Last time, we looked at a very simple atmospheric model known as the Lorenz equations, and saw it exhibit the ‘Butterfly Effect,’ in which even very small changes in initial conditions can dramatically effect which path the system takes. However, we also saw that the initial condition had a relatively small impact on the statistical properties of the system. Because climate is a statistical property of the earth system, asking
“How can we expect to predict future climate when we can’t predict the weather?”

“How can we claim to know the half-life of a radioactive element when we can’t predict when a given atom will decay?”

To those familiar with chaos, this shouldn’t come as a surprise. Lorenz didn’t just discover apparent disorder in his model, but a deeper, eerie structure lurking in the noise.

The Lorenz Attractor: wibbly-wobbly mess of the millenium. Three simulation runs (red, green, blue) are shown; they start close together but quickly spin off on different trajectories, demonstrating sensitivity to initial conditions. Nonetheless, the trajectories quickly converge on an intricate structure in the phase space, called an 'attractor'. The attractor doesn't vary with initial conditions, but is instead a feature of the Lorenz equations themselves. Image generated with code from TitanLab - click to check them out 🙂

You may remember that the Lorenz equations relate three variables (X, Y, Z), which vary over time. In the above image, I’ve plotted the evolution of three runs of the Lorenz model by putting a dot at each (X(t), Y(t), Z(t)) coordinate, at every time t in the given interval. The three runs start very close together in this three-dimensional ‘phase space’, but quickly diverge.

However, despite their different individual behaviors, these runs are confined to a structure in phase space, known as the Lorenz attractor – an attractor, because all trajectories converge on it, regardless of their initial conditions. If you perturb the system by bouncing it off the attractor, it quickly settles back into the same loops through phase space. Lorenz (1963) described it:

“Additional numerical solutions indicate that other trajectories, originating at points well removed from these surfaces, soon meet these surfaces. The surfaces therefor appear to be composed of all points lying on limiting trajectories.”

Like the statistical measures we saw last time, this attractor reflects the system parameters, and is independent of initial conditions.

Even if the models of fluid turbulence exhibit the Butterfly Effect, it doesn’t follow that climate can’t be modeled. But do the weather models even exhibit that effect? Not necessarily. Dr. Jagadish Shukla ran weather simulations with different initial conditions and compared the results. He found that

“even for very large differences in the initial conditions, the model solutions for the seasonal mean tropical circulation and rainfall do not diverge as would be expected in a chaotic system but instead converge to nearly identical values.”

The initial values chosen were from historical weather scenarios; simulation outputs agreed well not only with each other but also with the observed weather in these scenarios*. More powerful a determinant than initial conditions was sea surface temperature; predicting atmospheric conditions becomes a problem of predicting SSTs. However, Shukla argues that this problem is fairly tractable:

“at least for certain regions of the tropics, the potential exists for making dynamic forecasts of climate anomalies several seasons in advance. […] It is now clear that certain aspects of the climate system have far more predictability than was previously recognized.”

From Shukla (1998). The top two figures are weather simulations with different starting points: one starts with atmospheric conditions from December 1988 (top) and one from December 1982 (middle). In spite of these different starting points, the models converged on very similar solutions. Moreover, both models agree fairly well with the observed weather in spring 1983 - probably because it includes sea surface temperatures from the end of 1982.

The discovery of extreme sensitivity to initial conditions launched a revolution in math and science, but it doesn’t preclude climate modelling. We may never be able to predict the weather very well far into the future, but perhaps that sort of precisely detailed forecast isn’t what we really want from climatology. In his history of chaos theory, James Gleick mused:

“Only the most naive scientist believes that the perfect model is the one that perfectly represents reality. Such a model would have the same drawbacks as a map as large and detailed as the city it represents, a map depicting every park, every street, every building, every tree, every pothole, every inhabitant, and every map. Were such a map possible, its specificity would defeat its purpose: to generalize and abstract. Mapmakers highlight such features as their clients choose. Whatever their purpose, maps and models must simplify as much as they mimic the world.” (Gleick p.278-279)

oOo

Shukla, J. (1998). Predictability in the Midst of Chaos: A Scientific Basis for Climate Forecasting Science, 282 (5389), 728-731 DOI: 10.1126/science.282.5389.728

Lorenz, Edward N. (1963). Deterministic Nonperiodic Flow Journal of the Atmospheric Sciences, 20 (2)

James Gleick. Chaos: Making a New Science. 1987.

Lorenz equations modeled with software adapted from that available from TitanLab.