About Chaos (or why you should carry an umbrella with you tomorrow)

It is an amazing, yet often overlooked, feeling to go out of your house, fully clad in summer clothes, look up to a blue sky and a bright sun and knowing it will be like that all day. That a storm will not suddenly pop up and ruin your grilling and make you walk soaked to your house, right?

Well, we owe that nice feeling to the countless meteorologists that devote their lives to studying the weather and also try to apply that science to everyone’s everyday life. This is a fundamental property of any scientific theory: prediction. Or in weather-like slang, forecast.

But how is this forecast done, and how is it related to many body physics?

The connection is in fact quite straightforward. The atmosphere is basically made out of air, other gases and water, which are all fluids. This makes the whole system also a fluid. A very complicated fluid, that’s for sure, but a fluid anyway, and therefore, ruled by the laws of Classical (as in not quantum) Fluid Dynamics (1).

If now you are probably thinking: “But wait… if the theory that rules it is classical and therefore deterministic (as in random stuff plays no roll, also called not probabilistic), then if we knew the complete state (as in knowing every measurable quantity, everywhere) of the atmosphere at an arbitrary time, then we would also know its past and future, making forecasts readily available and usable to plan events on the short and long run!”, then I am sorry to tell you that you have to hold your horses. The first problem is that a description of the full state of the atmosphere (being such a huge system) is a very ambitious task, if not completely impossible. This even if satellite measurements have indeed made it way more reachable than a century ago. The second problem, however, is a bit more fundamental and it is related to a discovery made during the last century called Chaos.

CHAOS IS A VERY FUNNY THING.

It can be defined as the extreme sensibility of a system to the initial conditions. That is meant in the context of how we use models (2) and their equations to predict how they will behave in the future, given an initial — measured — state. But what does that mean in reality? Remember those blocks and balls falling to their demise through ramps in Physics 101 (or high school physics)? If you push them a bit before letting them go, or if you moved them to the right, or left of the ramp, the qualitative movement would change a lot, right? Now imagine that surface has a lot of bumps and holes and all sorts of obstacles. Any slight shifting of the initial position and speed of the ball could result into a big change in the outcome of the experiment, right? That is what that sensitivity means, and that is what chaos is like. But it is funny, because it can happen in seemingly simple systems.

A very intuitive analogy for what chaos is comes from a very simple thought experiment. A ball slides in a surface. In the left, the surface is smooth, and any slight shift in the initial conditions of the ball will result in a very similar fall. In the right, however, with a lot of bumps and possible bifurcations, the ball may be a lot more sensitive to where it starts, to know precisely where it will end.

Chaos can exist in a wide variety of systems. A double pendulum, two joined bars oscillating wildly, partners in crime, are in fact chaotic, as you can see in this video. Two uncannily similar initial configurations for the pendulum can very fast become extremely different. This is a good example of the sensitivity we told you before. Another example is three planets with similar masses orbiting around each other. There are configurations which give beautiful choreographies, but a lot of initial conditions are chaotic (If you are interested in this, we suggest you play Orbit, a game about gravitation. You will get a very good gut feeling of what we mean with all of this. It is in any case, challenging and fun to play). Sometimes, even starting with a number and applying a formula to it repeatedly can become, in fact, chaotic. This is the case of the logistic map, a simple model about the dynamics of population. It can be everywhere you look, regardless of the apparent simplicity.

The three body problem. Although great scientists were able to find solutions for special cases,the general solution can become chaotic depending on the parameters (masses, initial speeds, initial positions, etc)

SO, WHERE DOES THE WEATHER COME INTO THIS MESS?

Chaos was actually discovered thanks to Edward Lorenz’s research on atmospheric convection, in 1963. all of this happened because his model yielded some equations which could no be solved without a computer (3). He was interested how a fluid flows in between a cold (up) and hot (down) layers. This is called, a convective flow, which by the way, can be used also as an artistic technique (4). For his system, he grabbed the full Fluid Mechanics equations and simplified them for the case of a convective cell. When he ran the simulation several times, nothing matched properly. A very small shift in parameter gave ridiculously different numbers! However, whenever he ran exactly the same numbers, it matched perfectly. How could that be? For a long time, he thought he made some mistake in the computer, until he realized there was something deep inside those equations and worked hard to understand the structure inside this mess. He called his discovery Chaos Theory.

Left. Convection cell. In simple words, a convection cell is a “bubble” in which fluid motion happens because heavy fluids sink. Pretty straightforward, right? When you heat air or water, they rises their volume, without changing the mass. This decrease in density is the reason of the commonly known saying: hot air rises. In the atmosphere, or in your house, or in a hot air balloon, what makes the hot air rise is the the sudden decrease of density compared to its colder versions. Right, top: Convective cells in the atmosphere close to Peru (from NASA). Right, bottom: Convective cells in the artistic technique called Acrylic Pouring, originally developed by mexican realist painter David Alfaro Siqueiros (called Accidental Painting by him).

So to forecast the weather, meteorologists need a bit of understanding of the state of the atmosphere. For that they measure a lot of quantities, like pressure, humidity, temperature, wind velocity, among others. They can use a wide range of instruments for this task, which can be ground based, or going up the atmosphere — like the ones posted in weather balloons -, and of course the measurements modern satellites can carry on, at large scales of the atmosphere.

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

EDWARD NORTON LORENZ

After gathering all this information, they use it as the initial input to run simulations (Fluid Dynamics, or a model, depending on what they are interested on) of the gigantic fluid system that is, basically, everywhere. This is where it gets tricky. The sensitivity we talked about blurs the lines of what will happen and what will not. Imagine you measure some quantity, say temperature, to be 24°C but you have an uncertainty of 0.5°C. That means the “real” temperature lies somewhere between 23.5°C and 24.5°C. In a non-chaotic system, these borders would remain more or less the same throughout a simulation. In the chaotic case, they don’t. In fact, they would explode exponentially away, inserting more and more uncertainty into the prediction. The more initial uncertainty, the faster the explosion of the error. If we started with a temperature between 23.5°C and 24.5°C, could easily become in one or two weeks into a temperature between -50°C and 50°C! It would be very hard to believe any forecast coming from that, right?

Evolution of a measured point and its uncertainty according to a non-chaotic system (left) and a chaotic one (right). One can see that the uncertainty in the chaotic systems blows up fast thanks to the sensitivity to initial conditions.

To avoid this, the measured quantity is ran in the simulation, but also a lot of different artificial ones. They are randomly chosen to lie inside the trusted accuracy of the measurement. All the simulations are put together, and using algorithms, scientists can quantify when this simulation is valid and realistic. This method is called ensemble forecasting. In this, simulations are weighed depending of the probability of finding the atmosphere in every different state. This method is very complex and there are many ways of doing this, but what is important is the realization (by Edward Epstein in 1969) that one run will not give a trustable forecast. Using these tools they can give very good short-term forecasts (hours), and good medium term (couple of days). The absolute limit of predictability is now around 14 days, so better not trust that one for your upcoming vacation trip so much.

At this point, you may be wondering: “So what is the take-home message from all of this?” That chaos and uncertainty kill our ways of understanding and adapting to our environment using science and knowledge? No, chaos is not plain mess. It is the realization that even the most simple things may be way more complex than they look. That even in our deterministic world, or tiny pieces of it, we can’t predict the future without uncertainty. That even if we hone our measuring skills, there will be always some pieces that elude our understanding and control. But there is hope on this. Chaos is also the structure between the disorder. That fascinating calm between the noise. The hope that even in the midst of this impotence , we can also know to what extent we can understand and trust.

(1) Quantum fluids are very much fluids in their own right. They are made of little pieces, which one can call particles (or quasiparticles) which have freedom to move around, and statistically create emergent properties for the fluid. However, in quantum fluids, quantum effects are important, which basically means that instead of Newton’s Laws, these particles fully follow the rules of Quantum Mechanics. This gives some amazing, and sometimes mindbending properties for these liquids and gases. (A post on this Coming Soon to your closest internet outlet).
(2) A model in science is an idealization or simplification of a system which yields a mathematical way to fit to experimental data and make predictions (see more here).
(3) What one means with the phrase solving equations with a computer is something very simple. One gives either an educated guess (in the case of finding, for example the zeroes of an equation) or the initial conditions (in the case of evolution equations, or how does a quantity changes with time). The computer then iterates over these numbers and tells you the final answer in a given time. We would of course be able to find the solutions for everything by hand, a.k.a analytically, but some equations are just literally unsolvable in that way.
(4) Incidentally, this chaotic convective flow (or Rayleigh-Bérnard convection) is present also in acrylic pour techniques, which can create beautifully messy textures. An interesting scientific article about this can be found here.