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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 so

by dexen 15y ago

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-argument-that.html http://scientio.blogspot.com/2011/06/chaos-theoretic-argumen... ---- 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 http://en.wikipedia.org/wiki/Stefan–Boltzmann_law [2] http://en.wikipedia.org/wiki/Planck_law http://en.wikipedia.org/wiki/Planck_law [3] like http://en.wikipedia.org/wiki/Gulf_Stream 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 http://en.wikipedia.org/wiki/Snowball_earth ---- And a great animation of a somewhat chaotic system: http://en.wikipedia.org/wiki/Swinging_Atwood_Machine http://en.wikipedia.org/wiki/Swinging_Atwood_Machine

scarmig 15y ago
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?
- jerf 15y ago
  "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.

by dexen 15y ago

scarmig 15y ago

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?

jerf 15y ago
"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.