# Who discovered the chaos theory

## Chaos Theory - Part 1: The Butterfly Effect

There are limits to the weather forecast. What chaos and the flapping of a butterfly's wing have to do with it, we show in a short introduction to chaos theory.

We keep receiving inquiries like: "We want to get married in two months. How will the weather be on our wedding day?" In such cases we unfortunately have to disappoint the wedding couple, because weather forecasts are not possible for such a long period of time. But why are there limits to the forecast period and where are the limits of predictability? To answer this question, we need to look at weather as a chaotic process. In fact, the founder of chaos theory, Edward N. Lorenz, was a meteorologist. He came across chaos theory in 1963 while researching convection currents in shallow liquids and gases. (In one experiment, a gas that was heated by a heating plate rose, cooled on the surface and then sank down the sides again. In the process, rolls or so-called convection cells formed.) Edward Lorenz described these flows with a predictive model, in which he related the temperature and the convection rate in a system of equations. To solve these equations, he used what is now a relatively simple computer. The discovery of the chaotic behavior of this system was more of a coincidence. When he calculated his model a second time, he wanted to save computing time and entered the initial conditions with only three decimal places instead of the previous six decimal places. Although the initial conditions hardly differed from each other, Lorenz came to completely different results after a certain time. Tiny variations in the initial conditions do not have hardly any effects in some systems, as expected, but can lead to large deviations. The following section goes into more detail and can also be skipped for a general understanding. As a graphic solution of the equation, one obtains the structure in the figure, which is also called the Lorenz attractor. The axes X, Y and Z stand for the calculated variables of the equations, the line shows the development over time (course) of the respective variables and is called the trajectory. It is noticeable that the trajectory is not a chaotic path, but rather a certain order. It circles around two different orbits and never cuts its own orbit. This structure is also called a strange attractor. What is chaotic, however, is the change from one orbit to the other, which does not take place after a certain period. Whether the trajectory "tips" from one orbit to another depends heavily on the initial conditions. In chaos theory one speaks of a "bifurcation". Transferred to the weather forecast, such bifurcations are more common in borderline weather conditions. Then different weather model runs show two different weather conditions (which can be made clear by the change between the two different orbits.) The forecast then often jumps back and forth between these two solutions. In the current medium-term forecast, for example, most of the weather models show a high over north-eastern Europe for mid-June. This would result in a very warm east current. However, a very few model solutions show this high a little further to the east, so that from the west Atlantic low rivers could spill over to Germany, which would bring cooler sea air instead of warm air unpredictable and random behavior of systems is meant. Chaotic systems are quite predictable. That is why one speaks of deterministic chaos. There is also a certain order inherent in them. The key message is that nonlinear systems, like weather, are very sensitive to small changes in the initial conditions that have very large effects. Edward L. Lorenz illustrated this effect with a metaphor that crossed his mind as he looked at the shape of his Lorenz attractor (pictured): "Can a butterfly flapping its wings in Brazil cause a tornado in Texas?" Today this is known as the so-called "butterfly effect". Lorenz's work had a major impact on our view of the world and still influences the weather forecast today. Because the initial state of the atmosphere cannot be determined as precisely as desired for the weather models. On the one hand, measurements are not available for every point in the atmosphere; on the other hand, all observations are flawed to a certain extent. Furthermore, some of the equations in the weather models are only approximations. The model calculations become more and more uncertain as the forecast time increases. Where the limits of predictability lie and how one can extract information from the chaos in the forecast is the subject of the second part. There will probably be more about this tomorrow.

Dipl.-Met. Christian Herold

German Weather Service Forecast and Advice Center Offenbach, June 8th, 2020