More chaos than you can shake a stick at.
Guest post by Andi Cockroft
(Anyone familiar with the Moody Blues should recognise the title – from “Higher and Higher”)
As some will have learned by now, I do not possess the scientific skills of the regular WUWT contributors to engage in in-depth evidential-based posts. Rather I like to think that just like the great unwashed masses, I have an intellect and an enquiring mind, and want here to share my musings and seek feedback to better help me (and hopefully other readers) understand some of the more complex subject matter. For my shortcomings I apologise. For raising questions requiring answers I do not!
Way back in January 2011, Phil Salmon posted here on WUWT “Is the ENSO a nonlinear oscillator of the Belousov-Zhabotinsky reaction type?” – His alternate title “Standing on the shoulders of Giant Bob” Perhaps my alternate title should be the same. I have read and re-read Phil’s definitive article, and as Isaac Newton was believed to have complained of the three-body problem:-“his head never ached but with his study on the moon”. So in my own way, I want to raise issues associated with what I see as true chaos, and whilst following (albeit very slowly) Phil’s work, that was nonetheless focused towards ENSO and away from the general Climate Models that I want to investigate.
Why a butterfly? Most students of chaos theory will know the “butterfly effect”, whereby it is asserted that in a chaotic non-linear system, a butterfly beating its wings in say Brazil could inject such feedback as to disrupt the airflow sufficiently that many years later a tornado would form over Texas.
Similarly, travelling back in time and moving the butterfly a few centimetres would cause the tornado not to form, or form elsewhere.
Also note that the Quantum version of the Butterfly Effect (the Quantum Butterfly), is a different beast altogether – although some parallels are drawn.
Ascribed to Edward Lorenz, the US mathematician and meteorologist who sadly passed away a few years ago. Initially Lorenz referred to a Seagull flapping its wings forever changing the weather, but later in a 1972 speech entitled “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas”, the butterfly analogy was born.
It is Lorenz’s work on meteorological chaos that I want to look at specifically.
In or around 1961, vacuum-tube computers arrived at Lorenz’ disposal, and not surprisingly the first weather model was born as a set of a dozen or so differential equations involving such things as temperature, pressure, wind velocity etc.
During a re-run of this early model, Lorenz is believed to have restarted the program in the middle of its run by entering a variable to 3 decimal places – to his surprise the results were completely at odds with what was achieved earlier.
Restarting and re-entering the variable to its full 6 decimal places produced a repeat of the initial results – from this Lorenzo drew the inevitable conclusion that with his dozen or so equations, even a miniscule variation on input is capable of creating massive change in output.
But why such a radically different outcome for such a miniscule difference in input?
It transpires that the equations were non-linear, with a so-called “great dependence on initial conditions” – change the initial conditions even slightly and a large change in output is observed in a later state.
The conclusion that Lorenz drew, was that given that such small variations can create such massive variation in output, it was impossible to “model” a weather system.
This “sensitive dependence on initial conditions” was destined for higher things however.
Around this time, Lorenz published perhaps his most important paper “Deterministic Nonperiodic Flow”, and with it was born Chaos Theory, and the concept of the Lorenz Attractor.
(For those wanting to investigate the Lorenz Attractor further, I would recommend reading Phil’s original article here, or Wikipedia here )
I’m not sure if there is any relationship here to Peter, but James Gleick’s bestseller, ‘Chaos: Making a New Science’ built on Lorenz’ musings, became the mantra for many to follow in all walks of life:- finance, science and even time-travel effects in science-fiction. It is now bandied around in insurance, marketing and business boardrooms throughout the world.
Being quite a reserved individual, Lorenz was taken aback by the devotees to his speech. ‘I was just trying to determine why we didn’t have better luck with our weather forecasts, I never reached a point where I believed the butterfly was a scientific fact. At most, it’s a hypothesis. I never expected it to become so huge outside meteorology.”
So my question now is, given that we cannot even begin to measure things such as SST to anywhere near 3 decimal places, how can we expect low-accuracy and highly volatile data, entered into chaotic models of even more chaotic systems to produce anything of significance?
If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?
Andi
If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?
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Simple – hubris and an agenda that relies on propping up AGW to sustain your ability to make a living as a “scientist”
Once again, we find that laymen’s use of the mathematics of chaos is a problem. The shame is that people think that if they understand an analogy of something then they understand that thing. Sorry folks but to understand chaos you have to understand the mathematics of chaos. Also, if one does not understand the mathematics, don’t tell people you do understand it.
Specifically the X-Y plot showing the initial conditions which look close to one another leading to four different final states, i.e. A, B, C, and D, does not show chaos. In fact, it shows the opposite of chaos, i.e. final states that have zero dimention. Chaos is not “sensitivity to initial conditions.” Chaos is that DESPITE a sensitivity to initial conditions, the trajectories tend trace out shapes that look very similar, i.e. Lorenz’s butterfly plot of his three variable system. No matter what initial condition is chosen (with the exception of the point 0,0,0, IIRC) the trajectories trace out shapes that look very much the same. The strange attractor is INSENSITIVE to the initial conditions.
“If Lorenz gave up when he discovered weather to be totally chaotic and unpredictable beyond a few days, what makes the Climate Modellers believe they can do better forecasting years or even decades ahead?”
Because the Climate Modelers believe that weather is statistical and Global Warming is not. It’s that simple. They analyze climate and think of weather as a perturbation. One could do all the analysis of weather systems and still not convince the GCM modelers that they are wasting their time.
“The conclusion that Lorenz drew, was that given that such small variations can create such massive variation in output, it was impossible to “model” a weather system.”
No. That is not what he said. We model weather systems all the time. There is a website called weather.com that receives millions of visits each day to get weather forecasts. They are quite successful. What Lorenz said was that LONG TERM modeling may be impossible. It is that kind of precision of thought which is not possible if one doesn’t understand the mathematics.
I am plagiarizing the information others have taught me, but here are some observations. Chaos is a form of stability. You are stable within the attractor, but cannot predict where within the attractor you will be. It is possible to leave the attractor and enter another with a different “shape” if the system conditions change sufficiently. For me, the primary question regarding deterministic chaos and climate is “What is the nature of climate sensitivity?” Paleo records are frequently used as evidence that sensitivity is significantly positive. So why do we believe sensitivity exiting the Younger Dryas was the same as it is now. Is sensitivity deterministically chaotic? I would be very surprised if it was either a constant, or linear and predictable. Therefore, I would be very surprised if we can ever model climate. We will never be able to do anything except make educated guesses based on our belief about the shape of the attractor.
@Kasuha – Nice post.
“We can determine the probabilities of it being found within particular bands of the regions that it may occur within.”
No, that cannot be done. Probability theory and Chaos theory have nothing to do with each other because that doesn’t work. Back in the late eighties/early nineties when ‘Chaos: Making a New Science’ came out, I got very excited about that idea, went to one of my applied mathematics prof’s and asked him about it. He said no but I took a PhD level prob and stats class anyway just to figure it out for myself. What I got was a good understanding of theoretical prob and stats and no help what so ever in modeling chaotic systems.
At a more fundamental level “the regions that it may occur in” can’t be defined and hence one can’t properly box the strange attractor. It’s the classic Mandelbrot “How long is the coast of Britain” problem.
Too many things left off the table – for example:
1. fundamental difference between temporal chaos & spatiotemporal chaos.
2. strange nonchaotic attractors.
3. global constraints indicated by EOP …begs questions about aggregation spatiotemporal scales – very serious stuff that (very unfortunately) flies over, around, or through (in one ear out the other) the heads of ~99.999% of the people involved in the climate discussion.
Going severely wrong is as simple as making an untenable base assumption. FOR SURE there are mainstream misconceptions which have run wildly unchecked in an abstract vacuum. Spatiotemporal chaos in a shape-shifting box – where solar & lunisolar cycles DO constrain the shape-shifting in a previously underappreciated manner – might be the closest thing to common ground that can be used as a practical conversation starting point. (It’s foolish to waste so much as a single breath trying to frame terrestrial climate as temporal chaos.)
I think the reason these models “work” (ie don’t show chaotic outputs) is that they do something like this:
Let i=1,…n denote a cell (ie position on the earth’s surface) and let T(i,t) be the temperature in cell i at time t. The equation for CO2 (concentration C, same for all cells) “forcing” is something like
T(i,t+1) = T(i,t) + kf(C)
for some parameter k and function f. If heat capacity is the same for all cells the mean global temperature G(t) is given by a simple mean over all i=1,n of the T’s and
G(t+1) = G(t) + kf(C)
Now apply a “mixing equation” which stirs things up by shufffling the i’s about (eq T(i,t) becomes the new T(j,t) etc).; more generally each T(i,t) is some linear combination of the others). If we now take the mean of the new temperatures we are getting the mean of the same numbers as before, we have just changedd the order). So the mean global temperature G is the same as before mixing.
That is, mixing does nothing to obscure the important part, kf(C). So if you incorporate some sensible constraints (eg conservation off energy) to prevent chaotic collapse it does matter what mixing equation is used (remember, they make no claim to local prediction, only global)
A fractal model is what is required – but the computing power required would be enormous. Might as well get some mice to build another planet just for testing. Just make sure we get to the Vogon planning office in time.
So when a butterfly sneezes, does his proboscis pop out like a little party favor ?
@davidc – Google “Hadley Cell”. On a side note, the Hadley cell on an integral scale is intellectually similar to Lorenz’s cell on a differential scale. Lorenz simplified the “mixing equation” to a simple three variable system. He then tried to numerically integrate it and found that he could not get stable solutions, i.e. no points A, B, C and D.
fredb says:
March 15, 2012 at 7:35 am
Weather is not climate. To conflate the initial condition dependency problem of weather forecasting with the boundary condition problem of climate simulation is fundamentally erroneous.
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If the climate models are solving boundary value problems, i.e., the solution is a function of the spatial variables, then it’s mathematically impossible for the climate models to predict the future.
The Navier-Stokes equation is a boundary value problem with initial conditions.
In fact, you can view the Lorentz equation (the 3 mode truncation of the Benard heat convention problem) as a metaphor for the Navier-Stokes equation in 2 spatial dimensions.
More formally, the effect of mixing on global mean temperature:
T(i) = temperature in cell i befoer mixing; n cells
H(i)=CpT(i) = heat content of cell i before mixing; Cp=heat capacity, same for all i
G = global mean temperature before mixing = (1/n)[sum over i of]H(i)/Cp
After mixing the heat content of cell i, Hmix(i) is
Hmix(i) = [sum over j of]L(j,i)H(j)
where L(j,i) is the fraction of the heat content of cell j transfered to i (including itself; ie heat retained by cell i).
Conservation of heat during mixing:
[sum over i of]L(j,i) = 1;
that is the total of heat transfered from cell j to all other cells (including itself; ie heat retained) is the original heat content of cell j.
The global mean temperature after mixing, Gmix, is
Gmix = (1/n)[sum over i of]Hmix(i)/Cp
=(1/n) [sum over i of][sum over j of]L(j,i)H(j)/Cp
= (1/n)[sum over j of][sum over i of]L(j,i)H(j)/Cp; reverse of of the two summations
= (1/n)[sum over j of] H(j) [sum over i of]L(j,i)/Cp; H(j) independent of i
= (1/n)[sum over j of]H(j)/Cpas G before mixing
since by conservation of heat [sum over i of]L(j,i)=1.
But this expression for Gmix is the same as the expression for G (before mixing; except for the index j in place of i).
I can’t think of a way to describe mixing such that the outcome is not a linear combination of the original values, but maybe someone can point out my error. If I’m right then whatever equation they use for mixing will have no impact on the global mean temperature calculated. So it is determined solely by the assumed forcing equation for the effect of CO2 on temperature.
Same result with a weighted mean (weights W(i))
G = global mean temperature before mixing = (1/n)[sum over i of]W(i)H(i)/Cp
with n = [sum over i of]W(i)
provided the same weight are applied before and after mixing. So if weights are applied to the model to reflect quality of sites, it’s the same.
Perhaps I should point out that by “mixing” I mean something like the application of some GCM.
If the models are written with internal “constraints” to stop them leaving the rails, then how much of this mathematics becomes a moot point? Protein-folding models are far simpler systems yet exhibit the same problems.
@davidc – Get Lorenz’s 1963 paper. All that you are looking for is right there. IIRC, he even has a few words on the discretization.
Dinostratus, thanks for the reference. From Wikipedia this 1963 one is on atmospheric circulation (I haven’t found the original yet). My point is that if the sole object of the modelling is to predict the mean global average temperature (which I believe the IPCC says; no modelling of local effects) the method of modelling the atmospheric circulation (“mixing”) does not matter. Whether this is by Lorenz, Navier-Stokes or anything else it all comes out the same. The key equation is of the form
G(t+1) = G(t) +kf(C)
where the advance of the global average fro time t to t+1 depends only on the assumed temperature change in response to CO2. I’m assuming a computational scheme:
1. Calculate temperature change in individual cells (I mean cells in the grid used for a finite difference calculation)
2. Calculate the new distribution of heat by mixing/circulation
3. Calculate mean global temperature
4. Goto 1
My point is that Step 2 (eg Lorenz, Navier-Stokes) makes no contribution to the final result for is smoke and mirrors: this is too hard for you to follow, just trust me.G.
If you think I’m trying to establish an awesome generality for the models, I’m trying to establish the opposite: that they are trivial. For example over the period from about 1970 to about 1998 the reported global average temperature increased approximately linearly as did CO2 levels. So in place of the general term kf(C) we have a linear term to give something like
G(t+1) = G(t) + a + b.deltaC
where deltaC is the change in C over the time interval (t,t+1). So, simple linear regression is what’s needed to find a and b. I think the only function of the GCM is smoke and mirrors (this is too hard for you, just trust me).
I’m going to add three decimal points to every temp I’ve recorded for the last 25 years, go out and buy 4 sets of Russian dolls, and complete a research grant request. I’m ready to theorize!
michael hart says:
March 16, 2012 at 2:48 pm
If the models are written with internal “constraints” to stop them leaving the rails, then how much of this mathematics becomes a moot point? Protein-folding models are far simpler systems yet exhibit the same problems.
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The motion is determined by the driving and diffusive forces.
It’s also a plot in phase space.
The major deficiency of the Lorentz model is the extreme truncation to 3 modes.
There are 14 modes so the attractor has a set of rich dynamics.
It turns out the most sensitive parameter of the Lorentz equation is the number of modes.
“this is too hard for you to follow, just trust me.G.”
No problem. I first saw the NS equations in their full glory over twenty years ago. That’s not enough time to understand them so I’m still working on it.
“My point is that if the sole object of the modelling is to predict the mean global average temperature (which I believe the IPCC says; no modelling of local effects) the method of modelling the atmospheric circulation (“mixing”) does not matter.”
That may be one small step for you but it is a giant leap for mankind. Your assumption is that the “mixing” doesn’t matter. Your equations then show that it all comes down to other variables then you conclude that the mixing doesn’t matter. In fact, the integration of the mixing terms is nontrivial.
What Al Gore’s professor, Dr. Revelle, showed was that for a zero dimensional model such as yours, global warming happens as their is an amplification cycle of human generated CO2 to water vapor, Since then everyone’s been arguing over the water vapor issue. More detail has been added to the models and hence fewer assumptions need to be made, i.e. mixing models, etc.
“the method of modelling the atmospheric circulation (“mixing”) does not matter”
That is the fundamental question that Lorenz was asking. He modeled a differential buoyancy driven element to determine if the mixing could be integrated. He thought not (obviously) but others have come along making assumptions that overcome Lorenz’s difficulties. Whether those assumptions are valid or not is up for debate. Sometimes we come up with very good mixing models and sometimes we don’t. Over the years I’ve noticed that when simplifications of the NS equations retain a coupling between the elliptical terms and parabolic terms, our ability to model the flow is poor.
But what do I know?
Dinostratus,
An analogiy might help to make my point. I am drinking a cup of cofee and I have added a teaspoon of sugar and stirred it with the teaspoon. I could in principle apply the Navier-Stokes equation to describe what happens as I stir (tricky, time dependent boundary conditions), or some other model. As a scientist I insist that my model (of stirring/mixing) be tested against observations. But I also insist that the observation that is to be used to test my model is the average concentraion of sucrose (sugar) over the entire cup. Using concentration units of teaspoons/cup we know what the answer will be: 1 teaspoon/cup. To get this answer the stirring/mixing model has to do just one thing: satisfy conservation of mass. If it does this by any means at all (Navier-Stokes, something to do with Lonenz, patterns in tealeaves constrained somehow to satisfy conservation of mass) then when I add up all the little bits of mass being stirred around I get 1 teaspoon. The volume of the cup remains at 1 cup. The average concentration (total mass/volume) is then 1 teaspoon/cup. So the model is validated (all of them, and any stirring model that satisfies conservation of mass that you or anyone else might come up with).
I think I understand. The sugar in your example is called a passive scalar. “Passive” because it does not affect the mixing mechanics. What you describe is called a “Stirred Reactor”. The simplest reactor model is a “Perfectly Stirred Reactor.” The test to see if a reactor is perfectly stirred is the Damkohler Number. If the reaction rate is slow compared to the mixing rate we say the Dh<<1 and the reactor is perfectly stirred and the fluid mechanics don't need to be modeled. We just assume that the reactor is homogenous.
The giant leap you are making is the assumption that the sugar is a passive scalar.
Temperature, in the case of the atmosphere, is NOT a passive scalar. It is the temperature gradient that drives the buoyancy driven flows (not directly but through density). This is important because it is the temperature at the surface that drives the liberation of CO2 and water vapor from the oceans. The faster the convection (or "advection" in the language of atmospheric scientists), the cooler the surface and the less liberation of CO2. Moreover, the higher the surface temperature, the more water vapor is loaded into the atmosphere and then carried up to make clouds.
Spencer has spent a lot of time on the last issue and it is that water vapor / cloud model that is (and always has been) the weakest link of GCM models.
Using your analogy,if the sugar is an active scalar then the sugar is also a spoon and makes sugar (or not) as it settles to the bottom. One can't say for sure that there is only one teaspoon of sugar.
PS – If you want to read something really whack, read the comments on global warming by Octave Levenspiel, the author of "The Chemical Reactor Omnibook" which is the seminal reference on reactor modeling.
Dinostratus, Yes, the analogous quantity should be heat, but since the heat capacity will be constant this doesn’t alter the argument in a significant way.
But you are misssing my main point which is the use of the climate models to estimate global mean temperature, not local temperatures. We are told that the models don’t predict local effects. Some of us might think it’s odd that you could calculate a global mean temperature for the model without knowing local values (after all that is how they estimate the so-called “observed” global mean temperature). My suggestion here is that that is how it’s estimated in the models but the local effects in the models don’t correspond to any real local effects. But – main point – if you stick to global mean temperature as the value reported it is irrelevant whether the model works at that level or not (provided it satisfies some conservation of heat constraint).
Back to the coffee. Suppose you added the sugar as a syrup containing the equivalent of one teaspoon of solid sugar, carefully layered on the bottom. The appropriate model for “mixing” would then be the diffusion equation. The distribution of sugar in the coffee is quite different from the stirred example. But if you define the average concentration in the cup in the obvious way as amount/volume, solved the diffusion equation for some value of time, integrated the concentration-position profile to get an average concentration, you would get … 1 teaspoon/cup. If you did the calculation with the wrong diffusion coefficient it wouldn’t matter, still 1 teaspoon/cup. If you made a mistake and used 1/D where it should be D, again doesn’t matter, still 1 teaspoon/cup.
The only way to see which model is correct is by sampling local concentrations (say, by drinking the coffee). The stirred model would clearly be wrong when you find that the first few mouthfuls were not sweet enough, the last few much too sweet (but on average, just as you like it).
PS There is a mistake in my algebra in my earlier post.
If temperature is sugar then the assumption you are making is that the sugar/temperature is a conserved passive scalar. In the case of the atmosphere, temperature is not conserved. It is increased or decreased depending on the temperature gradient. It is not passive. Buoyancy drives the mixing rate. Energy is conserved. Not temperature.
Just because temperature can be integrated doesn’t mean it is conserved. One can integrate the velocity in the coffee cup too. It integrates to zero (assuming you let it sit on a table and haven’t thrown it at me). Velocity however is not conserved.
You really need to read Lorenz’s paper. You’re asking good questions. Essentially he tries to numerically integrate a differential cell. He finds that he can’t because of the deterministic non-period flow that the model suggests.
Again, using the syrup example, the syrup (temperature) gradient creates (or destroys) syrup (temperature). The strength of the temperature gradient either liberates or traps heat, depending. It’s not the instantaneous integration that is the problem. It’s the time varying evolution of those integrations that causes the problem. Taking your equation, G(t+1) = G(t) +kf(C), the problem is that k depends on f(CO2, H2O, T, dT, etc.). k depends on clouds, H2O(l) & H2O(s), and clouds are not modeled well at all.
As far as the CGM models, no they don’t get local results right. (Curry has gained an undeserved reputation as an skeptic because she cautions not to predict local weather changes using CGM’s.) However the modelers ASSUME that the differences are statistical in nature. That is, they average out to zero (the Central Limit Theorem applies). Do they? Lorenz says no. Tennekes has a good paper on that issue.
I do agree with one of your main points. The CGM folks use models that are insensitive to individual cell variations. Cell variations are damped out to the point that the computers can make projections. New computers are then purchased, the models are lessened up a bit and then re-run on faster computers. It’s a nice budget scheme for National Lab and University managers. They did the best they could with what they had and would like more capability to do better. Cha-ching.
OK, the Lorenz paper next (algbraic mistake fixed, same result – ie if mean global temperature is the quantity you want to predict, the circulation part of the model makes no significant contribution)