Math without physics, is not physics.
There’s yet another mathturbation post at WUWT. This one, by Andy Edmonds, argues that because weather is chaotic (in the mathematical sense), it’s impossible to model climate. In fact that’s the whole argument — a lot of words, but it boils down to nothing more.
The idea that a chaotic system can’t be predictable is right — and it’s wrong. It certainly can’t be predicted, at least not long-term, because of the phenomenon of “extreme sensitivity to initial conditions.” The slightest change in the starting conditions eventually (usually sooner rather than later) leads to a drastic change in the state of the system. Even if you know the exact “equations of motion” and the exact starting condition, to predict long-term you’d need infinite computing power and to calculate with an infinite number of significant digits. That can’t be done. And that makes prediction, frankly, impossible.
At least, detailed prediction over the long term is impossible. Weather is like that: long-term detailed prediction is beyond our ability and probably always will be. We can predict it short-term, up to about a week or maybe two at the most, but we have no hope of predicting, with any accuracy, whether or not it’ll rain in Boston on April 1st, 2095. In fact the study of mathematical chaos was jump-started by Edward Lorenz when he studied computer models of weather.
But even though a chaotic system can’t be predicted, its statistical properties often can be. The statistical properties of weather — it’s long-term average and variation — are referred to as climate. Those who believe that chaos in weather makes climate modeling, or climate prediction, impossible, have failed to comprehend the difference between weather and climate.
Allow me to illustrate. Let’s use a simple chaotic system: the logistic map. It’s a simple function of a single argument
I’ll actually use an equivalent form
This is the same as the previous equation, except I’ve replaced the old x variable by a new one, equal to the old one minus one half.
For values between 0 and 4, if we start with an x value between and , then apply the logistic map to get a new value of x, the new value will also be between and . We’ll then apply the logistic map to that new value to get an even newer value of x, etc., repeating the process as many times as we wish. By doing so, we can generate a time series of values.
The logistic map with is chaotic (actually chaos begins about ), so the time series can’t be predicted long-term. It’s not random — it’s perfectly deterministic — but the values will certainly seem random and will indeed defy prediction. Weather is like that. With this value of the parameter, it’s
I’ll use the parameter (the same as used by Andy Edmonds in his post) to generate a time series of x values which we’ll pretend are monthly temperature anomaly. They don’t really behave like temperature anomaly in detail (they follow a different distribution) but they’ll illustrate the point. Then we’ll compute annual averages to get a time series of yearly average temperatures. This time series will also be chaotic.
I did so for 1000 simulated years, and got this:
At least visually, it’s plausible as a time series of annual average temperature. It appears to exhibit random year-to-year variation, but it’s truly chaotic. But what about the climate? Here are 10-year averages (with error bars) of this temperature time series:
Although the simulated weather is unpredictable, the simulated climate is not. It’s stable. Almost all of the 10-year averages are within their error limits of zero, and those that aren’t are no more frequent nor more extreme than we’d expect from random fluctuations. If you test this series for significant change (using either the analysis of variance, or the non-parametric Kruskal-Wallis test since they don’t follow the normal distribution), there’s no climate change — not even close. The “weather” in this system is unpredictable, but the climate is not: it’s stable.
Computer models which simulate climate, do so by simulating weather. Nobody seriously expects them to get the weather right because nobody seriously maintains that weather isn’t chaotic. But predicting its long-term average and variation — the climate — is not in vain. That’s why climate models are usually repeated, doing as many “runs” as possible (within the limits of computing time), so that we can see as many weather simulations as practical and get a better handle on the long-term statistical properties — the climate.
The logistic map is chaotic, but its long-term statistics are not. They’re stable. The real point of climate change is not the apparently random fluctuations due to the chaotic dynamics of weather, but the fluctuations of climate due to changes in the dynamics of weather. 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.
Allow me to illustrate. Let’s write the logistic map in a more general form:
If we set , , and , we get the previous version of the logistic map. Let’s use this to simulate the weather, but this time we’ll allow the dynamics (the parameter values) to change over time. We’ll keep and , but we’ll let start at zero and increase linearly at a rate of 0.017 deg.C/year (just like the climate is doing presently). Simulating 100 years, I got this:
Lo and behold — with stable dynamics we got a stable climate, but with changing dynamics we got a changing climate.
Ironically, the “example” which Edmonds gives of a chaotic system exhibiting drift is this one:
Here’s something else to stimulate thought. The values of our simple chaos generator in the spread sheet vary between 0 and 1. If we subtract 0.5 from each, so we have positive and negative going values, and accumulate them we get this graph, stretched now to a thousand points.
… The point I’m trying to make is that chaos is entirely capable of driving a system itself and creating behaviour that looks like it’s driven by some external force. When a system drifts as in this example, it might be because of an external force, or just because of chaos.
All he’s doing is taking a pseudo-random time series and accumulating it, to create a pseudo-”random walk.” His “drift” has nothing to do with chaos, I could get the same behavior by accumulating truly random numbers in a random walk. It’s just more mathturbation.
I’m sure Andy Edmonds is a pretty smart guy. I’m also sure he’s pretty dumb — it’s all too common to find both in the same person. As for when he manages to be one rather than the other, I’m not sure whether that’s predictable, random or chaotic.