since I managed to zone in for the best part of a decade on likely evolution of snowfall patterns in the European Alps when I was a mountain goat for a decade back in the day, successfully predicting times of the season to ski in different parts of the alps, at least before a shift in the PDO came along), given the ability to understand and measure the right input variables, and, if so, down to what geographical range (probably easier to make average predictions for the whole of the US corn belt than it is to predict precise patterns for East Anglia in the UK, after all).
I was most interested in the analyses a year or so back comparing the climate system to an electronic circuit – it seemed to me to be one of the best analogies.
What is perhaps a more interesting theoretical question is this: ‘how much previous data do you need to input into your system before an apparently chaotic one becomes a non-chaotic one? Or vice versa (if a singularity comes along)?’
It would provide scientific robustness in answering the question of how long scientists need to keep on collecting primary data before rigorous modelling is likely to bear fruit……..
Dr Norman Page
I have addressed your comment (October 23, 2015 at 9:08 pm) above at (richard verney October 24, 2015 at 2:33 am). dbstealey at (October 23, 2015 at 5:16 pm ) makes the valid point:
“The ‘climate’ consists of many parameters. Those parameters have been wide ranging over the past century or so. But the current climate parameters are not unusual, and they are certainly not unprecedented. They are normal and natural; every climate parameter observed now has been exceeded in the past, repeatedly, and to a much greater degree than now.”
What he is describing is just what Climate is. You say that “the climate changes continually”. I would say continuous change is simply what Climate is, such that change of any one parameter is not in and of itself Climate change. As I say change (by which I mean change within bounds as described by dbstealey) is not Climate change, it is only once the natural bounds of variability have been exceeded for a protracted period of time that one may be witnessing Climate change. That may be subtle, but it is important.
In your comment (October 23, 2015 at 1:54 pm) you ask: “Don’t you think that ,for purposes of discussing the climate system as a whole over geological time scales, the system is obviously stable.”?
I find that difficult to answer since it depends upon what you mean by stable and the precise context in which that is being considered. Over the geological timescale you mention (the past 700 million years or so), the Climate has varied over relatively narrow bounds bearing in mind the possible extremes that may have been available. That is probably the consequence of the fact that responses are dampened and with a large buffer because the planet is a water world with most of that water being on the surface.
So personally, I would not say that the Climate was stable, but rather that it is bounded between a relatively narrow range and that there is a net negative feedback such that the Climate always stays within these bounds. Depending upon context, the difference between us may be little more than semantics
Richard You say” So personally, “……. it is bounded between a relatively narrow range and that there is a net negative feedback such that the Climate always stays within these bounds. Depending upon context, the difference between us may be little more than semantics” I agree – Norman.
Richard Better posted this way
You say ” it is bounded between a relatively narrow range and that there is a net negative feedback such that the Climate always stays within these bounds. Depending upon context, the difference between us may be little more than semantics” I agree – Norman.
This conclusion:
is true.
It is also trivial if the question is Is Earth’s climate system “chaotic”? [in the strict sense of Chaos Theory/Non-linear Dynamical Systems].
That the observational datasets exhibit strong stability over time is evidence that the climate system is chaotic.
When forcings are within certain values for a non-linear dynamic system, it forces first stability, then as forcing changes (usually increasing, but can be decreasing), one begins to see periodic results, rapidly doubling and re-doubling, leading to “chaotic” results.
“In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system.”
Stability in spite of perturbation, the rapid return to the stable value after perturbation, is a indicator that we are dealing with Chaos Theory’s “strange attractors” (later on, when things gets wild the attractors can indeed be strange) — kick a value out of whack with a decade of volcanism, for example, and it rapidly (relatively) returns to the “preferred” value range (the “attractor” of the system).
In the case above, the value had been programmatically perturbed way out of the stable range. Within a dozen or so iterations (days, years) it has returned to the point of stability. This type of behavior is what I (and many others) see in Earth’s climate today, “those six observational datasets” can only be perturbed short-term and are, in the Chaos Theory sense, “attracted” back to their point of stability given the existing range of forcings.
In due time, I will post my Chaos Part II essay, which is on this particular point.
Mr, Hansen,
“That the observational datasets exhibit strong stability over time is evidence that the climate system is chaotic.”
Any chance the stat/math folks have gone and picked the wrong term? Strong stability over time is pretty much the opposite of chaotic, in English, ya know?
If a biologist said; That the observational datasets exhibit strong stability over time is evidence that the living system is chaotic . . wouldn’t you question the wisdom of using that term in particular, to describe this sort of . . effectual stability over time? Or I said; That catcher sure is good with wild pitches, the runners rarely have a chance to advance, his ball handling systems are chaotic . . would those be “technically” correct in the sense your sentence was?
(If so, I’m just sayin’ ; )
Reply to JohnKnight ==> First you have to have the right definition of Chaotic — you must know and use the term in its own frame of reference, which is Chaos Theory — w’s essay, and all these comments, are based on the premise that we are talking Chaos Theory when ever the words “chaos” and “chaotic” are used. This idea stems from the IPCC’s early admission about the Earth’s Climate System: “The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible.” TAR – Working Group I: The Scientific Basis
If you are not passingly familiar with chaos and chaotic in that sense, the Wiki has a nice primer .
In comments to my first essay on Chaos and Climate, at least one engineer stated that engineers depend on this stability in non-linear dynamic systems (which are, in the vast majority of cases, chaotic) to hold things together and are very careful to avoid parameters which would push their systems past the known points of stability. Engineers love this feature, as if something twangs the bridge, say, the bridge ‘wants’ to get back to stability.
If after reading the Wiki primer and following some of the links, if you still have questions, ask me.
Unless, of course, you were pulling my leg, teasing me, etc…..
Kip Hansen,
Thanks . . and yeah, I get the idea, it’s just a very poor choice of term to my mind. Complex order is not chaos . . it’s reality ; )
Reply to JK ==> It is worse than you think….
Chaos Theory is not a theory, there is no theory.
Chaos Theory refers to systems that are strictly, thoroughly deterministic but not at all predictable once they enter the chaotic realm.
Chaos Theory predicts the stability I mention above.
The Chaos comes from a deterministic system, like a simple formula, whose results are easily determined, but only one step at a time, it is impossible to know what the result ten steps ahead will be without going through the intermediate nine steps. Edward Lorenz found that one could fairly accurately predict tomorrow’s weather given enough correct information about today’s weather…but not next weeks (except in a general way) — results are very sensitive to initial conditions, the tiny changes in initial input may cause big changes in output down the line (and the may is important, they may not do so).
Fun stuff — an introductory understanding of Chaos Theory can change one’s appreciation of the physical and mathematical world.
.
Kip.,
I don’t doubt that an introductory understanding of so called Chaos Theory can change one’s appreciation of the physical and mathematical world . . but it would help the whole appreciation drive if mathematicians would appreciate the language realms a bit more, and avoid using terms for their ideas/concepts that generate contra-logical language snafus when the ideas are spoken of outside the shop, so to speak.
I’ve given it some thought (yer welcome math/stat heads ; ) and a bit of word searching, and I suggest ‘chaosh’.
Our largely water and less so moving land masses planet with its highly variable atmosphere is the source of both long term and jagged temperature trends between the extremes of the various forms of Milankovitch cycles. I can well imagine an orbital tilt of one variety or another causing changes in our semi-permanent and permanent oceanic/atmospheric systems, along with the way discharge and recharge regimes work.
http://www.pnas.org/content/112/12/E1406.full.pdf
M Courtney October 22, 2015 at 2:58 pm
Thanks, M. I wrote this post to engender discussion and encourage questions.
Rather than saying climate has a “strange attractor”, I’d say that the climate has a number of emergent thermoregulatory phenomena operating that keep those variables consistent.
Which brings up an interesting question:
Is a thermostatically controlled system chaotic?
I’d say “Could be, but not necessarily so”. I’d also say that if the thermoregulatory phenomena are independently mobile (e.g. dust devils, thunderstorms) that chaoticity would be almost assured. But like I said … what do I know, I was born yesterday.
w.
Dinostratus October 23, 2015 at 11:26 pm
Indeed, random and chaotic are different. I showed the difference in Figure 1.
As to the ocean being “random”, I’d disagree. The ocean has tides, resonances, whirlpools, currents, bifurcations, tide rips, and waves from the surface down to the bottom. While you could describe it as “chaotic” or “turbulent”, I don’t think “random” even begins to touch the nature of the ocean.
Not true in the slightest. For example, the bottles will never go to certain areas, because the areas are upwelling and thus the current goes outwards in all directions. As an example, the Argo floats rarely sample parts of the normal location of the ITCZ, because of the upwelling there. So no, the movement of a floating bottle does NOT end up filling that 2D space, and thus, a bottle doesn’t walk randomly about the surface of the ocean.
w.
Thank you for being stubborn and unwilling to learn from your betters. I was concerned that you might figure something out then start to misinform and confuse people at a higher level.
Dino, normally a scientific discussion starts with someone making a scientific claim. Then someone else disagrees. Then the first person comes back and defends their view, pointing out errors and providing support for their view. Then the second person does the same, clarifying and explaining. It is an interchange of ideas.
With you, on the other hand, you make a scientific claim. Then I disagree. And your response?
Rather than defending your claim, you jump right to the easy insult, that I’m “unwilling to learn from my betters”.
Dino, I learn from my betters all the time … but you are definitely not among them. I suspect you could be, but instead you spend all of your time attacking me instead of showing what’s wrong with my ideas.
Which is why I usually just let your attacks float on by. But in this case, I had hope, because you actually and most unusually made a scientific comment instead of your usual meaningless, uncited attack on my abilities, my style, or my knowledge.
But that didn’t last long. One comment from me on your scientific claims and you abandon them with no defense, instead you’re back to slinging mud …
Sorry, son, not impressed. Come back when you’ve grown up and you wish to discuss other peoples’ ideas and not their learning styles …
w.
The future climate is not chaotic:
It is always getting hotter, with 105% certainty, er, uh, I mean 95% certainty!
The past climate is very chaotic:
It is changing every year, sometimes more than once a year, and as a result of these adjustments, re-adjustments, and re-re-re-adjustments, the historical average temperature data are in a permanent state of chaos .. or perhaps I should say in limbo.
So many half thoughts. When you aren’t specific you come across as someone with a thesaurus on his knees.
My seat of the pants thinking is this: There is a millennial scale cycle that rules thusly: based on orbital mechanics solar insolation shifts also shifts the ITCZ important global oceanic-atmospheric teleconnected systems driving whether or not there is a continued net disgorgement of heat from the oceans (which occurs rapidly), or a jagged recharge which results in the typical downward steps observed in the paleo-reconstruction due to the continued recharging of ocean heat, leaving us without benefit.
In terms of what puny humans experience in our life times, it sure feels chaotic. But over the course of 10’s of 1000’s of years, it is anything but chaotic. The metronome of Earth’s orbital mechanics makes it non-chaotic.
http://www.climategeology.ethz.ch/publications/2014_Schneider_et_al.pdf
Earth’s climate system is random and chaotic is more correct then saying the earth’s climate is stable.
Chaotic is in the eyes of the beholder. The earth’s climate having the ability to go from a glacial state to a glacial state and have abrupt climate changes at times satisfies chaotic for my way of thinking.
A non chaotic climate would be a global temperature changed over 1000’s of years that varied by less then 1C which we know is not the case although the overall global temperature change from glacial to inter glacial periods is perhaps ONLY between 4-7C.
That being deceiving due to the fact that the tropics relatively small temperature changes from glacial versus inter- glacial periods of time tempers the global total temperature change.
In closing a measure of how chaotic the climate is should be what impact does it have upon humanity and from this approach I would say the climate changes sufficiently enough to have a very big impact on humanity which makes the climate chaotic regardless.
The climate is random due to the fact it is subject to random extra terrestrial events and random terrestrial events.
It is chaotic because the climate is always a reflection of everything that is effecting it at a given time and all the forces that effect the climate at a given time are never the same ranging from land/ocean arrangements, to solar variability , to Milankovitch Cycles, geo magnetic field strength, lunar influences, the ice dynamic etc etc.