Forecasting the weather
Even with very powerful computers we all know that we cannot forecast the
weather more than a few days in advance. Over just a few days forecasting the
weather still gives us nasty surprises. This is because the equations which govern
the weather are nonlinear – they involve the variables multiplied together, not
just the variables themselves.
From meteorology to mathematics
The discovery of the butterfly effect happened by chance around 1961. When
meteorologist Edward Lorenz at MIT went to have a cup of coffee and left his ancient
computer plotting away he came back to something unexpected. He had been aiming to
recapture some interesting weather plots but found the new graph unrecognizable. This
was strange for he had entered in the same initial values and the same picture should
have been drawn out. Was it time to trade in his old computer and get something more
reliable?
After some thought he did spot a difference in the way he had entered the initial
values: before he had used six decimal places but on the rerun he only bothered with
three. To explain the disparity he coined the term ‘butterfly effect’. After this discovery
his intellectual interests migrated to mathematics.
The theory behind the mathematics of weather forecasting was worked out
independently by the French engineer Claude Navier in 1821 and the British
mathematical physicist George Gabriel Stokes in 1845. The Navier–Stokes
equations that resulted are of intense interest to scientists. The Clay Mathematics
Institute in Cambridge, Massachusetts has offered a million dollar prize to
whoever makes substantial progress towards a mathematical theory that unlocks
their secrets. Applied to the problem of fluid flow, much is known about the
steady movements of the upper atmosphere. But air flow near the surface of the
Earth creates turbulence and chaos results, with the subsequent behaviour largely
unknown.
While a lot is known about the theory of linear systems of equations, the
Navier–Stokes equations contain nonlinear terms which make them intractable.
Practically the only way of solving them is to do so numerically by using powerful
computers.