Today I want to take on a question that I have not been asked, but that I have seen people asking – and not getting a good answer. It’s how can scientists predict the climate in one hundred years if they cannot make weather forecasts beyond two weeks – because of chaos. The answer they usually get is “climate is not weather”, which is correct, but doesn’t really explain it. And I think it’s actually a good question. How is it possible that one can make reliable long-term predictions when short-term predictions are impossible. That’s what we will talk about today.
Now, weather forecast is hideously difficult, and I am not a meteorologist, so I will instead just use the best-known example of a chaotic system, that’s the one studied by Edward Lorenz in 1963.
Edward Lorenz was a meteorologist who discovered by accident that weather is chaotic. In the 1960s, he repeated a calculation to predict a weather trend, but rounded an initial value from six digits after the point to only three digits. Despite the tiny difference in the initial value, he got wildly different results. That’s chaos, and it gave rise to the idea of the “butterfly effect”, that the flap of a butterfly in China might cause a tornado in Texas two weeks later.
To understand better what was happening, Lorenz took his rather complicated set of equations and simplified it to a set of only three equations that nevertheless captures the strange behavior he had noticed. These three equations are now commonly known as the “Lorenz Model”. In the Lorenz model, we have three variables, X, Y, and Z and they are functions of time, that’s t. This model can be interpreted as a simplified description of convection in gases or fluids, but just what it describes does not really matter for our purposes.
The nice thing about the Lorenz model is that you can integrate the equations on a laptop. Let me show you one of the solutions. Each of the axes in this graph is one of the directions X, Y, Z, so the solution to the Lorenz model will be a curve in these three dimensions. As you can see, it circles around two different locations, back and forth.
It's not only this one solution which does that, actually all the solutions will end up doing circles close by these two places in the middle, which is called the “attractor”. The attractor has an interesting shape, and coincidentally happens to look somewhat like a butterfly with two parts you could call “wings”. But more relevant for us is that the model is chaotic. If we take two initial values that are very similar, but not exactly identical, as I have done here, then the curves at first look very similar, but then they run apart, and after some while they are entirely uncorrelated.
These three dimensional plots are pretty, but it’s somewhat hard to see just what is going on, so in the following I will merely look at one of these coordinates, that is the X-direction. From the three dimensional plot, you expect that the value in X-direction will go back and forth between two numbers, and indeed that’s what happens.
Here you see again the curves I previously showed for two initial values that differ by a tiny amount. At first the two curves look pretty much identical, but then they diverge and after some time they become entirely uncorrelated. As you see, the curves flip back and forth between positive and negative values, which correspond to the two wings of the attractor. In this early range, maybe up to t equals five, you would be able to make a decent weather forecast. But after that, the outcome depends very sensitively on exactly what initial value you used, and then measurement error makes a good prediction impossible. That’s chaos.
Now, I want to pretend that these curves say something about the weather, maybe they describes the weather on a strange planet where it either doesn’t rain at all or it pours and the weather just flips back and forth between these two extremes. Besides making the short-term weather forecast you could then also ask what’s the average rainfall in a certain period, say, a year.
To calculate this average, you would integrate the curve over some period of time, and then divide by the duration of that period. So let us plot these curves again, but for a longer period. Just by eyeballing these curves you’d expect the average to be approximately zero. Indeed, I calculated the average from t equals zero to t equals one hundred, and it comes out to be approximately zero. What this means is that the system spends about equal amounts of time on each wing of the attractor.
To stick with our story of rainfall on the weird planet, you can imagine that the curve shows deviations from a reference value that you set to zero. The average value depends on the initial value and will fluctuates around zero because I am only integrating over a finite period of time, so I arbitrarily cut off the curve somewhere. If you’d average over longer periods of time, the average would inch closer and closer to zero.
What I will do now is add a constant to the equations of the Lorenz model. I will call this constant “f” and mimics what climate scientists call “radiative forcing”. The radiative forcing is the excess power per area that Earth captures due to increasing carbon dioxide levels. Again that’s relative to a reference value.
I want to emphasize again that I am using this model only as an analogy. It does not actually describe the real climate. But it does make a good example for how to make predictions in chaotic systems.
Having said that, let us look again at how the curves look like with the added forcing. These are the curves for f equals one. Looks pretty much the same as previously if you ask me. f=2. I dunno. You wouldn’t believe how much time I have spent staring at these curves for this video. f=3. Looks like the system is spending a little more time in this upper range, doesn’t it? f=4. Yes, it clearly does. And just for fun, If you turn f up beyond seven or so, the system will get stuck on one side of the attractor immediately.
The relevant point is now that this happens for all initial values. Even though the system is chaotic, one clearly sees that the response of the system does have a predictable dependence on the input parameter.
To see this better, I have calculated the average of these curves as a function of the “radiative forcing”, for a sample of initial values. And this is what you get. You clearly see that the average value is strongly correlated with the radiative forcing. Again, the scatter you see here is because I am averaging over a rather arbitrarily chosen finite period.
What this means is that in a chaotic system, the trends of average values can be predictable, even though you cannot predict the exact state of the system beyond a short period of time. And this is exactly what is happening in climate models. Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels.
This video was sponsored by Brilliant, which is a website that offers interactive courses on a large variety of topics in science and mathematics. In this video I showed you the results of some simple calculations, but if you really want to understand what is going on, then Brilliant is a great starting point. Their courses on Differential Equations I and II, probabilities and statistics cover much of the basics that I used here.
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