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January 29, 2009
Strange Attractors
I was reading Hendrik Tennekes on climate models and learned some very interesting mathematics. Let me walk you through it. It isn't hard and it turns out to be very beautiful. First off let me give you the flavor of the man. Here is something he says that I really like: "Physicists dream of Nobel prizes, engineers dream of mishaps." So true. When ever an aircraft goes down I want to know if it was something I worked on. If the answer is yes I say to myself: "Pray to God it wasn't something I did." OK. Climate models. "The constraints imposed by the planetary ecosystem require continuous adjustment and permanent adaptation. Predictive skills are of secondary importance."Next up we start getting into ideas from the mathematics of chaos. The math was first found by by Edward N. Lorenz a meteorologist who founded chaos theory and found the Lorenz attractor. A bit on chaos theory is in order. In mathematics, chaos theory describes the behaviour of certain dynamical systems - that is, systems whose states evolve with time - that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.Here is a look at one version of a Lorenz Attractor: Lorenz wrote a book on the subject that will be helpful to those that want to get deeper in the subject: The Essence of Chaos. Ok. Now that you have some background lets continue on with weather and climate models. Back to Lorenz. Complex deterministic systems suffer not only from sensitive dependence on initial conditions but also from possible sensitive dependence on the differences between Nature and the models employed in representing it. The apparent linear response of the current generation of climate models to radiative forcing is likely caused by inadvertent shortcomings in the parameterization schemes employed. Karl Popper wrote (see my essay on his views):The short version: climate models can't predict anything as they currently stand because they are to coarse to properly model the phenomenon in question. When they get fine enough they won't be able to predict anything because chaos of the climate system and the models will take over. I agree with Hendrick on the solution to the climate problem: preparation for adaptation to what ever happens is effort well spent. Trying to hold back the tides is a waste of time, effort, and accumulated capital. H/T icarus at Talk Polywell Cross Posted at Power and Control posted by Simon on 01.29.09 at 11:58 PM |
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It's really hard to get systems of nonlinear equations to converge. When they do, we suppose we have the right answer. We might be wrong, because sometimes (as the article points out) small differences in initial values can lead to very different convergences, or to some other result entirely.
But having the right answer to our system of equations only means that the model works as designed, not that it matches reality.