The Lorenz Model

The father of chaos theory was a meteorologist himself, Edward Lorenz. In 1963 he published a number of observations regarding the feasibility of long term weather prediction in the Journal of Atmospheric Science, an article which like the butterfly effect he described, seemed to cause only slight ripples of reaction at the time of its publication but since then has come to foment the most gigantic turbulence in many areas of intellectual enquiry.

In 1960 Lorenz had been experimenting with numerical weather simulation, using an early electronic computer. It was during one of these experiments that he serendipitously discovered that the output of his highly-simplified weather model was sensitive to even the tiny changes in initial conditions caused by the truncation of decimal places from the numbers he input. In order to try to understand the fundamental essence of this effect, he turned from the weather to a simpler set of equations, designed to model convection. This set of equations has come to be known as the Lorenz model.

Simple convection of a fluid in a box.

The diagram above shows the physical situation which the Lorenz model approximately represents. You have a box which contains a fluid. When the bottom of the box is heated, the warm fluid rises and the cooler fluid sinks, and the tendency of the motion is to form two cylindrical rolls as shown, with the warm fluid rising up the middle, cooling as it goes, then sinking back down the sides.

When the temperature is increased, it might be expected that this simple convective motion would be retained, albeit rotating more quickly. However there are complications: if a parcel of fluid rises quickly enough it may begin to descend the side of the box, following the path of the convection, before it has lost all of the heat that drove it upwards. In this case the fluid will feel a buoyancy opposite to the direction of its motion as it descends, and in this way the whole pattern of flow becomes more complicated.

In reality the convective motion of a fluid is more complicated than the Lorenz model suggests, but the non-linearity of the Lorenz equations is sufficient in itself to make the flow unpredictable above a certain temperature. In the next page we will examine the actual form of Lorenz's model equations, before going on to examine the surprising complexity of the results that the model produces.