Why do small errors make it impossible to predict the weather system with a high degree of accuracy?
Beyond two or three days, the world's best weather forecasts are speculative, and beyond six or seven they are worthless.
The Butterfly Effect is the reason. For small pieces of weather -- and to a global forecaster, small can mean thunderstorms and blizzards -- any prediction deteriorates rapidly. Errors and uncertainties multiply cascading upward through a chain of turbulent features, from dust devils and squalls up to continent-size eddies that only satellites can see.
The modern weather models work with a grid of points of the order of sixty miles apart, and even so, some starting data has to be guessed, since ground stations and satellites cannot see everywhere. But suppose the earth could be covered with sensors spaced one foot apart, rising at one-foot intervals all the way to the top of the atomosphere. Suppose every sensor gives perfectly accurate readings of temperature, pressrue, humidity, and any other quantity a meteorologist would want. Precisely at noon an infinitely powerful computer takes all the data and calculates what will happen at eachpoint at 12.01, then 12.02, then 12.03 ...
The computer will still be unable to predict whether Princeton, New Jersey, will have sun or rain on a day one month away. At noon the spaces between the sensors will hide fluctuations that the computer will not know about, tiny deviations from the average. By 12.01, those fluctuations will already have created small errors one foot away. Soon the errors will have multiplied to the ten-foot scale, and so on up to the size of the globe.