Friday, January 30, 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."

Today I still feel that way. I cannot bring myself to accept any type of prediction paradigm, and choose a adaptation paradigm instead. This brings me in the vicinity of Roger Pielke Sr.'s emphasis on land-use changes and Ronald Brunner's modest bottom-up alternatives. It goes without saying that I abhor such dogmas as various claims to Manage The Planet or Greenpeace's belief in Saving the Earth. These ideologies presuppose that the intelligence of Homo sapiens is capable of such feats. However, I know of no evidence to support such claims.
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.

Chaotic behaviour is also observed in natural systems, such as the weather. This may be explained by a chaos-theoretical analysis of a mathematical model of such a system, embodying the laws of physics that are relevant for the natural system.
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 method of science depends on our attempts to describe the world with simple models. Theories that are complex may become untestable, even if they happen to be true. Science may be described as the art of systematic oversimplification, the art of discerning what we may with advantage omit."

If Popper had known of the predictability problems caused by the Lorenz paradigm, he could easily have expanded on this statement. He might have added that simple models are unlikely to represent adequately the nonlinear details of the response of the system, and are therefore unlikely to show a realistic response to threshold crossings hidden in its microstructure. Popper knew, of course, that complex models (such as General Circulation Models) face another dilemma.

I quote him again: "The question arises: how good does the model have to be in order to allow us to calculate the approximation required by accountability? (…) The complexity of the system can be assessed only if an approximate model is at hand."

From this perspective, those that advocate the idea that the response of the real climate to radiative forcing is adequately represented in climate models have an obligation to prove that they have not overlooked a single nonlinear, possibly chaotic feedback mechanism that Nature itself employs.

Popper would have been sympathetic. He repeatedly warns about the dangers of "infinite regress." As a staunch defender of the Lorenz paradigm, I add that the task of finding all nonlinear feedback mechanisms in the microstructure of the radiation balance probably is at least as daunting as the task of finding the proverbial needle in the haystack. The blind adherence to the harebrained idea that climate models can generate "realistic" simulations of climate is the principal reason why I remain a climate skeptic. From my background in turbulence I look forward with grim anticipation to the day that climate models will run with a horizontal resolution of less than a kilometer. The horrible predictability problems of turbulent flows then will descend on climate science with a vengeance.
The short version: climate models can't predict anything as they currently stand because they are too 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 Classical Values

1 comment:

tonemcd said...

I studied chaos theory for a while - long enough to realise that the maths involved was well above the level I was able to handle, so I dropped back to theoretical physics ;)

And now I'm wondering why I didn't see this earlier - of course the models as they exist today are too coarse. But what I completely missed (to my shame) is the sensitive dependence on initial conditions angle. Which makes the whole modelling/prediction of the climate impossible - even in principle.

Thanks for this!