Predicting the weather has never been easy. That is, making weather forecasts of pinpoint accuracy in an age of super computers and advancing technology. Of course, we had to go and make life harder by burning a load of fossil fuels (and we still aren’t learning that much after 2022), but why was it difficult, no, impossible in the first place? The answer to that lies in chaos theory, something made to explain the unpredictable.

Now most things in mechanics (a part of science which explains how things move) are fairly predictable. Whether it be the trajectory of a catapult, or how fast the Earth orbits the Sun, everything seemed easy to explain with mechanics. In fact, at the time when Newton was working on his laws of motion, he could find an explanation for how most objects moved, relying on something called determinism – that is, the future is entirely predictable, but we just have to wait for it to happen. For example, if a car were to accelerate at 1 ms ^{-2} for 2 seconds (assuming no air resistance and that the force applied is constant), the car hopefully would be travelling at a velocity of 2 ms ^{-1}.

However, there was one problem Newton did face. It was called the three-body problem, and involved finding the velocities and positions of 3 objects, namely the Earth, Moon and the Sun at any given time. The problem was that it was impossible for Newton to find the relationship between these three objects, when it was surprisingly easy to find the relationship for two.

It wasn’t until a scientist called Pierre Laplace came along, that more anomalies began to appear in Newton’s idea of total determinism. Laplace deduced that, assuming the velocity and position of each particle in the entire universe could be known at one given moment, the state of the universe at any time could be known using Newton’s Laws since the universe supposedly followed the idea of total determinism. However, for this to even happen, someone would have to take a snapshot of the universe from the outside (this person is commonly referred to as Laplace’s Demon). Of course, the idea of this demon even existing happening was scientifically absurd, but would that imply the universe did not follow total determinism?

It seemed not all systems followed the laws of classical mechanics predictably, notably the universe and the solar system in the three-body problem, but the question was: why was it unpredictable? Would it be fair to say these systems were “random”? The answers to both questions became apparent in Edward Lorenz’s paper titled the “Predictability; Does the Flap of a butterfly’s wings in Brazil set off a tornado in Texas?”. Strangely enough, the weather it seemed behaved unpredictably too, something now we describe as a chaotic system.

Edward Lorenz was a MIT professor, who, in simple terms, programmed a small model of convection currents in the atmosphere. He used a set of mathematical equations, known as differential equations, which use rates of changes as variables. All the factors that would affect how the warm rising air would oscillate such as humidity, pressure, wind speed and so on were determined by twelve differential equations. The computer churned out numbers for each of the factors, which corresponded to a type of weather you might see. One day, Lorenz inputted a set of numbers into the program, and waited, before quickly going to grab a coffee. When he came back, he inputted the numbers from the halfway point to save time, but upon returning he found to his surprise, the results were completely different. That is, the exact same initial conditions gave out completely different weather descriptions. He tried to fix the computer, and even reduced his twelve equations to a set of three differential equations, but he got the same result every time. What would later come apparent to him, is that the initial results he inputted into the program were to 3 more significant figures to the ones he inputted halfway. That is, there was a small difference of less than 1/1000 between the inputs for both programs.

So how can such a small change create such a great effect? This idea, which applied to all chaotic systems, was known as the Butterfly Effect, or scientifically as the Sensitive Dependance to Initial Conditions. Essentially, without knowing the initial conditions to an infinite precision, you can never predict a chaotic system. And that makes sense, doesn’t it? Since it is impossible to know the exact velocities and positions of each particle in the universe, it is impossible to predict the state of the universe at any time.

Fascinatingly, our understanding of chaotic systems can be visually expressed in phase space. Take a pendulum for example. Pendulums can be relatively easily explained using classical mechanics but allow us to introduce a more complex dimensional field to “draw” chaotic systems. If we have a graph of velocity against the angle of the pendulum and we let it go, it would go round in an ellipse and eventually move to the origin, where it stops. We call this point an attractor. If we assume there is no friction, then the pendulum would move in a closed loop, since its speed would never reach 0 ms ^{-1}.

With even two pendulums, the system behaves chaotically. Unless the initial conditions are met exactly, the two pendulums will swing in the same way for a small time, before deviating completely. Rather surprisingly, the same thing happens if we plot Edward Lorenz’ differential equations we met earlier. In fact, if we take a large collection of these initial conditions, then we get a 3-D attractor, quite coincidentally, in the shape of a butterfly. The term “strange attractor” was soon coined and led to the beginning of fractals (each line in the attractor is of an infinite line in a finite space, which has similar properties to a fractal).

As a result, predicting the weather to an infinite precision is impossible just because we don’t know the initial conditions accurately enough. That’s why forecasting the weather even a week from now on will give completely inaccurate predictions, and at that point, it’s more accurate to use the average of all the readings taken for that day. Chaotic systems are in fact deterministic, but because we can never know the initial conditions to a high enough precision, we see its random characteristics. Nowadays, forecasts are made by changing the initial conditions slightly and by using an ensemble of all the forecasts, you can make rough predictions.

What’s Global Warming got to do with this though? Even if we can’t predict the weather accurately, wouldn’t this mean climate (average weather readings) also act chaotically? Our climate changes mostly because of heat transfers between the ocean and the atmosphere (aside from the smaller impact of volcanic eruptions, solar cycles and airborne particle numbers). However, in the ocean heat transfer patterns are slightly chaotic, and this was showed by a simple model of the El Nino by Eli Tziperman in 1994. Therefore, it appears that even dynamic systems overtime will exhibit some chaotic effects, however small.

The increase in atmospheric temperature and number of carbon sinks has pronounced the butterfly effect to an extent, and whilst climate may take more chaotic expels of carbon into the atmosphere, it is certainly becoming harder to predict the weather.