CHAOS 8- The Lorenz waterwheel
Listen to Ed Lorenz telling us
about his concept of chaos in three points:
«lf a single flap of a butterfly’s wings
can be instrumental in causing a tornado,
so also can all the previous
and subsequent flaps of its wings,
as can the flap of the
millions of other butterflies,
not to mention the activities of
innumerable much more powerful
creatures, including our own species? »
(We have already seen that the
general public understands this first point well)
If a flap of a butterfly's wings
can be instrumental in causing a tornado,
it can equally well be instrumental in preventing it.»
Yes!
But the third point
is the most important
and this one provides work for scientists.
«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 tornadoes.
The most they may do
is to modify the sequence
in which these events occur."
Let’s return to the Lorenz model.
Remember that each point in space
represents a state of the atmosphere.
In some areas the weather is fine
and in others a hurricane blows!
Suppose that the area
corresponding to a hurricane
is this little ball.
Let’s take an initial climate condition
and let’s observe the trajectory.
Occasionally,
a hurricane occurs,
when the trajectory enters the ball.
Let’s measure the proportion of the time
between zero and T during which a hurricane rages.
It seems difficult
to guess at which specific moments
the trajectory enters the ball,
but the average time inside
the ball converges to a limit
when T becomes infinitely long.
Here it is 5.1%.
We now take another initial condition
and do the same calculation.
Well, again the trajectory
spends a certain amount of its time
inside the ball but
Oh! How about that,
it's the same as before: 5.1%!
Lets’ try it with yet another trajectory...
It works!
We obtain the same limit!
…and yet, the trajectories
are very different, they are sensitive
to initial conditions
on the wings of the butterfly...
We add another ball,
pink maybe, and a third one, green...
Perhaps periods of heat waves or snow?
When a trajectory develops,
it traverses the balls in a certain order.
To see what happens,
let’s take three trajectories
starting from three initial conditions.
Look.
On the trajectories,
the situations hurricane / snow / heat wave
alternate incomprehensibly,
and in a different way for every trajectory.
Impossible to understand.
But the proportions of yellow, green and pink
become the same for all of them
Here they are, 5.10, 14.03 and 7.40%
It is as if we picked yellow,
green or pink balls at random
with well-defined probabilities.
Listen to Lorenz again:
«Small changes do not increase
or diminish the frequency
of weather events like tornadoes.
The only thing they can do
is to change the order
in which these events occur."
This is a scientific statement!
The purpose of the forecast has changed.
We now try to predict
averages, statistics, probabilities.
The idea is that these statistics are - perhaps...
insensitive to initial conditions and to the
flapping of the wings of Brazilian butterflies!
This hypothesis - unproven...
the possible coexistence of a
meteorological chaos where the
future movements are sensitive
to initial conditions, and a statistical stability,
insensitive to initial conditions,
took a long time
to be formulated
mathematically.
Lorenz was probably a little embarrassed
by the simplistic (to say the least!)
theoretical side of his 3-parameter atmosphere.
With the help of two physicists,
Howard and Markus,
he developed a "real physical system",
even if it is still simplistic and
far removed from the true weather phenomena!
Here is a mill.
It consists of a wheel,
with buckets suspended all around it.
The buckets have a hole in the bottom of course,
and the water can run out, all the more
quickly if the water level in a bucket is higher..
We open a valve at the top
and watch the movement.
The mill turns sometimes to the right,
sometimes to the left and
we seem to be totally incapable of
predicting which way it will turn
in a few seconds.
The movement seems totally
erratic, unpredictable, chaotic.
In fact, is there a relationship
between the mill and the Lorenz attractor?
Let’s choose three numbers to
describe the mill, for instance the angular velocity
and the two coordinates of its center of gravity.
The evolution of these numbers traces
a yellow curve in space.
Amazing is it not?
Our mill is moving ...like a butterfly!
Let’s change the
initial position imperceptibly.
All buckets are initially empty
and the left wheel is turned by 2 degrees,
while the right wheel
is turned only 1.9996 degrees.
(we would need a good
microscope to notice the difference).
Yes, there clearly is a
"sensitive dependence on initial conditions"!
After a while, the two mills
have completely different behaviors.
Let's see if Lorenz’
statistical statement is true for the mill.
Our two mills take off
from almost the same position.
Let’s measure their speed:
for instance 25 times per second
during 5000 seconds,
so 125 000 observations.
At each observation,
we note the rotational speed of the wheel.
A simple idea is to do
what statisticians do:
a bar chart to illustrate the distribution.
We have 35 equal intervals
for the speed
and we count the number of times
that the measured speed
falls in each interval.
Here is the result.
Some intervals appear to
be visited more than others.
Our two mills behave
in very different ways.
There is sensitive
dependence on initial conditions.
But the observation is
that the two sets of data,
although different,
are statistically identical.
The bar charts tend to become identical.
It works!
Lorenz seems to be right!
When the statistics of a trajectory’s future
are insensitive to initial conditions,
we say that the dynamics have a
Sinai-Ruelle-Bowen measure, or SRB measure.
The goal of a forecaster is then
to determine these statistics.
For example, let’s observe
the evolution of temperature over time,
one of the coordinates in Lorenz’s equation.
We draw another bar chart,
as we did with the mill.
For each temperature range,
we note the proportion of observations
that fall within this range.
Even if we consider trajectories
that are quite different from one another,
after a while the diagrams
tend to become identical.
The Lorenz attractor has an SRB measure.
It is as if the temperature
was changing at random
but with very specific probabilities.
But there is a lot left to do
mathematically and physically.
We must find these probabilities
and these distributions in order to
be able to say something useful!
Let’s hope we have done justice
to Lorenz who did not say merely
that the future depends heavily on the present.
Many people before him had said that.
But his contribution is also,
and perhaps most importantly,
to show that by refocusing
on statistical issues,
Science can still make predictions.