Chaotic systems and randomness | Hacker News Reader

Chaotic systems and randomness

tamino.wordpress.com

31 pointswhiskers7 comments

There are major problems with the logic in the article. He's basically trying to argue that the average of a variable in a chaotic system over some finite period of time can't drift or change dramatically due to chaos alone. His only support for this is that it doesn't happen in the simplest possible one dimensional discrete chaotic system. What about the Lorenz attractor? If you aren't familiar with it, it basically acts like two well behaved oscillators that you switch between after some "random" amount of time. If you take a time average of a space coordinate with the Lorenz attractor you'll see chaotic swings between periods of relative stability in the two oscillators. As you make increasingly complicated chaotic systems with more and more variables you open up the possibility of having all sorts of longer term behavior that would effect what he calls "climate". He is absolutely wrong about this.

When you increase greenhouse gases and therefore inhibit heat loss, you change the dynamics — the “equations of motion” as it were — and that will change the climate. In ways that are predictable. He's wrong here as well. From bifurcation theory we know that tiny changes in the parameters of a dynamical system can have huge seemingly unpredictable effects on the behavior. Subtle differences in models can have huge impacts on where and how bifurcations occur. We have only various educated guesses that might very well be dramatically wrong.

By the way, I'm not arguing against climate change. I just think that this article is terribly wrong.

As someone who doesn't particularly believe in the Standard Dogma of warming, I believe it's my turn to trot out an old chestnut: "If you don't understand [chaos theory], you shouldn't be arguing about it. [Chaos theoreticians] have spent a lot of time working these things out and are very smart, do you really think you're going to outsmart them?"

(And I do accept the credentials of the original poster on WUWT; you can't help but learn about chaos if you're in that career path. It's staring you in the face every day.)

The really interesting thing in the original post is the characterization of the Lyapunov exponent in terms of bits lost per time unit. If you understand information theory and what that really means, it is an extraordinarily powerful argument that climate is truly, fundamentally not predictable, on a deeply, profoundly mathematical level.

Part of why I don't really accept the Standard Dogma is precisely my impression that the weather simulators have no idea what realm they are operating in; they seem to have the same paper-thin understanding of chaos theory as this poster, and I have seen no evidence that they understand information theory or even know it exists, since I have never seen it mentioned by them or any of their proponents, despite the powerful, mathematical upper bounds it puts on their efforts, ones which are rather tighter in nature than the predictions they so breathlessly make.

I'm not arguing "against" climate change, I'm arguing against the value of any attempts to predict it beyond the upper limits established by mathematics.

There's some good stuff in the comments there. Your argument "I'm arguing against the value of any attempts to predict it beyond the upper limits established by mathematics." is ignoring the physics, which limits the problem greatly.

I'm curious (genuine question): we know that if we dropped the Earth into the Sun, the climate would change substantially. And yet this is a prediction about gross properties of a chaotic system that no one would dispute.

How does your point of view account for this prediction from physics versus predictions of greater CO2 concentrations causing an increased greenhouse effect, which also derives from physics? Is it simply the margins of error for different parameters?

Fragments of the mentioned post [1]; it's worth reading in whole as it explains the matter very neatly.

> Chaotic systems are not entirely unpredictable, as something truly random would be. They exhibit diminishing predictability as they move forward in time, and this diminishment is caused by greater and greater computational requirements to calculate the next set of predictions.

> Computing requirements to make predictions of chaotic systems grow exponentially, and so in practice, with finite resources, prediction accuracy will drop off rapidly the further you try to predict into the future.

> The difference between the initial conditions is minute, but the two series diverge for all that. This illustrates one of the key things about chaos. This is the acute sensitivity to initial conditions.

> If you try to make predictions from your model, any minute inaccuracies in your guess of the initial conditions will result in your prediction and the result diverging dramatically.

> This divergence grows exponentially, and one way of measuring this is called the Lyapunov exponent

> It also gives us a bound on the quality of predictions we can get if we try to model a chaotic system.

[1] http://scientio.blogspot.com/2011/06/chaos-theoretic-argumen...

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And my long ramblings (I'm already sorry for posting that):

The temperature of Earth stabilizes (`thermal equilibrium') when energy input equal energy output. There are two serious sources of energy for Earth: Solar radiation and radioactive decay in Earth's body (minor sources would be lunar tides and chemical reactions, but I guess the former varies little and later is of limited capacity and wouldn't matter much in the longer run).

If Earth was just large uniform sphere, with no magnetic field and no atmosphere, predictions of surface temperature would be simple: from [1], the temperature would have to be adequate to match thermal input. The temperature would have to provide energy output equal to: internal heating + solar radiation hitting earth - reflected radiation (the part of solar radiation that did not transfer to the Earth).

The laws [1][2] of thermal radiation are simple for case of idealized black body with uniform temperature, however Earth is not such one (unless dropped onto the Sun ;-)). We have several features that reflect some solar energy back to space, absorb solar energy, or emit energy from Earth's surface. The temperatures on Earth are not uniform, and there are large active energy transfer mechanisms [3]. Some of the features add or subtract energy emission; those can be estimated quite reliably.

There are threshold effects, like possible methane release from hydrates on oceanic floor, or large areas of vegetation disappearing because of temperatures dropping too low. Those are harder to estimate reliably, as very small variation in temperature will change wide areas of Earth's surface or atmosphere. We probably don't know all of them yet.

There are some negative and positive feedback loops [4], like desert reflecting more than vegetation, and also icecaps reflecting more than vegetation; deserts grow with temperature increase, icecaps grow with temperature decrease. Positive feedback are exceptionally hard to estimate reliably.

Some of the features multiply (!) effects of other features; take clouds for example; they block a lot of solar energy from reaching whatever feature is below them; yet they depends on several climatic factors -- some of them increase, othres decrease formation. Now that's got power to throw simulations off-track even further.

To wrap it up: complex systems with many intermingled nonlinear factors; in effect very sensitive to input parameters. And there are countless input parameters...

[1] http://en.wikipedia.org/wiki/Stefan–Boltzmann_law

[2] http://en.wikipedia.org/wiki/Planck_law

[3] like http://en.wikipedia.org/wiki/Gulf_Stream that causes Warsaw to have pretty much same temperatures Toronto has, only that it's 8.5 degrees further to the north.

[4] there's speculation they may lead to runaway effects like http://en.wikipedia.org/wiki/Snowball_earth

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And a great animation of a somewhat chaotic system: http://en.wikipedia.org/wiki/Swinging_Atwood_Machine

Thanks for the thoughtful writeup. But I don't get specifically how this doesn't prove too much--even though climate is a complex system, we know (right?) that throwing it into the center of the sun would increase mean global temperatures. But if I said that chaos and information theoretic arguments prevent us from making any predictions, I'd rightly be laughed off the stage.

What is the differentiating factor that makes those arguments apply to models looking at CO2 concentrations as exogenous forcers but not being-in-the-center-of-the-sun?

"But if I said that chaos and information theoretic arguments prevent us from making any predictions, I'd rightly be laughed off the stage."

"Information theory bounds our ability to predict" is not the same as "Information theory prevents us from giving any predictions". You could predict the temperature of "the Earth" (for some suitable definition) within some bounds, but if you tried to do it to within a tenth of a degree Celsius for a hundred years you would continue to encounter difficulty... rather more than you have right now, actually, I'm quite confident the Sun's Lyapunov constant will strip bits from you much faster than Earth's current climate does.

Also, you're engaging in a bit of tautology without realizing it. The entire concept of "forcing" is one divorced from a chaos-informed concept of weather, it's part of the reason I really don't think they understand chaos. The "forcings" are themselves subject to the prediction problem, so you may be able to predict what the system will do immediately after the application of them, but you don't know what will happen later. In Earth's specific example, what if the hotter climate ends up triggering Yellowstone? Or that methane clathate catastrophe? Or something entirely unexpected? Much more realistically, what if the "forcings" tend to cancel themselves out over longer times? There's been a couple of good papers that suggest that all the water on Earth (oceans, etc) are very, very strongly effecting the climate by virtue of cloud production in the tropics such that it is extremely strongly forcing the climate to look the way it does now, and all your "forcings" may be nothing next to it. Or that may be totally wrong. By the time you're done caveating "forcings" in light of what a chaotic system can do, they aren't a very useful approach to the problem.

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