Guest essay by Anthony R. E.
![climate-model-1[1]](https://wattsupwiththat.files.wordpress.com/2013/09/climate-model-11.jpg?resize=576%2C576&quality=83)
Recently, I read a posting by Kip Hansen on Chaos and Climate. (Part 1 and Part 2) I thought it will be easier for the layman to understand the behavior of computer models under chaotic conditions if there is a simple example that he could play. I used the attached file in a course where we have lots of “black box” computer models with advance cinematic features that laymen assume is reality.
Consider a thought experiment of a simple system in a vacuum consisting of a constant energy source per unit area of q/A and a fixed receptor/ emitter with an area A and initial absolute temperature, T0 . The emitter/receptor has mass m , specific heat C, and Boltzmann constant σ. The enclosure of the system is too far away such that heat emitted by the enclosure has no effect on the behaviour of the heat source and the emitter/receptor.
The energy balance in the fixed receptor/emitter at any time n is:
Energy in { q/A*A= q} + energy out {-2AσTn 4 } + stored/ released energy {- mC( Tn+1 – Tn )} = 0 eq. (1)
If Tn+1 > Tn the fixed body is a heat receptor, that is, it receives more energy than it emits and if Tn > Tn+1 it is an emitter, that is, it emits more energy than it receives. If Tn = Tn+1 the fixed body temperature is at equilibrium.
Eq (1) could be rearranged as :
Tn+1 = Tn -2AC Tn 4 /mC +q/mC eq(2)
Since 2AC/mC is a constant, we could call this α, and q/mC is also a constant we could call this β to facilitate calculations. Eq (2) could be written as:
Tn+1 = Tn – αTn 4 + β eq.(3)
The reader will note this equation exhibits chaotic properties as described by Kip Hansen in this previous post at WUWT on November 23, 2015, titled “Chaos & Climate –Part 2 Chaos=Stability”. At equilibrium, Tn=+1 = Tn , and if the equilibrium temperature is T∞ then from equation (3)
T∞ 4 =β/α or α = β / T∞4 if eq. (4)
And eq (3) could be written as
Tn+1 = Tn – βTn 4 /T∞4 + β or Tn+1 =Tn +β(1-Tn4 /T∞4 ) eq (5)
Eq (5) could be easily programmed in Excel. However, there are several ways of writing T4 . One programmer could write it as T*T*T*T, another programmer could write it as T ^2* T ^2, another programmer could write it as T*T ^3 and another could write as T^4. From what we learned in basic algebra, it does not matter as all those expressions are the same. The reader could try all the variations of writing T4 . For purposes of illustration, let us look at β= 100, T∞ =243 ( I am using this out of habit that if it were not for greenhouse gases the earth would be -30o C or 2430 K but you could try other temperatures) and initial temperature of 300 K. After the 17th iteration the temperature has reached its steady state and the difference between coding T4 as T^4 and T*T*T*T is zero. This is the non-chaotic case. Extract from the Excel spreadsheet is shown below:
| beta= | 100 | ||||
| Iteration | w T^4 | w T*T*T*T | % diff. | T ∞= | 243 |
| 0 | 300.00 | 300.00 | 0.00 | ||
| 1 | 167.69 | 167.69 | 0.00 | ||
| 2 | 245.01 | 245.01 | 0.00 | ||
| 3 | 241.66 | 241.66 | 0.00 | ||
| 4 | 243.85 | 243.85 | 0.00 | ||
| 5 | 242.44 | 242.44 | 0.00 | ||
| 6 | 243.36 | 243.36 | 0.00 | ||
| 7 | 242.77 | 242.77 | 0.00 | ||
| 8 | 243.15 | 243.15 | 0.00 | ||
| 9 | 242.90 | 242.90 | 0.00 | ||
| 10 | 243.06 | 243.06 | 0.00 | ||
| 11 | 242.96 | 242.96 | 0.00 | ||
| 12 | 243.03 | 243.03 | 0.00 | ||
| 13 | 242.98 | 242.98 | 0.00 | ||
| 14 | 243.01 | 243.01 | 0.00 | ||
| 15 | 242.99 | 242.99 | 0.00 | ||
| 16 | 243.00 | 243.00 | 0.00 | ||
| 17 | 243.00 | 243.00 | 0.00 | ||
| 18 | 243.00 | 243.00 | 0.00 | ||
| 19 | 243.00 | 243.00 | 0.00 | ||
| 20 | 243.00 | 243.00 | 0.00 |
If β is changed to170 with the same initial T and T∞, T does not gradually approach T∞ unlike in the non chaotic case but fluctuates as shown below. While the difference in coding T4 as T^4 and T*T*T*T is zero to the fourth decimal place, differences are really building up as shown in the third table.
| beta | 170 | ||||
| Iteration | w T^4 | w T*T*T*T | % diff | T ∞ | 243 |
| 0 | 300.0000 | 300.0000 | 0.0000 | ||
| 1 | 75.0803 | 75.0803 | 0.0000 | ||
| 2 | 243.5310 | 243.5310 | 0.0000 | ||
| 3 | 242.0402 | 242.0402 | 0.0000 | ||
| 4 | 244.7102 | 244.7102 | 0.0000 | ||
| 5 | 239.8738 | 239.8738 | 0.0000 | ||
| 6 | 248.4547 | 248.4547 | 0.0000 | ||
| 7 | 232.6689 | 232.6689 | 0.0000 | ||
| 8 | 259.7871 | 259.7871 | 0.0000 | ||
| 9 | 207.7150 | 207.7150 | 0.0000 | ||
| 10 | 286.9548 | 286.9548 | 0.0000 | ||
| 11 | 126.3738 | 126.3738 | 0.0000 | ||
| 12 | 283.9386 | 283.9386 | 0.0000 |
By the 69th the difference between coding T4 as T^4 and T*T*T*T is now apparent at the fourth decimal place as shown below:
| 69 | 88.6160 | 88.6153 | 0.0008 |
| 70 | 255.6095 | 255.6088 | 0.0003 |
| 71 | 217.4810 | 217.4824 | 0.0007 |
| 72 | 278.4101 | 278.4086 | 0.0005 |
| 73 | 155.4803 | 155.4850 | 0.0030 |
| 74 | 296.9881 | 296.9894 | 0.0004 |
The difference between the two codes builds up rapidly that by the 95th iteration, the difference is 4.5 per cent and by the 109th iteration is a huge 179 per cent as shown below.
| 95 | 126.5672 | 132.2459 | 4.4866 |
| 96 | 284.0558 | 287.3333 | 1.1538 |
| 97 | 136.6329 | 125.0047 | 8.5105 |
| 98 | 289.6409 | 283.0997 | 2.2584 |
| 99 | 116.5073 | 139.9287 | 20.1029 |
| 100 | 277.5240 | 291.2369 | 4.9412 |
| 101 | 158.3056 | 110.4775 | 30.2125 |
| 102 | 297.6853 | 273.2144 | 8.2204 |
| 103 | 84.8133 | 171.5465 | 102.2637 |
| 104 | 252.2905 | 299.3233 | 18.6423 |
| 105 | 224.7630 | 77.9548 | 65.3169 |
| 106 | 270.3336 | 246.1543 | 8.9442 |
| 107 | 179.9439 | 237.1541 | 31.7934 |
| 108 | 298.8261 | 252.9321 | 15.3581 |
| 109 | 80.0515 | 223.3877 | 179.0549 |
However, the divergence is not monotonically increasing. There are instance such as in the 104th iteration, the divergence drops from 102 per cent to 18 per cent. One is tempted to conclude T^4 ≠ T*T*T*T.
Conclusion:
Under chaotic conditions, the same one line equation with the same initial conditions and constant but coded differently will have vastly differing results. Under chaotic conditions predictions made by computer models are unreliable.
The calculations are made for purposes of illustrating the effect of instability of simple non-linear dynamic system and may not have any physical relevance to more complex non-linear system such as the earth’s climate.
Note:
For the above discussion, a LENOVO G50 64 bit computer is used. If a 32 bit computer is used the differences would be noticeable at a much earlier iterations. A different computer processor with the same number of bit will also give different results.
What is the essential difference between to two treatments?
The second graph, which represents the non-chaotic region of the formula (the first data set in the essay above) — the ‘forcing’ factor being the ß which coincides with the “r” term in the examples in my Chaos and Stability essay.
At a ß of 100, the system tends to stability.
In the top graph, Anthony R.E. has increased the ß to 170, at which value the system becomes chaotic.
The two traces used show what happens when one performs the mathematical rendering of a non-linear chaotic dynamic system on modern computers — even if one does not change the values at any point in the way that Edward Lorenz did, a serendipitous event as it lead to the popularization of Chaos as a topic of scientific interest — because non-linear chaotic systems are extremely sensitive to initial conditions, any change in any one value, at any point, no matter how slight, will grow to change the outcome in unpredictable ways. Now, Anthony R.E. did not change values mid-stream — but the differences in the programming, the coding of the two formulas, though mathematically the same, introduce differences in the output values — which then set off the chaotic sensitivity to initials conditions, and we get not only wildly different values for specific iterations, but the behavior of the two systems have diverged and are dissimilar.
Why has Anthony R.E. mixed these two different lessons together? It is not to confuse you — it is to show that even the slightest differences in these non-linear calculations lead to wildly different results — and these slight differences can be introduced by the computers and their internal workings themselves.
To even approach comparable results between GCMs, a line by line evaluation would be required by specialists in the underlying code and machine languages.
The above is exactly why GCMs produce spaghetti graphs of one thousand runs, initiated with almost exactly the same initial conditions, that predict everything from Fire Ball Earth to Ice Ball Earth.
The idea that these chaotic results can then be averaged to produce a valid rational projection is beyond absurd.
This is simply not true. A particular model run with slightly different initial conditions but exactly the same forcings and run over, say, 100 years or more of rising greenhouse gas levels produce “spaghetti graphs” that all show different details (e.g., of ENSO and such) but show about the same overall rise in temperature.
Over shorter time scales of, say, 10 or 15 years, then yes, you will see different trends…which is (one of reasons) how we know that such trends are not reliable or robust.
Reply to joeldshore ==> Sir, you are either being disingenuous or you are unaware of how the IPCC uses GCMs to make projections for its various scenarios. I suggest reading, say, Dr. Robert Brown, from Duke, quoted in this essay (not mine) : http://wattsupwiththat.com/2013/06/18/the-ensemble-of-models-is-completely-meaningless-statistically/
Research how GCM ensemble results are used in arriving at a projection. Actually look at graphed results from a ensemble of runs…read the IPCC’s explanation of why they use ensembles instead of simply running the program once.
The outlying runs, those at the top and the bottom out a hundred years, literally predict/project almost any climate, bounded only by the parameters fed in to the system by its administrators.
Kip:
Bull
Kip Hansen: The link you gave me provides no support for the claim that Robert Brown makes. He makes some vague reference to some unnamed thread by Spencer. He also talks about an “ensemble of models”, i.e., he is not talking about how one particular model behaves but rather how the whole range of models behaves. (I believe that range has model sensitivities varying between about 1.5 and 4.5 C per CO2 doubling, so there is no great surprise to find a range of projections.) Even then, Robert Brown made no claim that they “predict everything from Fire Ball Earth to Ice Ball Earth”.
Reply to Mike ==> Matbe I haven’t been clear.
Surely you are aware of the individual ensemble runs? And that they are produced by using slightly different initial conditions? And that they then discard “outliers” (those implausible results at the edges), and then use averaging to pick a mid-line, and parameters to set the upper and lower limits for each emissions scenario. None of this is controversial or in question — the IPCC explains it very clearly as the process.
It is the discarded outliers that seem to “predict” Fire Ball Earth and Ice Ball Earth — and because they easily acknowledge that such results are ridiculous, they throw them out. I would too!
It is the simple, IPCC-stated fact that Earth’s climate is a coupled non-linear chaotic dynamic system that causes the individual ensemble members to vary so much, despite careful parameterization (which tends to set bounds).
“None of this is controversial or in question — the IPCC explains it very clearly as the process”
Is there any other field with such not “controversial or in question” procedures?
Great, so then you can presumably supply us with a link to the IPCC discussion?
The only time I have heard about such outliers was in some early climateprediction.net set of runs, where they had an occasional issue due to the simple ocean model that they were using…and these outliers were clearly identifiable. [In that modeling experiment, they were not (just) varying initial conditions but actually varying parameters to see what sort of range of climate sensitivities they could produce by varying the parameters within physically-plausible ranges.] Doing this was still not ideal, but it was pretty straightforward to justify which ones were completely unrealistic due to a known problem.
Reply to joeldshore ==> Start with this:
https://www.ipcc.ch/pdf/assessment-report/ar5/wg1/WG1AR5_Chapter09_FINAL.pdf
There are few graphics and the text is a summary.
Thanks for the link to the 100+ page chapter in the IPCC report, but what I was hoping that you would provide was the specific text or graphics that justifies your claim ,”The above is exactly why GCMs produce spaghetti graphs of one thousand runs, initiated with almost exactly the same initial conditions, that predict everything from Fire Ball Earth to Ice Ball Earth.”
What does it matter if they’re spaghetti graphs? They’re wrong. All of them. That’s what matters:
http://l.yimg.com/fz/api/res/1.2/u.i.A9hIbX2Ql7L7LC5_jg–/YXBwaWQ9c3JjaGRkO2g9Njk5O3E9OTU7dz0xMDE1/http://notrickszone.com/wp-content/uploads/2013/09/73-climate-models_reality.gif
dbstealey: Since you claim to be a skeptic, you must have looked skeptically at the graph you have pasted. Given that, why don’t you explain for us exactly what the graph shows, i.e.,
* What temperature data is plotted and what are the known limitations of that temperature data set?
* How does that compare to the temperatures that the models are simulating?
* How did the authors choose to align the graphs for the different temperature series?
It is not hard to post lots of graphs that have never passed peer review and never could because they are created to push a certain point-of-view rather than to inform.
joelshore,
Give it up. Your models are NFG. They fail. All of them.
dbstealey: So, in other words, you know nothing about the details of that graph, but since it shows what you want to believe, you uncritically accept it. (And, without a bit of irony, you call yourself a “skeptic”.)
I will give you a little hint: The temperature data shown there is what is called “TMT” or mid-tropospheric temperature data. Some of the problems with that data include the fact that the weighting function for what that satellite channel samples includes a tail extending into the stratosphere and hence it is contaminated by stratospheric cooling. Attempts to deal with that and other issues in the analysis is why the trend for the mid-tropospheric data differs by a factor of 3 between UAH and RSS, a fact that has been covered up in that graph by having one of the two plots that say “Reality” being an average of the UAH and RSS. (The other plot is an average of, I believe, 4 different analyses of balloon data. That data has similar difference between different analyses and even between different versions of different analyses. Not sure what versions of each analysis were used in producing the average here, but for a while some skeptics refused to use the latest version of one of the analyses because it showed a significantly larger trend than the earlier versions.
So, in other words, the fact that the data that is labeled “Reality” is really shows considerable variation from one analysis to the next has been hidden in these plots by doing some sneaky averaging, so that you are fooled into believing that this data is reliable. So, a discrepancy between models and data that serious scientists are trying to understand is instead presented in such a way that the problems with the data and the analysis that went into producing the data are covered up.
This one graph is a nice illustration of the sort of differences one sees between real science presented in scientific journals and pseudoscience presented for the consumption of gullible “skeptics”.
Weather is what’s happening at a point in time. Climate is a statistical profile of weather. If the global circulation models were any good they’d at least get the statistical profile close to right even as chaotic feedback caused them to diverge from actual weather.
But anyone following this site regularly knows how badly the models diverge from even the statistical profile.
No, it is not the system that is unstable it is iterative method this is unstable.
Both you and Anthony R.E are making the same mistake. Nick Stokes seems to be the only one who gets it right.
The system described by the initial ODE is perfectly stable, linear and non-chaotic. The unstable result comes from incorrect maths. The problem starts with the very first equation where we see:
Firstly Stephan-Boltzman does not give “energy” it gives the power emitted.
The energy is the integral of the power over the time step interval. Using the calculations with a step of
one this is not seen but what that term is really saying is :
So now we see the error. The area under the T^4 curve is being approximated as the rectangle with the height calculated at Tn, ignoring the fact that T changes in the interval. This assumption is not stated and clearly the author has not realised he is doing it, otherwise he would not be suggesting using it in a situation where beta makes the iterative steps very large and such an approximation becomes nonsensical.
This will work for small enough dt but when you get to large time intervals this becomes first crudely wrong, leading to oscillations around the correct solution, as the interval gets even greater it become totally chaotic. Increasing beta has the same effect as increasing dt for similar reasons.
Since T^4 is relatively straight locally ( around the 300K region being discussed ) Nick Stokes’ suggestion of a trapezoidal approximation will be far better and produces stable results.
The whole of the rest of this article is a false lemma which was caused by talking of energy instead of power, mis-specifying the original iterative equation and not stating the assumptions being made.
The ensuing discussion, however, was very interesting , especially the comments by Nick and Leo.
“Chaotic” is just another word for random or kinda-random. It only exists in the mind and on paper; not reality. It is caused by round-off and truncation and lack of truly knowing the inputs to required precision. Nothing more.
If this was correctly framed as an issue related to the need to keep the steps small enough, it may be relevant to GCMs. But the instability here is due improperly specifying the iteration, not the rounding errors.
No, it is not the system that is unstable it is iterative method.
Both you and present author are making the same mistake. Nick seems to be the only one who gets it right.
The system described by the initial ODE is perfectly stable, linear and non-chaotic. The unstable result comes from incorrect maths. The problem starts with the very first equation where we see:
Firstly Stephan-Boltzman does not give “energy” it gives the power emitted.
The energy is the integral of the power over the time step interval. Using the calculations with a step of
one this is not seen but what that term is really saying is :
So now we see the error. The area under the T^4 curve is being approximated as the rectangle with the height calculated at Tn, ignoring the fact that T changes in the interval. This assumption is not stated and clearly the author has not realised he is doing it, otherwise he would not be suggesting using it in a situation where beta makes the iterative steps very large and such an approximation becomes nonsensical.
This will work for small enough dt but when you get to large time intervals this becomes first crudely wrong, leading to oscillations around the correct solution, as the interval gets even greater it become totally chaotic. Increasing beta has the same effect as increasing dt for similar reasons.
Since T^4 is relatively straight locally ( around the 300K region being discussed ) Nick Stokes’ suggestion of a trapezoidal approximation will be far better and produces stable results.
The whole of the rest of this article is a false lemma which was caused by talking of energy instead of power, mis-specifying the original iterative equation and not stating the assumptions being made.
The ensuing discussion, however, was very interesting , especially the comments by Nick and Leo.
Of course, you and Nick Stokes are right about this not being an example of chaos. Even the simple linear equation dy/dt = -(beta)*y will exhibit the sort of numerical instability shown here if you use the forward Euler method (which is what is being used) and you make (beta*timestep) too large.
No, it is not the system that is unstable it is iterative method.
Both you and the author are making the same mistake. Nick seems to be the only one who gets it right.
The system described by the initial ODE is perfectly stable, linear and non-chaotic. The unstable result comes from incorrect maths. The problem starts with the very first equation where we see:
Firstly Stephan-Boltzman does not give “energy” it gives the power emitted.
The energy is the integral of the power over the time step interval. Using the calculations with a step of
one this is not seen but what that term is really saying is :
So now we see the error. The area under the T^4 curve is being approximated as the rectangle with the height calculated at Tn, ignoring the fact that T changes in the interval. This assumption is not stated and clearly the author has not realised he is doing it, otherwise he would not be suggesting using it in a situation where beta makes the iterative steps very large and such an approximation becomes nonsensical.
This will work for small enough dt but when you get to large time intervals this becomes first crudely wrong, leading to oscillations around the correct solution, as the interval gets even greater it become totally chaotic. Increasing beta has the same effect as increasing dt for similar reasons.
Since T^4 is relatively straight locally ( around the 300K region being discussed ) Nick Stokes’ suggestion of a trapezoidal approximation will be far better and produces stable results.
The whole of the rest of this article is a false lemma which was caused by talking of energy instead of power, mis-specifying the original iterative equation and not stating the assumptions being made.
The ensuing discussion, however, was very interesting , especially the comments by Nick and Leo.
So now we see the error. The area under the T^4 curve is being approximated as the rectangle with the height calculated at Tn, ignoring the fact that T changes in the interval. This assumption is not stated and clearly the author has not realised he is doing it, otherwise he would not be suggesting using it in a situation where beta makes the iterative steps very large and such an approximation becomes nonsensical.
This will work for small enough dt but when you get to large time intervals this becomes first crudely wrong, leading to oscillations around the correct solution, as the interval gets even greater it become totally chaotic. Increasing beta has the same effect as increasing dt for similar reasons.
Since T^4 is relatively straight locally ( around the 300K region being discussed ) Nick Stokes’ suggestion of a trapezoidal approximation will be far better and produces stable results.
The whole of the rest of this article is a false lemma which was caused by talking of energy instead of power, mis-specifying the original iterative equation and not stating the assumptions being made.
The ensuing discussion, however, was very interesting , especially the comments by Nick and Leo.
Since T^4 is relatively straight locally ( around the 300K region being discussed ) Nick Stokes’ suggestion of a trapezoidal approximation will be far better and produces stable results.
The whole of the rest of this article is a false lemma which was caused by talking of energy instead of power, mis-specifying the original iterative equation and not stating the assumptions being made.
Since T^4 is relatively straight locally ( around the 300K region being discussed ) Nick’s suggestion of a trapezoidal approximation will be far better and produces stable results.
The whole of the rest of this article is a false lemma which was caused by talking of energy instead of power, mis-specifying the original iterative equation and not stating the assumptions being made.
The ensuing discussion, however, was very interesting , especially the comments by Nick and Leo.
Sorry for the piecemeal posting , something was causing my comment to be binned and I had to break it down to find the problem : apparently we are not allowed to mention N.S. by name !!
Odd, he can post be we must not refer to him.
[Moderators]
LOL, someone has kindly dug my blocked posts out of the bin now without realising all the dupes.
Mike, agree and nicely explained.
The constant β has no physical meaning. It is necessary to keep it small.for the series to converge. It is no chaos problem but simple mathematics
Anthony R. E. and Kip Hansen are right.
Their critics—of I showed myself above to be one—insist on talking about what I will call “inherently” chaotic systems. In such a system, there is a time t for which one can ensure that any states x(t) and x’(t) resulting from different initial conditions x(0) and x’(0) will differ by less than some (perhaps small but often a nearly as large as the entire state range) value delta only by making x’(0) differ from x(0) by less than some epsilon that is smaller than the resolution to which we can physically measure the initial condition. That large difference in subsequent state results even if there is no limit on the time and state-value resolutions of our calculations; it is caused by measurement limits. It’s true that the system that Anthony R. E. calculated values for is not such a system.
In practice, though, there are indeed limits on our calculations’ time and state-value resolutions. And that means that a different kind of chaos can arise from the calculations themselves, even if the physical system being modeled isn’t itself inherently chaotic. This is what Anthony R. E. demonstrated, by way of an extreme example. He quantized the time resolution very coarsely so that the state-value resolution would make the computational process chaotic–even though the modeled system was not inherently so.
Now, in his example we could dispel the chaos by increasing the calculation resolution. But in climate-system calculations the extent to which that is possible is limited.
Reply to Joe Born ==> The reason I write about Chaos here at WUWT is the same reason that Edward Lorenz was stunned by his first attempt to use computers to predict weather, even with his very simplistic toy model — he realized that one can not “dispel the chaos by increasing the calculation resolution”. One can improve short range weather prediction and they have succeeded at this since Lorenz’s time. What they can’t do is predict weather or climate long range. They can’t because you can’t “dispel the chaos” — it is, as you say, inherent in the system.
It is the inherent Chaos that matters — we will eventually overcome the problems of rounding errors etc in computing, I believe. That part can be solved, the slight alterations to calculated values introduced by the computing process itself.
Resolution improvements stave off the advent of Chaos in calculation, but not in Nature. Nature is resolution to the atomic scale, and yet Chaos ensures in non-linear chaotic systems, such as the atmosphere and the ocean circulations.
But the problem of Chaos, which is built-in to the natural system, can not be reduced nor obviated. It is there to stay.
I have no opinion about all the hoopla here about the formula for energy balance to equilibrium given here.
You realize this implies effect without cause or at the very least that not all causes are known. Once all are known, the only thing preventing prediction would be lack of knowledge of precise values.
It’s possible that randomness is inherent in physical processes but is devilishly hard to prove.. In the meantime, avoid confusing mathematics and mathematical models and out lack of complete understanding with reality.
Reply to DAV ==> Alas, the existence of Chaos (in the Chaos Theory sense) has long since been proven by experiment in many fields — not only to exist in the mathematics of the field but to exist in the results of real world physical experiments with real material dynamical systems.
The cause of Chaos is known — that is what the study of non-linear dynamics consists of.
If you haven’t yet, please read my two essay on Chaos here and here.
If you still are not convinced, start with the Wiki page on Chaos.
My two essay include a reading list if you wish to dig deeper.
Kip Hansen:
Exactly.
As you say, although it was quantization errors in the calculations that put us (or, more accurately, Lorenz) wise to the problem, the problem that we thereby recognized arises not from calculation limitations but rather from measurement limitations.
That said, I think it’s more than plausible that as a practical matter calculational-resolution problems could prevent us from computing the behavior even of systems that are not inherently chaotic. E.g., even if it were true that we could know the climate’s initial state accurately enough in principle to narrow the range of the end of the century’s possible states adequately, arguments that Robert G. Brown has often made at this site incline me to believe that running the calculation in a reasonable amount of time still would not be possible even with the computing power whose availability a few decades into the future one might infer from Moore’s Law.
But I could be wrong.
DAV:
Sure: the system is deterministic. But the whole concept of chaos is that although the system is deterministic those initial conditions can’t be known well enough: chaos is sensitivity to initial conditions so high as to beggar our ability to measure those conditions with enough precision to constrain the future state.
Now, that inability may be “merely” a practical question. But I’ve heard–although I haven’t verified this for myself–that guys who know this stuff has shown that there are no “hidden variables” behind quantum mechanics, that, e.g., even in principle it is impossible to know the initial conditions well enough to compute the climate system years into the future. So the initial conditions may be unknowable not only in practice but also in principle. Or maybe not. Either way, what you have is still chaos.
Joe Born: Sure: the system is deterministic. But the whole concept of chaos is that although the system is deterministic those initial conditions can’t be known well enough: chaos is sensitivity to initial conditions so high as to beggar our ability to measure those conditions with enough precision to constrain the future state.
Well, it’s either deterministic or not. Chaos is an epistemological artifact. It doesn’t exist anywhere except in our minds and is evidence of our limitations. To claim otherwise is to say the physical system can change without cause. Now, this may be possible but it would be impossible to prove it did so. If a system is deterministic it means that it has defined outputs for given inputs. That doesn’t mean those outputs can be practically determined because of any number of reasons.
Kip Hansen: Alas, the existence of Chaos (in the Chaos Theory sense) has long since been proven by experiment in many fields
Er, no it hasn’t. If you think it has then how? It’s a mathematical concept and not a physical one. If anything “chaos” is just a word that means a certain kind of “random” and “random” merely means “unknown”. It is not a physical property. Again, if physical reality were “random” and “chaotic” then you are accepting effect without cause. Any model that exhibits chaotic behavior (that is, it doesn’t predict well) is incomplete.
Reply to DAV ==> I do not have the time to attempt to school you individually in a topic which is the subject of an entire library of books.
If you wish to know more:
A google search for “online courses on Chaos Theory” returns lots of opportunities.
Kip, really?
You actually believe chaos has been empirically found to be an inherent physical attribute? That it is ontological instead of epistemological and that it is an artifact arising from our knowledge limitations?
Do you know the difference between the two? If not, I would not presume to educate you.
When you say things like The problem of Chaos, which is built-in to the natural system … it seems you don’t. If the natural system were really chaotic then when a time comes where two or more possible outcomes are possible, what makes the selection? If the answer is “nothing, it just happens” then you are espousing effect without cause. Is that a fair summary of your thinking?
If, OTOH, you are saying models are incomplete; some exhibit chaotic results; and likely always will — then I agree with you. But, in no way would a chaotic model imply chaos in a natural system.Doing that would be engaging in reification where the model is also the reality.
Reply to DAV ==> Check back with me when you’ve at least read the Wiki article.
You seem to still think that “chaos” means entirely random activity or results….which it does not in Chaos Theory, which is what the rest of us are talking about here. In Chaos Theory, chaotic behavior is entirely deterministic, but unpredictable.
In that sense, Chaos is a natural feature, an inherent attribute, of real world physical non-linear dynamical systems in certain regions of their range. It has been found to be so by physical experiment in many and varied fields of endeavor. They have been studying it for fifty years.
Read Ian Stewart’s book, Does God Play Dice? for a load of real world, real physical proofs of the existence of chaotic period doubling leading to chaotic results and other Chaos Theory behaviors.
This is science, not philosophy class. You can rail and rant all you want in philosophy, but in science, you are required to read the studies and see the results before deciding that they are not so. Take a look at the evidence, even the Wiki has quite a list, if very incomplete, and get back to me again.
No. I thought I made it clear that “random” means “unknown” and so does “chaotic”. There is no random behavior in reality unless you want to believe in effect without cause. Nor could you ever show reality is chaotic regardless of your sense of the word. See below.
I agree that increasing the precision may not work. If would do so only if the model were faithful to reality. In fact, any claim that a chaotic model demonstrates chaotic tendencies in a is claiming the model IS an accurate representation of reality. You apparently want to claim you effectively know all the causes so any aberration from your predictions must be because the underlying system is chaotic. How do you know you have listed all the causes? Merely because you can’t think of any others?
Think about it. You can NEVER show that a mathematically chaotic model represents anything. How would you do so? Not by comparing predictions. They may match for a time but will eventually depart. When would you admit the model is not quite right and needs correction? What would it take? Besides, if the models predictions don’t match reality what purpose do they serve?
The only way you could show that your chaotic model might truly represent reality is for it to make accurate predictions while being chaotic. Something you admit can’t be done. That’s how models are validated. You are claiming your chaotic model is an accurate description of the underlying system despite not being able to validate it.
You don’t seem to know the difference between these two as well. Science does not actually prove anything. It only provides probable explanations which are useful. These may or may not accurately represent reality. If the explanations are useful, they may be doing so, but in the long run it’s the usefulness which matters. Whether they are showing the true underlying structure of reality (What Truly Is) or not is a matter for philosophy. Science can’t provide the answer.
When you start making statements about what really is and how we know what we know — which is exactly what you are doing when you claim chaotic systems exist in reality — then you are treading into a subject area you demonstrably know little about. After which, while still in the subject, you wave your hand in dismissal. How silly. Follow your own advice. It wouldn’t hurt if you tried to learn more about the subject you are attempting to employ and simultaneously dismiss because you don’t understand it.
As far as reading your links, the Wiki one links to a branch point and it’s entirely unclear which of those you think I should follow. Even if they ended up where you thought they might, reading what someone else thinks will not tell me why you think the way you do. Only you can do this and you want to run away instead of giving an answer.
Last reply to DAV ==> Now you’re just trolling.
If you want to learn, I have given you the references.
If you just want to rail and rant — take it elsewhere
I won’t be spending anymore time feeding your needs. .
How sad. I was asking why you think chaotic response is inherent in a natural system. You don’t want to reveal you reasoning or even narrow down a web search for possible ones. You make a pile of references and hope I find within them the kernels of your reasoning. IOW: a needle in a haystack. Maybe it’s all a muddle to you and you aren’t confident you could state why in a couple of sentences. I have said why I believe that your claim is incorrect and somehow that’s ranting and trollish. At least you didn’t play the Hitler card.
But, OK, bye.
Energy in { q/A*A= q} + energy out {-2AσTn 4 } + stored/ released energy {- mC( Tn+1 – Tn )} = 0 eq. (1)
and
Tn+1 = Tn – βTn 4 /T∞4 + β or Tn+1 =Tn +β(1-Tn4 /T∞4 ) eq (5)
“I thought it will be easier for the layman to understand the behavior of computer models under chaotic conditions if there is a simple example that he could play”.
You’re ‘aving a larf, aintcha?
https://thepointman.wordpress.com/2011/01/21/the-seductiveness-of-models/
Pointman
DAV,
I really admire your stamina. With patience like that you could become a good scientist. The first thing you could study, was a river’s motions. You could for example choose the river ‘Sjoa’, in Norway, a popular rafting site. You could find a good place to stand on one of its shores and observe the motions of water and try to predict the motions in a couple of scales (for example metric and decimetric) over a period of, say one year. Try to make weekly and monthly predictions. I think you would find out that there is chaotic (non-predictable) behaviour in the river. But take care, the river is infamous as it takes a few lives per year…
What many fail to understand is that chaos is only associated with the path from one equilibrium state to another and has no bearing on what the next equilibrium state will be consequential to some change. We see this clearly in the data where the longer we average the apparent chaos, the measured behavior of the planet converges to the requirements of the Stefan-Boltzmann LAW and the inferred sensitivity.
“The calculations are made for purposes of illustrating the effect of instability of simple non-linear dynamic system and may not have any physical relevance to more complex non-linear system such as the earth’s climate.”
An excellent self effacing comment. The arguments that climate is chaotic are by analogy. They are not mathematical proof.
The joke is that the “simple non-linear dynamic system” was …. a linear system.
The non-linear ‘chaotic’ part was an unstable algo based on mathematics and physics errors.
The large values of beta correspond to a very fast settling exponential solution of the ODE. On the scale of the numerical method it is almost a step function. The numerical error in using Euler’s rule to solve the integral is effectively introducing a significant lag in the system response that means that it does not model the original linear system as intended. The strong feedback (large beta ) plus the erroneous lag is what leads to the instability.
This almost certainly is non-linear but has little to do with the original equation.
The main lesson to be drawn from this article and discussion is how one can be led to incorrect conclusions and false attribution by computer models and numerical methods if you are not well competent in using such methods.
There are certainly implications for the many home-spun, ad hoc methods that get used in climatology. This applies to many of out hubristic “nobel prize winners”.
Indeed so. I have posted my own analysis of this problem here.
Anthony R.E.,
‘computer models with advance/d/ cinematic features that
‘laymen assume is reality.’ :
____
computer models with advanced cinematic features
only impress obamas, popes, swartzeneggers, merkels …
-laymen LIVE IN reality differing to Hollywod and tabloid pulp.
____
talking ‘assume’ pls. 1st refer 2Urself + convinient peers.
Regards – Hans
You really assume laymen
clinton, ban ki moon, obama, trump, sahra palin, trudeau, J.C.Junker, Putin, La Guarde
____
mismatch reality with
‘Consider a thought experiment of a simple system in a vacuum consisting of a constant energy source per unit area of q/A and a fixed receptor/ emitter with an area A and initial absolute
temperature, T0 . The emitter/
receptor has mass m , specific
heat C, and Boltzmann constant σ.’
Hans