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COMPUTING SCIENCE

The Weatherman

Brian Hayes

Leapfrog Integration

In general, the system of differential equations describing the behavior of the atmosphere cannot be solved exactly. Bjerknes advocated a graphical method of finding approximate solutions, but Richardson favored numerical techniques. Specifically, he adopted a finite-difference method, replacing continuously varying fields with changes calculated over discrete intervals. For example, consider the calculation of the winds in a given M cell. In the discrete model the change in the east-west component of the wind depends on the difference between the pressures in the P cells to the east and the west; in the same way the north-south component is calculated from the difference in pressure values to the north and south.

Boundary conditions present an annoying complication. If the winds in an M cell depend on the four surrounding P cells, what happens at the edge of the array, where some of the P cells do not exist? The ideal solution would be to cover the entire globe, so there are no edges, but that option was beyond Richardson's reach. Instead he ducked the question by making his prediction only for two squares (one M cell and one P cell) in the middle of his diamond-shaped array. Thus he made use of the initial values in the boundary cells but did not try to calculate new values there.

Time as well as space is discrete in Richardson's scheme. He chose a basic time unit of Δt=3 hours. He also employed a "leapfrog" method of numerical integration, where events at time t+1 depend on the state of the system both at the present time t and at the previous moment t–1. But the leapfrog process hardly got started in Richardson's calculation. He carried out only a single step of the integration, computing the "initial tendencies"—the rates of change in pressure, momentum, etc.—and used these tendencies to compute the changes in the weather over a period of 2Δt, or six hours, centered on 7 a.m.

Here are the results, as Richardson reported them in his book Weather Prediction by Numerical Process. In the central M cell, which covered the area surrounding Nurenberg and Weimar, the surface wind freshened somewhat over the six hours of the forecast, while the stratospheric winds increased more than tenfold. An even more dramatic change was seen in the P cell to the south, over Munich. According to the model, barometric pressure in the lowest stratum rose by 145 millibars to 1,108 millibars. If this surface pressure had been correct, it would have been a world record; the actual barometric reading was nearly steady on that day.




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