Where’s my umbrella? Thunder, Chaos, and Lorenz Attractors

Okay, you live in California, but there must have been at least one or two occasions when you’ve had to endure complaints against the weather–and the weather app. For example, perhaps an umbrella-less friend expected a sunny weekend before the skies decided to open itself on him as punishment for his presumption. What should one do in such situations? After nearly two decades of careful observation, I have concluded that the socially acceptable response is to join that unhappy friend in uttering invectives against the grey clouds, the National Weather Service, or Zeus. Indeed, as you hurl your curses to the skies and wipe away the drops of rain sliding down your burning cheeks, it is only natural to wonder why our weather prediction services are so poor. Are the prediction models wrong? Are the meteorologists insufficiently paid? We will concentrate our ruminations on the former possibility–the latter is one we shall reserve for a future investigation.

So how exactly do we predict the weather? If today is sunny, how will I know if I will be battered by hail tomorrow? The answer lies, as is often the case, with a certain set of equations. Specifically, meteorologists use what they call the “primitive equations”, which are a set of non-linear partial differential equations that use the laws of physics to predict how air flows in the atmosphere. Basically, if you feed your model with parameters like the current air pressure, humidity, temperature, and wind speed across the globe, it will return those parameters at a later time. In fact, the model is entirely deterministic, which means that if you give it the same initial parameters every time, it will output the same set of final parameters every time. That seems consistent–what could possibly go wrong? 

The answer is that these equations are non-linear–and the important thing to remember about non-linear equations is that sometimes they admit what are called “chaotic” solutions. Chaotic solutions to differential equations are solutions that are not periodic (i.e. they have no repeating patterns) and are highly sensitive to initial conditions, so small deviations lead to differences that blow up. In other words, if the temperature yesterday at 2 p.m. at the base of the Campanile were 20.38 degrees Celsius instead of 20.39 degrees Celsius, the blue skies of today may have been covered with clouds instead. Therefore, meteorologists need data that is very, very precise–and that is often difficult to procure; and since tiny errors blow up for chaotic solutions, our weather predictions can admit tremendous amounts of inaccuracy. 

As it turns out, the weather model can be simplified into the structure of a Lorenz system–a mathematically famous set of three differential equations (Lorenz equations)² that are used across physics, biology, and engineering. Example solutions are plotted below. 

Some paths circle and diverge, and some paths, like the one on the bottom right, seem to be attracted to certain points–those solutions are called “Lorenz Attractors” and exhibit chaotic behavior. Let’s probe a little deeper.

Suppose we have two trajectories for our Lorenz Attractor (plotted in yellow and in blue), which start just barely apart from one other. As you can see, the blue line can scarcely be distinguished from the yellow at the beginning.

What happens just one second later?

And after another second?

Again, those two paths follow the exact same equation! The only difference is that they started at points that were barely apart from each other. This phenomenon is that which is often referred to as the “butterfly effect”: the idea that a butterfly flapping its wings in the Southern Pacific Ocean could cause a hurricane in Kansas a few weeks later.

 

While this seems very abstract, there are many manifestations of analogous behaviour that you observe all the time. For example, suppose you drop a die at a specified orientation at a specified distance above a specified table. The laws of physics are deterministic, so if you manage to do precisely that every time 100 times, the die will land on the same face every time 100 times. However, it is obvious that if you rotate the die by just half a degree, it will bounce around the table in an utterly different trajectory and likely settle with a different face up. That’s why people say dice are “random”: technically, their trajectories can be predicted with complete and “infinitely precise” information about their initial orientation, but their movements are so sensitive to initial conditions that without this “infinitely precise” information, our prediction efforts would be in vain. 

But really–what’s so bad about a little rain? For all the terrible sins you’ve committed, I wouldn’t blame Zeus if he decided to turn you into his pet goat. When that happens, you’ll have more things to worry about than the weather. You’d need to think about things like the State of the Grass. But then again, you wouldn’t need to trouble yourself about midterms anymore. 

References:

1. “Lorenz System.” Wikipedia, Wikimedia Foundation, 29 Mar. 2025, en.wikipedia.org/wiki/Lorenz_system. 
2. Taylor, John R. (John Robert), 1939-. Classical Mechanics. Sausalito, Calif. :University Science Books, 2005.
3. Blalock, Lindsay. 9 Feb. 2018. Umbrella: Weapon You Can Use on Dog Walk, https://tailoredpetservices.com/2018/umbrella-day/. 

Image References

1. Header Image: Blalock, Lindsay. 9 Feb. 2018. Umbrella: Weapon You Can Use on Dog Walk, https://tailoredpetservices.com/2018/umbrella-day/.
2. Figure 1:  “Lorenz System.” Wikipedia, Wikimedia Foundation, 29 Mar. 2025, en.wikipedia.org/wiki/Lorenz_system.
3. Figure 2:  “Lorenz System.” Wikipedia, Wikimedia Foundation, 29 Mar. 2025, en.wikipedia.org/wiki/Lorenz_system.
4. Figure 3:  “Lorenz System.” Wikipedia, Wikimedia Foundation, 29 Mar. 2025, en.wikipedia.org/wiki/Lorenz_system.
5. Figure 4:  “Lorenz System.” Wikipedia, Wikimedia Foundation, 29 Mar. 2025, en.wikipedia.org/wiki/Lorenz_system.