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COMPUTING SCIENCE

The Weatherman

Brian Hayes

Checkerboard Europe

Richardson's forecast was actually a hindcast: He was "predicting" events that had taken place years before. His initial data described the state of the atmosphere over Germany and neighboring countries at 7:00 a.m. (Greenwich time) on May 20, 1910. His goal was to model the weather in this region over a span of six hours. He chose this particular time and place not because the weather was in any way unusual that Friday morning but rather because unusually good data were available. May 20, 1910, was one of a series of days on which weather observations were collected from coordinated balloon ascents all over Europe. The results had been tabulated and analyzed by the Norwegian meteorologist Vilhelm Bjerknes.

The observing sites for the Bjerknes project were scattered irregularly across the map of Europe, and so Richardson had to interpolate to produce a uniform grid of data points. The grid he chose was a checkerboard, with squares roughly 200 kilometers on a side. A pattern of 25 squares covered a diamond-shaped region extending from Denmark to Italy and from the English Channel to Poland. Vertically, he sliced the atmosphere into five layers, with boundaries at altitudes of roughly 2, 4, 7 and 12 kilometers. (The heights were chosen so that each stratum had about the same mass of air.) Thus the model divided the volume being studied into 125 compartments.

The pattern of squares laid out on the landscape was described as a checkerboard rather than merely a grid because different quantities were computed in the alternating black and white squares. In one set of squares (called P cells) Richardson recorded the barometric pressure in each of the five altitude layers, and also moisture amounts and the stratospheric temperature. In the other squares (M cells) he calculated the momentum of the atmosphere—that is, the wind speed and direction multiplied by the mass of the air.

It's not hard to see in a qualitative way how these variables would enter into a model of the atmosphere's dynamics. In particular, winds and barometric pressures have an obvious interconnection. Pressure is the force that drives the winds; air flows from place to place in response to differences in pressure. At the same time, the convergence or divergence of winds alters the pressure in a region, as air is either blown in or sucked out. A similar linkage connects pressure with temperature, since heat is generated when air is compressed. The model has to track all these relationships (and others) from moment to moment. Given initial values of all the variables at time t0, the model calculates the values at a later time t1; then these t1 values form the basis of a new calculation at time t2, and so on.

Figure 2. Data for Richardson’s forecastClick to Enlarge Image

The most important quantities to keep track of in Richardson's model turned out to be barometric pressure and three components of momentum (along north-south, east-west and up-down axes). All of these quantities vary from place to place and from moment to moment; in other words, they are functions of latitude, longitude, height and time. There are also crucial dependencies among them. For example, one of Richardson's equations states that the rate of change in the east-west component of momentum depends on the pressure gradient along the same axis. This relation is unsurprising; it says that air goes where you push it. But the full atmospheric equations of motion are more complicated. In addition to the pressure-gradient term, they also includes a term representing the Coriolis force, by which the earth's rotation twists the winds and thereby couples east-west and north-south velocities.

Vertical motions in the atmosphere are even more problematic. No vertical winds were included among the initial data. To calculate them, Richardson relied on a simple ground truth: The earth is solid, and therefore impervious to wind. It follows that if horizontal winds are converging in some ground-level cell of the model atmosphere, then air must be flowing upward out of that cell. By the same principle, divergent winds at ground level must be balanced by air sinking into the cell from above.





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