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Calculating the Weather

Brian Hayes

The Emergence of Numerical Weather Prediction: Richardson's Dream. Peter Lynch xii + 279 pp. Cambridge University Press, 2006. $75.

Lewis Fry Richardson was a meteorologist who made only one weather forecast, some 80 years ago, and it was utterly wrong. This might seem a dubious claim to fame, but Richardson was a key figure in the development of modern forecasting methods, and his ideas are in use every day in weather offices around the world. Peter Lynch of University College Dublin now gives us a careful analysis and reconstruction of Richardson's famous forecast. He finds the source of the fatal error, and even fixes it!

In Richardson's era, weather prediction was based mainly on a historical or analogical scheme. Meteorologists would gather reports of current conditions and draw up charts showing geographic patterns of barometric pressure, wind, temperature and other variables. Then they'd look for earlier occasions when similar patterns occurred and try to make inferences about future developments from what happened in the past.

Lewis Fry RichardsonClick to Enlarge Image

Richardson's approach was more direct: He set out to calculate the weather. He built a working mathematical model of the Earth's atmosphere, based on straightforward physical rules. For example, one rule says that if regions differ in barometric pressure, then air will start to flow along the gradient toward the lower-pressure area. Richardson filled in initial values of pressure, wind velocity and so on, and then traced the model's evolution over time.

Models based on essentially the same principles now run on supercomputers capable of trillions of operations per second. Richardson, however, worked entirely with pencil and paper, using a sheaf of forms he had printed up to guide the computations; his only aids to calculation were a slide rule and a table of logarithms. Furthermore, the circumstances in which he did all this arithmetic have made the project legendary. The time was World War I. Richardson, a pacifist, was serving as a volunteer with the Friends Ambulance Unit in the north of France. He performed his calculations between calls to carry wounded from the front. "My office," he reported, "was a heap of hay in a cold rest billet."

Richardson has gotten considerable attention in recent decades, not only for his work as a meteorologist but also for his later mathematical investigations of the causes of war and peace. A major biography by Oliver Ashford appeared in 1985, followed in 1993 by two volumes of collected papers. In The Emergence of Numerical Weather Prediction, Lynch gives a précis of the biographical background, but this book is not a general account of Richardson's life or even an overall assessment of his contributions to meteorology. The focus is on the problematic forecast.

Because Richardson's pencil-and-paper arithmetic could not possibly keep pace with the evolution of the weather itself, his "prediction" actually concerned events long past. His initial data were observations made across Europe at 7 a.m. on May 20, 1910. His aim was to calculate the barometric pressure and the wind three hours later for two points near the middle of the continent.

The inner mechanism of the mathematical model was a set of differential equations, relating quantities such as the instantaneous rate of change in air pressure to the horizontal and vertical components of the wind. Finding an exact or analytic solution of these equations was not feasible, and so Richardson sought a numerical approximation. It's worth noting that Richardson already had experience with such numerical methods, which were not then popular in the mathematics community. (They have since become a major industry.)

To apply his numerical techniques, Richardson had to replace the continuous space and time of the differential equations with discrete finite-difference equations. He divided the European area covered by the model into a lattice of 25 boxes, each about 200 kilometers on a side. Vertically, the atmosphere was sliced into five layers. Together, the horizontal and vertical subdivisions created a total of 125 three-dimensional cells. Within each of these cells, conditions were assumed to be spatially uniform. Outside the lattice of cells, all conditions were held constant.

In the temporal dimension, Richardson proposed to update the state of each variable at intervals of six hours. In the end, he completed his calculation only for a single interval. From his data for 7 a.m., he worked out the "initial tendencies"—the rates of change—and based on those rates "predicted" various quantities three hours earlier and three hours later. The barometric-pressure prediction is the one that both Richardson and later commentators have discussed at greatest length. At a certain point in Bavaria, the model has the surface air pressure rising by 145 millibars over six hours, an impossibly large change. In the real world, the barometer was nearly steady.

What went wrong? One popular explanation is numerical instability—the cascading growth of errors when the time step between calculations is too long. There's no doubt that Richardson's computation did violate an important stability criterion, but that can't be the cause of the error, because he didn't continue beyond the first time step. Richardson's own verdict about his failure put the blame on "errors in the initial data for winds," and he later came to believe that the problem could have been rectified by smoothing the initial data. Lynch concedes this is closer to the truth—wind errors were indeed present, and a significant source of trouble—but merely smoothing the measurements would not have redeemed the model.

The real root of the problem turns out to be a subtle and underappreciated property of our atmosphere. We tend to look on our weather as active and occasionally even violent, but things could be much worse. If gravity waves—disturbances analogous to the swells that move over the surface of the ocean—made any significant contribution to the weather, we would see storms careering across the continents in a few hours instead of a few days. Because such waves don't in fact arise, it's important that a mathematical model of the atmosphere also exclude or suppress them. But Richardson's model has imbalances between initial winds and initial pressures that act like a tightly wound spring, immediately setting off large gravity-wave oscillations. It's not enough just to smooth these fluctuations; pressure and wind have to be brought into balance.

Lynch, who was formerly deputy director of Met Éireann, the Irish Meteorological Service, is well equipped to give a modern practitioner's critique of Richardson's model. He goes beyond the critique and recreates the model in modern form, starting with the same data and equations but encoding them in a computer program rather than doing the calculations on paper forms. The first challenge for this procedure was to get the same wrong result. Doing so offers some reassurance that there are no additional unnoticed errors or misconceptions in Richardson's work. (Lynch did find that Richardson made some mistakes in arithmetic, but they were few and inconsequential.)

Having reproduced the incorrect forecast, Lynch then set out to improve it. To restore gravitational balance he preprocessed the input data with a filtering method called initialization, which changes the observed parameters only slightly but in a coordinated way that eliminates gravity waves. With this preliminary conditioning but no other changes, Richardson's basic model yields an essentially correct prediction for the weather on that morning in May in 1910. Richardson came closer to the answer than he ever knew.

The Emergence of Numerical Weather Prediction is the best single source available for understanding Richardson's forecast—better even than Richardson's own book, Weather Prediction by Numerical Process, although that work still has its special charms. (And it is about to be reissued by Cambridge University Press, with a foreword by Lynch.) Lynch's book is more than just a historical case study—he has provided an insider's guide to how weather prediction works. The final chapters bring the story up to date with accounts of the first computer forecasts in the 1950s and of the technological and theoretical innovations since then.

My one complaint is that with greater authorial or editorial effort, these ideas could have been made accessible to a wider audience. This is an essentially mathematical discourse, and no amount of sugarcoating will make the notation palatable to those who blench at the first sight of an equation. But a few extra words of explanation would have helped the book reach out to those who need a refresher on vector calculus or partial differential equations. Furthermore, even the mathematically adept may pause over some of the specialized jargon of meteorology.

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