Chaos is a fundamental feature of many physical systems – the atmosphere primary among them – that places a strict limit on the predictability of those systems. For the atmosphere this has obvious consequences, namely that we cannot accurately predict local weather beyond about a week. Edward Lorenz was the first to discover chaos in 1960, and all subsequent studies are based on his pioneering work.
Figure 1. Animation showing unstable, chaotic convection in an annulus as described in the text. The colours represent temperature and density – red is warm and less dense, and blue is the opposite. The graph in the centre shows the average direction that the fluid is moving, at the time given by the moving vertical line. When the blue line crosses the horizontal line at y=0, the fluid changes direction.
Chaos is essentially extreme sensitivity to initial conditions and this is brilliantly illustrated in Figure 1 (created by William F. Louisos, Darren L. Hitt and Christopher M. Danforth – see further reading). This animation shows a basic version of convection – a fundamental process that occurs in the atmosphere. The bottom half of a thin, vertical annulus of fluid is heated evenly from the edges, whilst the top half is cooled. Over time the fluid in the bottom half will become less dense and want to rise to the top, but this can happen in two ways – clockwise or anticlockwise. The direction that it chooses depends on the initial conditions – in this case the miniscule movement of the atoms in the fluid at the start of the simulation*. But of course it does not stop there – when the warm fluid rises, it is replaced by the colder fluid previously at the top which then also starts to heat, and can also go either way, and turbulence within the fluid means that it randomly flips between clockwise and anticlockwise motion. Which way the fluid goes at each turn is determined, ultimately, by the precise conditions of the fluid at the start, and therefore no two experiments will be the same – this is a chaotic system!
So, what exactly is causing this chaotic behaviour? I mean, sure, sensitivity to initial conditions, but why are some systems like this and not others? There are three main reasons. Firstly, the system has to be non-linear. All this means is that a change in one variable of the system, say x, can cause a big or small change in another, say y, depending on the specific value of x. This is probably easiest to see graphically in Figure 2; small movements along the horizontal axis (changes in x) lead to big or small vertical movements along the line (changes in y), depending on the value of x. The atmosphere has many non-linear processes. One example is convection similar to that in the annulus, where air that is heated by the ground or ocean can rapidly accelerate upwards, i.e. a slight change in temperature causes a large change in the upward speed of the air. Secondly, the system must be coupled, meaning that when one variable in the system changes, it influences how other variables change, which in turn influences how the original variable changes. An example of this is the ocean and the atmosphere. The ocean can heat the atmosphere from below, driving winds and ocean circulation, in turn changing the heat supplied to the atmosphere. It is this combination of non-linearity and coupling that can cause small changes to be rapidly amplified. However, these conditions are necessary but not sufficient for chaos to emerge, as there is a third crucial factor at play – turbulence. Turbulent eddies can be seen in Figure 1, most prominently at the boundary between heating and cooling, and it is the distribution of these after one turn (a result of the initial conditions) that determines which way the fluid will turn next. You only need to stand outside on a breezy day to know that the atmosphere is turbulent, and thus it has the three key properties needed for a chaotic system – non-linearity, coupling and turbulence!
The annulus example represents just one of the many chaotic processes that make up the real atmosphere, which has far more variables and possible interactions than this simple system. It is this magnification of tiny changes, and extreme sensitivity to the conditions at the start of a forecast, that make it so difficult to predict the weather, but equally make it a fascinating system to study. So next time the rain arrives early and spoils your party, maybe give the weatherman a break, and blame chaos instead.
*Technically, as this is a model simulation, the fluid can be totally at rest at the start and the initial direction is determined by the turbulent eddies that form, but were the experiment to be carried out in real-life, this would be the case.
Figure 3. As Figure 1, but for steady convection.
- The paper from which the animations were taken: Louisos, W. F., D. L. Hitt, C. M. Danforth, 2013: Chaotic flow in a 2D natural convection loop with heat flux boundaries. International Journal of Heat and Mass Transfer, 61, 565-576. http://www.sciencedirect.com/science/article/pii/S0017931013001300
- Lorenz’ seminal paper on chaos theory: Lorenz, E. N., 1963: Deterministic nonperiodic flow. Journal of the atmospheric sciences, 2, 130-141. http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2
- Plenty more information on chaos theory: https://en.wikipedia.org/wiki/Chaos_theory
Mind-blowing stuff: Consider two Earths, in particular the atmospheres of these two Earths, which are identical in every way. Now change the position of a couple of the atoms that make up the atmosphere on one of these Earths, and let the two evolve in time as they normally would. What you would see happen is that after a few weeks they would look completely different – the clouds would be in different places, as would the rain, all because they started off from slightly different states.
“In a nutshell my research interests include atmospheric/climate dynamics (i.e. how the atmosphere and climate system work), African rainfall and drought prediction. I am also known to dabble in a bit of storm-chasing over in the US. Oh and I like rollercoasters. A lot.”