BOOK REVIEW
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.
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.