Mittwoch, 11. April 2012

Chaos and Climate

Does Chaos impede predictions? Answer: NO

In discussions about climate and man-made climate change one assumption is often stated by climate change denialists. That is, that because there is chaos involved in weather and thus in climate, there is no way to make any prediction.

That's not true, and I'll show you why.

What does chaos mean?

Imagine two particles (e.g. O2 molecules) in a room starting close by, one beside each other and having roughly the same energy and the same momentum. In a non-chaotic system---although they move around in space---these particles will still be close to each other after a period of time. Maybe they get a little bit further apart, but they will stay rather close. Particles far apart from each other will stay far apart (in terms of position, momentum and energy) and close particles will stay close.

In a chaotic system on the contrary, these two particles get quickly far apart from each other, they will soon have very different locations, energies and momenta.

What does chaos NOT mean?

Some people think, that in order to get chaos, the implicated laws of nature have to be non-deterministic. This is wrong. Some people think, that in order to get chaotic behavior it is necessary to have a reduced knowledge about the interactions between the particles. Again wrong. It's very easy to construct mathematical models, where all the rules for the movement of the particles and for their interactions are perfectly known and you still get complete chaos. For instance just some little particles in a box might be enough.

Where is the chaos to be found?

Think of all the air molecules in a room. If you would try to follow one of these molecules you would most likely fail miserably. Let alone predicting where this one molecule will be in 10 seconds time, or even in 10 minutes. This is clearly a chaotic system. Two particles starting close by will be very far apart already after a short period of time.

How's that with the predictions?

Does that mean it is correct to say, that there is no prediction possible? NO, it is just correct to say that the prediction of the exact path of one particle in the system is not possible (or at least very very difficult). You can make predictions concerning the whole ensemble of particles. It actually much easier to make statements and predictions about the whole system just because the system is chaotic. Why that?

Let's dig deeper

Since in science the experiment trumps the theory I will provide first examples which show, that one indeed can make predictions. Then we'll dive into a little bit of theory.

The experiment

The "experiment" is an easy one. You are standing at the bar of a club drinking ice cold Mojito. On the other side of the club at a table, there is an attractive woman sitting (the scene is set for the experiment). One can state with little uncertainty, that the temperature on the other side of the bar in the area close the beautiful woman is about the same as on this side. And so is the air pressure. One can predict as well, that the temperature will not change suddenly unless the door is opened and cold air from outside streams in. We can also be fairly certain, that there will not be any sudden changes of air pressure. A movement of all the air molecules from the area around the woman towards  you (leaving her in a vacuum) or the other way round is a fairly unlikely event. It just will not happen tells us our experience.
Let's conclude from the "experiment": Although we cannot possibly predict the trajectory of any of the air molecules---let alone all of them (there are quite many of them flying around)---we can predict states of temperature and air pressure very accurately.

According to climate change denialists this is not possible---no way. But in real life it is.

OK, this was just a room. You were sitting just about 10 meters away from this nice looking woman. Let's make the predictions more difficult. Let's predict the average high temperature of Boston next June: I predict roughly 24 degrees Celsius. On the other hand, right in the neighborhood of Sarah Palin, in Petropavlovsk-Kamchatsky it will be around 11 degrees Celsius. And I predict the same temperatures for June in two years and June in three years.
Boston (CC BY-SA 3.0 by Henry Han)
Petropavlovsk-Kamchatsky (CC BY-SA 2.5)

I've never lived in any of these cities, neither have I traveled there. I didn't make an extensive study or deduced the numbers using a climate model. How did I do that?

A climate change denialist would say: "It's impossible to make any prediction, because weather is chaotic and the weather forecast for in three weeks is already as good as random guessing". The denialist obviously is wrong, since I am able to make a prediction and this prediction is of a fairly high confidence (I didn't quote an uncertainty, but give it +-3 centigrade and it'll be for sure two sigma fine, I am too lazy to derive the numbers exactly). I just had to look into the temperatures of the last years in June in these two cities and take this number as predictions for the temperatures for the next years in June. And you can do the same experiments for yourself for any of the places on earth you are interested in. You rather book a swimming-in-the-ocean-vacation in a the Caribbean or the Maldives than in Spitsbergen? Why? Probably because it will be hotter there. How do you know? Because you know something about the climate.

The key point is: We plan what we do the coming months and in one or two years or longer on basis of our private little climate predictions. And we are quite right with those.

Sounds paradoxical? One cannot predict the weather in four weeks, but one can predict the temperature in two years?
The paradox is solved by adding the omitted boundary conditions to the statement: One cannot predict the local weather in a short time slice in four weeks, but one can predict the average temperature of a large area in two years.The key words are local and short time slice for the weather prediction, and average and a larger area for the climate predictions. You can have a rainy 11 centigrade day in Boston in June, but it will not change much the average.


The thinking

While experiments are nice and only experiments can prove something to be true or false, it's theory which delivers more insight.

Let's start lightweight: Get back to the box with the many particles inside flying around. Imagine each of the particles to be a little glass sphere. Imagine there is no gravity for the moment. All these little spheres will happily fly around in a straight line until they hit a wall or another sphere. Hitting the wall gives a predictable change of motion: The sphere is reflected on the wall. Predicting the trajectory of the sphere hitting another sphere is a little bit more difficult, since the resulting trajectories, momenta and energies of the two implied spheres depend strongly on the impact parameter (a little bit left, a little bit right, etc.). The mean free path of air molecules at ambient pressure is 68 nm. The average speed of air molecules is about 500 m/s. Hence, one single particle hits another one about all 136 ps (on the average of course). After a second the particle will have hit other particles several billion times, loosing or gaining energy and changing direction after every hit.

Let's make the step from looking at one particle interacting with the wall or with another particle to observing the behavior of the whole group of particles. If you would record the speed of one particle every nanosecond for a longer period of time (let's say one minute or so) and put it into a graph, you'd get an approximation to the Maxwell-Boltzmann distribution.

Maxwell-Boltzmann distribution (CC-BY-SA-3.0-MIGRATED, Wikimedia Commons)

You could as well record the speeds of all particles at one specific point in time and put it into a graph with the same outcome, the Maxwell-Boltzmann distribution. By the way, you get the same type of distribution for the energies and for the momenta of the particles. The shape of the distribution depends on the properties of the particles and on the temperature.

Hence, only chaotic movement in the box, and a lot of it. But the whole ensemble together behaves very nicely.

Going deeper leads us to some mathematical concepts which I will just scratch a little, mainly providing the keywords which help you to dig into the matter. The first keyword denotes one further requirement for chaotic (dynamic) behavior, it's: topological mixing, and means, that any region of phase space (position and speed) will overlap with any other region at some time. In simple terms, once you put some gas into the wild on one end of the room, parts of it it will eventually reach the other end of the room. If it is an ugly smelling gas, you can hope for the gas to be diluted enough to fall below your level of odor bearablity very soon. The topological mixing will solve the dilution problem for you but you still might have to open up a window to give the mixing procedure enough space.

Further important keywords are: Statistical mechanics, Ergodic theory, there you find: microcanonical, canonical and grand-canonical ensemble:

Starting from just a bunch of particles (a big bunch actually), the forces acting upon them (e.g. gravity) and their interactions (their behavior at collisions) a statistical observation of which states are possible and what's their probability enables us to derive temperature, pressure, enthalpy, heat capacity, chemical potential etc.


Climate science is hard and difficult to get right. But the reason is not the chaotic movement involved in the interaction of the molecules of the air. The reasons are, that earth is a very large system which is not at all uniform. There are oceans which have their streams, there is air, there are mountains which influence the movement of the air, there is the earth's rotation, there is the formation of ice on the polar caps and there are the glaciers, there are volcanos, there is the sun which has it's cycles, there are clouds, there are several gases in the atmosphere which influence climate, there are plants which are growing, there are plants which are rotting when the permafrost defreezes, there are trees in the amazonas which are cut by men or a a power plant is built and large areas of rainforest are flooded, and not the least, there are humans which burn precious resources like oil or coal.

It is as well difficult to get all the data right, be it the measurements from satellite, from land, from regions which are difficult to access, proxy markers from ancient times. It's not easy to synchronize these and get a consistent picture of the temperatures, the concentration of certain gases, the flora and the fauna of the earth several thousand or even millions of years ago. 

With all these influences and uncertainties it is obvious, that it is not an easy task to get it all right. But scientists have done a good job and every new generation of climate models describes the earth's climate better. And every improvement of the models reduces the uncertainty of the results. And the results just let one conclusion. We (humans) are heating our earth like hell.

You are very welcome to leave your comments! Let me know what you think.

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