APPENDIX 1
The Butterfly
Effect
Portuguese
version
THE
FOLLOWING is the text of a talk that I presented in a session devoted
to the Global Atmospheric Research Program, at the 139th meeting of
the American Association for the Advancement of Science, in Washington,
D.C, on December 29, 1972, as prepared for press release. It was never
published, and it is presented here in its original form.
Predictability:
Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in
Texas?
Lest
I appear frivolous in even posing the title question, let alone
suggesting that it might have an affirmative answer, let me try
to place it in proper perspective by offering two propositions.
1. If a single flap of a butterfly's wings can
be instrumental in generating a tornado, so also can all the previous
and subsequent flaps of its wings, as can the flaps of the wings
of millions of other butterflies, not to mention the activities
of innumerable more powerful creatures, including our own species.
2. If the flap of a butterfly's wings can be
instrumental in generating a tornado, it can equally well be instrumental
in preventing a tornado.
More generally, I am proposing that over the
years minuscule disturbances neither increase nor decrease the
frequency of occurrence of various weather events such as tornados;
the most that they may do is to modify the sequence in which these
events occur. The question which really interests us is whether
they can do even this – whether, for example, two particular
weather situations differing by as little as the immediate influence
of a single butterfly will generally after sufficient time evolve
into two situations differing by as much as the presence of a
tornado. In more technical language, is the behavior of the atmosphere
unstable with respect to perturbations of small amplitude?
The connection between this question and our
ability to predict the weather is evident. Since we do not know
exact1y how many butterflies there are, nor where they are all
located, let alone which ones are flapping their wings at any
instant, we cannot, if the answer to our question is affirmative,
accurately predict the occurrence of tornados at a sufficiently
distant future time. More significantly, our general failure to
detect systems even as large as thunderstorms when they slip between
weather stations may impair our ability to predict the general
weather pattern even in the near future.
How can we determine whether the atmosphere
is unstable? The atmosphere is not a controlled laboratory experiment;
if we disturb it and then observe what happens, we shall never
know what would have happened if we had not disturbed it. Any
claim that we can learn what would have happened by referring
to the weather forecast would imply that the question whose answer
we seek has already been answered in the negative.
The bulk of our conclusions are based upon computer
simulation of the atmosphere. The equations to be solved represent
our best attempts to approximate the equations actually governing
the atmosphere by equations which are compatible with present
computer capabilities. Generally two numerical solutions are compared.
One of these is taken to simulate the actual weather, while the
other simulates the weather which would have evolved from slightly
different initial conditions, i.e., the weather which would have
been predicted with a perfect forecasting technique but imperfect
observations. The difference between the solutions therefore simulates
the error in forecasting. New simulations are continually being
performed as more powerful computers and improved knowledge of
atmospheric dynamics become available.
Although we cannot claim to have proven that
the atmosphere is unstable, the evidence that it is so is overwhelming.
The most significant results are the following.
1. Small errors in the coarser structure of
the weather pattern – those features which are readily resolved
by conventional observing networks – tend to double in about
three days. As the errors become larger the growth rate subsides.
This limitation alone would allow us to extend the range of acceptable
prediction by three days every time we cut the observation error
in half, and would offer the hope of eventually making good forecasts
several weeks in advance.
2. Small errors in the finer structure –
e.g., the positions of individual clouds – tend to grow
much more rapidly, doubling in hours or less.
This limitation alone would not seriously reduce our hopes for
extended-range forecasting, since ordinarily we do not forecast
the finer structure at all.
3. Errors in the finer structure, having attained
appreciable size, tend to induce errors in the coarser structure.
This result, which is less firmly established than the previous
ones, implies that after a day or so there will be appreciable
errors in the coarser structure, which will thereafter grow just
as if they had been present initially. Cutting the observation
error in the finer structure in half – a formidable task
– would extend the range of acceptable prediction of even
the coarser structure only by hours or less. The hopes for predicting
two weeks or more in advance are thus greatly diminished.
4. Certain special quantities such as weekly
average temperatures and weekly total rainfall may be predictable
at a range at which entire weather patterns are not.
Regardless of what any theoretical study may
imply, conclusive proof that good day-to-day forecasts can be
made at a range of two weeks or more would be afforded by any
valid demonstration that any particular forecasting scheme generally
yields good results at that range. To the best of our knowledge,
no such demonstration has ever been offered. Of course, even pure
guesses will be correct a certain percentage of the time.
Returning now to the question as originally
posed, we notice some additional points not yet considered. First
of all, the influence of a single butterfly is not only a fine
detail – it is confined to a small volume. Some of the numerical
methods which seem to be well adapted for examining the intensification
of errors are not suitable for studying the dispersion of errors
from restricted to unrestricted regions. One hypothesis, unconfirmed,
is that the influence of a butterfly's wings will spread in turbulent
air, but not in calm air.
A second point is that Brazil and Texas lie
in opposite hemispheres. The dynamical properties of the tropical
atmosphere differ considerably from those of the atmosphere in
temperate and polar latitudes. It is almost as if the tropical
atmosphere were a different fluid. It seems entirely possible
that an error might be able to spread many thousands of miles
within the temperate latitudes of either hemisphere, while yet
being unable to cross the equator.
We must therefore leave our original question
unanswered for a few more years, even while affirming our faith
in the instability of the atmosphere. Meanwhile, today's errors
in weather forecasting cannot be blamed entirely nor even primarily
upon the finer structure of weather patterns. They arise mainly
from our failure to observe even the coarser structure with near
completeness, our somewhat incomplete knowledge of the governing
physical principles, and the inevitable approximations which must
be introduced in formulating these principles as procedures which
the human brain or the computer can carry out. These shortcomings
cannot be entirely eliminated, but they can be greatly reduced
by an expanded observing system and intensive research. It is
to the ultimate purpose of making not exact forecasts but the
best forecasts which the atmosphere is willing to have us make
that the Global Atmospheric Research Program is dedicated.
Edward
Norton Lorenz, 1972.