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COMPUTING SCIENCE

The Weatherman

Brian Hayes

The Answer Is Blowin' in the Wind

What went wrong? The prime suspect in a case like this one is usually numerical instability. When the result of one step serves as the input for the next step, small errors can multiply and grow explosively. There is a rule for avoiding such a catastrophic loss of accuracy: The time step must be kept shorter than the time needed for information to propagate through the model—in this case between observing stations spaced every 200 kilometers. According to this criterion, Δt in Richardson's model should have been no more than about 30 minutes. Richardson can hardly be blamed for breaking this rule—the limit on Δt was not discovered until a decade later (by Richard Courant, Kurt Otto Friedrichs and Hans Lewy)—but the model is clearly in violation all the same.

The Courant-Friedrichs-Lewy condition has been cited as a possible cause of failure by several commentators on Richardson's work, including Sydney Chapman in the preface to a reprinting of Weather Prediction by Numerical Process. Nevertheless, numerical instability cannot be the source of the problem. Errors grow exponentially when the finite-difference procedure is iterated, with the output of each stage becoming the input to the next stage. Richardson stopped calculating after one step. If he had kept going, instability would doubtless have appeared, but it can't explain a large error in the initial tendencies.

Another possible culprit is the phenomenon called deterministic chaos, which Edward N. Lorenz described 35 years ago, also in the context of weather prediction. Lorenzian chaos is superficially similar to numerical instability, but a chaotic system remains unpredictable even if each step of the calculation is performed with perfect precision. The slightest change in the initial conditions is enough to produce a divergent result. There is no question that Richardson's calculation would be subject to this problem if it were continued long enough, but again chaos can't explain a failure in the first step.

The real cause of Richardson's forecasting error was identified by Richardson himself. At the end of his table of results, he wrote: "It is claimed that the above form a fairly correct deduction from a somewhat unnatural initial distribution." In other words, the problem lay not in the algorithm but in the input data. A close look at the data confirms this diagnosis and also shows clearly why weather prediction is such a hard problem.

The main troublespot is the part of the calculation where pressure is determined from the convergence (or divergence) of winds. Along each horizontal axis, the wind convergence is calculated as the small difference between two large numbers. Under these circumstances, even slight errors in the initial wind data can cause large variations in the computed convergence. For example, suppose the true east-west winds on either side of a P cell have magnitudes of 101 and 99 (in some appropriate system of units). Then the east-west convergence is equal to 2. If the measurement of either wind is off by just 1 percent, however, the convergence will change by 50 percent or 100 percent. This disastrous error in convergence will be reflected in the predicted barometric pressure.

Another property of the atmosphere compounds the sensitivity to measurement errors. If low-level winds in a region are converging, upper-level winds over the same area are usually diverging. This state of balance tends to reduce the effect that convergence or divergence would have on barometric pressure, but again the balance is easily upset by measurement errors.

And just as errors in wind measurement degrade the prediction of pressure, so also pressure errors confound wind predictions. Here too the culprit is the subtraction of two nearly equal quantities—the pressure-gradient term and the Coriolis term in the atmospheric equations of motion. Because the terms generally have about the same magnitude, small errors in observations yield large changes in predicted wind.

Richardson recognized these mechanisms and suggested smoothing the initial data as a possible remedy. Lynch points out that the problem goes deeper. To maintain the overall state of balance in the atmosphere, smoothing either the winds or the pressures is not enough; the two sets of observations have to be made mutually consistent.





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