August
2003 [ HOME ] Table of Contents for printer Arakawa's Computation Device
Climate science required the invention and mastery of difficult techniques. These
had
pitfalls, which could lead to controversy. An example of the ingenious technical work and
hardfought debates underlying the main story is Akio Arakawa's invention of a mathematical
method that solved a vexing instability in big computer models. For other examples, see the
essays on Uses of Radiocarbon Dating and on Temperatures from Fossil Shells
Norman Phillips's 1956 experiment was a great
success, but after a month it blew up. He had succeeded, for the first
time, in constructing a computer model of the general circulation
of the atmosphere that produced something looking much like real weather
maps. As the calculations stepped forward through time, however, after
twenty or so simulated days the pattern of flow began to look strange,
and by thirty days the numbers veered off into conditions never seen
on Earth.(1) 
 LINKS 
More discussion at
<=Models
(GCMs) 
At first Phillips and his colleagues thought
their problem was "roundoff error," a difficulty that computer modelers
were just beginning to come to grips with. A computer cannot produce
numbers with infinite precision (for example, 2.394376113... might
be truncated to 2.394). When one round of calculations was fed back
into the next round, and this was repeated thousands of times with
the numbers truncated each time, tiny discrepancies could add up to
a big difference. Eventually the solutions became unrealistic and
"exploded." Modelers were working out a variety of solutions to such
errors, ways to smooth out artificial bumps in the numbers at one
stage or another. It did not take long for Phillips to find that such
tricks would not solve his difficulty. There was a more fundamental
problem in the way the computer chopped up continuous equations into
a grid of numbers. The computer was getting confused by weather patterns
that were smaller than the spacing between the grid points.(2*) 
<=>Chaos
theory

Phillips's experiment fired the imagination of many, and not least
Yale Mintz. He had taken his Ph.D. in meteorology at the University
of California, Los Angeles, in 1949, and took up work with the new
digital computers. One of his key steps was
to recruit Akio Arakawa. Arakawa had graduated from the University
of Tokyo shortly after the end of the Second World War only to find
job opportunities for scientists severely limited, and took up a career
in the growing enterprise of practical meteorology. Like some of his
peers, he eventually found a better opportunity in the United States.
He was hired to help Mintz express Mintz's ideas
in specific mathematical formulas, and to convert the formulas into
computer code. Arakawa, however, had ideas of his own. He persuaded
Mintz that it would be futile to launch straight into programming
when Phillips's experiment had shown there could be instabilities
that might wreck the results. Although Mintz was reluctant to delay
his project, he gave way and allowed the young man to dig into the
fundamental question: how should meteorological equations be represented
in numerical calculations?(3) 
<=Models (GCMs)
<=>Models
(GCMs)

Following Phillips's lead, Arakawa recognized that the instability
was (to make a crude analogy) like the illusory patterns that appear
when you look through a window screen at another screen. Where the
screens' holes and wires line up, you may see large dark or light
blotches ("moiré patterns") that don't really exist beyond
the screens. Something like this had turned up in Phillips's equations.
Suppose the computer has gone through a complete step of calculating
the atmosphere. Among the countless simulated waves sloshing in the
simulated atmosphere, many would have wavelengths smaller than the
grid spacing. Some of these waves would reach a peak at each grid
point. These would look to the computer just like longer waves (in
mathematicians' jargon this is "aliasing"). When the computer calculated
another step, these particular waves would be selected for a wholly
artificial magnification, more at each repetition, until the whole
system went awry. 

Arakawa sought a way to defeat the problem. The key, he realized,
would be a mathematical scheme that described air flow in such a way
that certain particular combinations of the numbers representing the
flow would remain unchanged. One of these combinations was the kinetic
energy, the energy of the wind motion. In the real world, kinetic
energy is usually not kept strictly constant. But by finding a way
to do calculations that would hold it constant from one step to the
next, Arakawa could make sure that no unrealistic spike of wind speed
would grow exponentially step by step. He also decided the system
should conserve another, more abstruse combination of numbers describing
the flow of air, called "enstrophy." For this as for wind speed, calculations
at a set of discrete grid points tended to produce an erroneous, unlimited
growth. Again the solution would be a mathematical system that held
the quantity constant at each step, squelching the spurious magnifications.(4*)


Given these basic ideas, Arakawa still had the challenging task
of designing equations that would represent the atmosphere in a manner
suitable for computer calculation. His path lay through technical
mathematics that it is needless to describe here. Making this kind
of match between physics and mathematics requires a rare kind of insight,
but a flash of inspiration does not bring success without many days
of pencil and paper work exploring different avenues. Only after finishing
this tough task could Arakawa start the job for which Mintz had brought
him to California, coding a program for the computer. 

This too was no small job, for programming
a computer of the day meant writing page after page of instructions,
telling the computer to take the number found in register suchandsuch
and add it to register soandso. Then you fed your program into the
computer, waited patiently for the printout, and studied the result
— which was usually nonsense. There might be a subtle error in
the way you had set up the program, but more often you had just hit
a wrong key at some point in writing the code. You fixed it, got another
printout full of nonsense, and began to hunt down the next problem.(5)
Only a perfectionist, which Arakawa was, could persist in such labors
for years. Sometimes Mintz gave him a hand, using his intuitive understanding
of air flow to suggest where an error lurked, although Mintz never
wrote a line of code himself. 
=>Models
(GCMs)

Meanwhile, starting in 1962 Arakawa presented his ideas at conferences.
(Busy with his programming, and determined to develop general proofs,
he did not get around to publishing a journal article until 1966.
But the community of modelers was so small and closeknit that by
then everyone already knew about the ideas and was
using them.)(6) He later recalled that while some of his colleagues were
enthusiastic, he had a hard time convincing others that the approach
made sense. It seemed intuitively implausible, unlike anything they
had seen before, hard to grasp. Some objected that he was misguided
to use equations that forced the conservation of a quantity like enstrophy,
which nature itself did not conserve. He replied that he was not really
dedicated to holding enstrophy constant — it was only a device
to make the calculation behave more realistically. 
Arakawa ca. 1997
(Courtesy NASA)

The device worked magnificently. It became a basic component of
the UCLA model for the general circulation of the atmosphere. Later
this scheme for computing air flow was incorporated in many other
models that borrowed ideas and even computer code from the UCLA model,
directly or at second hand. Meanwhile, other
modelers found a variety of other ways to damp down spurious magnifications
and other instabilities. If there is a single best way to build a
computer climate model, nobody found it. The community of experts
perennially debated the merits and drawbacks of various calculation
schemes. Meanwhile Arakawa's basic technique (in technical terms,
the numerical conservation of a squared quantity, especially kinetic
energy) remained in widespread use.(7) 
=>Models
(GCMs)

A notable example of a model that borrowed
Arakawa's scheme was the GISS General Circulation Model, begun in
the 1970s by a group under James Hansen at NASA's Goddard Institute
for Space Studies in New York City. While Arakawa pursued improvements
in the elusive mathematics and physics, Hansen wanted to churn out
rough results on practical questions as soon as possible, even without
fulltime access to a supercomputer. To keep the number of computations
manageable, Hansen's group used a grid as much as a thousand kilometers
on a side, averaging over all sorts of different weather patterns.
After all, it wasn't weather he was interested in, but climate. The
model produced a surprisingly realistic simulacrum of atmospheric
circulation, including even a jet stream (the real jet stream is often
much narrower than a thousand kilometers). That was possible because
Arakawa's computation scheme suppressed spurious irregularities in
local weather.(8) 
=>Models
(GCMs)

Down through the 1990s,
Hansen's group used this model for highly influential studies that
showed how various factors worked upon climate.(9) They were regularly criticized for using such a coarse grid,
in which, for example, the entire contiguous United States was divided
into barely a dozen cells. As computers got faster and faster, other
modelers minced the planet into ever finer slices. Hansen had to defend
his choice of a grid against criticism by the anonymous referees who
commented on the proposals he sent to funding agencies and on the
papers he submitted for publication. He insisted that making smaller
cells would be wasteful, for there were plenty of other things more
worth spending his computer time on, factors that did far more to
limit the model's validity. He could get away with the coarse geographical
resolution thanks to the power of Arakawa's computation device.(10) 
=>Models (GCMs)
=>Aerosols


RELATED:
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General Circulation Models of the Atmosphere
1. Phillips (1956). BACK
2. To be precise, "the grid system cannot resolve wave lengths
shorter than about 2 grid intervals" and "interprets them incorrectly as long waves." Phillips (1959), p. 501.
BACK
3. Here and below I use chiefly Arakawa (2000); see also Johnson
and Arakawa (1996), pp. 321718.
BACK
4. Ideas on conserving a squared quantity involving
energy had been offered by Lorenz (1960). Arakawa,
personal communication, 2002, patiently helped me remove some particularly
bad errors from this oversimplified account. BACK
5. This passage is based on my own experiences in the 1960s. BACK
6. Arakawa (1966); Lilly (1997).
BACK
7. Lilly (1997). BACK
8. Hansen et al. (1983). BACK
9. E.g., Hansen et al. (1992); Hansen et al. (1984).
BACK
10. Hansen et al. (1983);
Hansen, interview by Weart, Oct. 2000, AIP and Hansen et al.
(2000).BACK
copyright © 2003 Spencer Weart &
American Institute of Physics
