The next example is a mathematical model which allows the user to enter values into 6 cells which represent the values in a set of three differential equations:

where x, y and z are variables to be observed and a, b and c are constants. The outputs of the equations change through time (dt); each new value influences the outcome of the next. A very small change in one of the observed variables can, after a few cycles, have a large effect on later outputs. This kind of result is known as the ‘butterfly effect’- the beat of a butterfly's wing in a distant forest can cause, a few days later, a raging storm thousands of miles away.

This is an extreme way of describing the behaviour of the atmosphere but it explains why it is so difficult, even with the world's most powerful computers, to predict the weather more than a few days in advance. Meteorologists collect readings for variables such as pressure, humidity and temperature from weather stations and build complex mathematical models of the atmosphere which incorporate these ‘starting values’. The weather system is so sensitive, however, that tiny differences in the readings of the variables make huge differences in the predicted weather after just a few hours of development. It is virtually impossible to be sufficiently accurate in the computer model to predict the weather accurately, though the use of super computers does help.

The meteorologist Edward Lorenz described this ‘butterfly effect’ in 1961 and his work was later recognised as a founding part of ‘Chaos Theory’. Weather systems are now viewed as 'chaotic' in that their patterns are complex, non-linear and not susceptible to accurate prediction by any group of equations yet devised. This can be shown in a spreadsheet model as follows:

The equations are entered into cells A5, B5 and C5, based on the contents of B1:B3 and D1:D3. The formulae in row 6 and after incorporate the value from the cell immediately above and are copied into the rowsd below. There are three equations so a three dimensional chart is used - a surface chart or a column chart will do. Small changes in the starting values can have huge effects on the values in subsequent cycles of the models (rows in the spreadsheet). Try changing the values yourself and watch the charts change.

This three-dimensional column chart shows the output for a=1, b=2, c=3; x=0.5, y=0.6, z=0.5.

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