

Chaotic systems and randomness - whiskers
http://tamino.wordpress.com/2011/06/14/chaos/

======
foob
There are major problems with the logic in the article. He's basically trying
to argue that the average of a variable in a chaotic system over some finite
period of time can't drift or change dramatically due to chaos alone. His only
support for this is that it doesn't happen in the simplest possible one
dimensional discrete chaotic system. What about the Lorenz attractor? If you
aren't familiar with it, it basically acts like two well behaved oscillators
that you switch between after some "random" amount of time. If you take a time
average of a space coordinate with the Lorenz attractor you'll see chaotic
swings between periods of relative stability in the two oscillators. As you
make increasingly complicated chaotic systems with more and more variables you
open up the possibility of having all sorts of longer term behavior that would
effect what he calls "climate". He is absolutely wrong about this.

 _When you increase greenhouse gases and therefore inhibit heat loss, you
change the dynamics — the “equations of motion” as it were — and that will
change the climate. In ways that are predictable._ He's wrong here as well.
From bifurcation theory we know that tiny changes in the parameters of a
dynamical system can have huge seemingly unpredictable effects on the
behavior. Subtle differences in models can have huge impacts on where and how
bifurcations occur. We have only various educated guesses that might very well
be dramatically wrong.

By the way, I'm not arguing against climate change. I just think that this
article is terribly wrong.

~~~
jerf
As someone who doesn't particularly believe in the Standard Dogma of warming,
I believe it's my turn to trot out an old chestnut: "If you don't understand
[chaos theory], you shouldn't be arguing about it. [Chaos theoreticians] have
spent a lot of time working these things out and are very smart, do you really
think you're going to outsmart them?"

(And I do accept the credentials of the original poster on WUWT; you can't
help but learn about chaos if you're in that career path. It's staring you in
the face every day.)

The really interesting thing in the original post is the characterization of
the Lyapunov exponent in terms of bits lost per time unit. If you understand
information theory and what that really means, it is an extraordinarily
powerful argument that climate is truly, fundamentally not predictable, on a
deeply, profoundly mathematical level.

Part of why I don't really accept the Standard Dogma is precisely my
impression that the weather simulators have no idea what realm they are
operating in; they seem to have the same paper-thin understanding of chaos
theory as this poster, and I have seen no evidence that they understand
information theory or even know it exists, since I have never seen it
mentioned by them or any of their proponents, despite the powerful,
mathematical upper bounds it puts on their efforts, ones which are rather
tighter in nature than the predictions they so breathlessly make.

I'm not arguing "against" climate change, I'm arguing against the value of any
attempts to predict it beyond the upper limits established by mathematics.

~~~
scarmig
I'm curious (genuine question): we know that if we dropped the Earth into the
Sun, the climate would change substantially. And yet this is a prediction
about gross properties of a chaotic system that no one would dispute.

How does your point of view account for this prediction from physics versus
predictions of greater CO2 concentrations causing an increased greenhouse
effect, which also derives from physics? Is it simply the margins of error for
different parameters?

~~~
dexen
Fragments of the mentioned post [1]; it's worth reading in whole as it
explains the matter very neatly.

 _> Chaotic systems are not entirely unpredictable, as something truly random
would be. They exhibit diminishing predictability as they move forward in
time, and this diminishment is caused by greater and greater computational
requirements to calculate the next set of predictions._

 _> Computing requirements to make predictions of chaotic systems grow
exponentially, and so in practice, with finite resources, prediction accuracy
will drop off rapidly the further you try to predict into the future._

 _> The difference between the initial conditions is minute, but the two
series diverge for all that. This illustrates one of the key things about
chaos. This is the acute sensitivity to initial conditions._

 _> If you try to make predictions from your model, any minute inaccuracies in
your guess of the initial conditions will result in your prediction and the
result diverging dramatically._

 _> This divergence grows exponentially, and one way of measuring this is
called the Lyapunov exponent_

 _> It also gives us a bound on the quality of predictions we can get if we
try to model a chaotic system._

[1] [http://scientio.blogspot.com/2011/06/chaos-theoretic-
argumen...](http://scientio.blogspot.com/2011/06/chaos-theoretic-argument-
that.html)

\----

And my long ramblings (I'm already sorry for posting that):

The temperature of Earth stabilizes (`thermal equilibrium') when energy input
equal energy output. There are two serious sources of energy for Earth: Solar
radiation and radioactive decay in Earth's body (minor sources would be lunar
tides and chemical reactions, but I guess the former varies little and later
is of limited capacity and wouldn't matter much in the longer run).

If Earth was just large uniform sphere, with no magnetic field and no
atmosphere, predictions of surface temperature would be simple: from [1], the
temperature would have to be adequate to match thermal input. The temperature
would have to provide energy output equal to: internal heating + solar
radiation hitting earth - reflected radiation (the part of solar radiation
that did not transfer to the Earth).

The laws [1][2] of thermal radiation are simple for case of idealized black
body with uniform temperature, however Earth is not such one (unless dropped
onto the Sun ;-)). We have several features that reflect some solar energy
back to space, absorb solar energy, or emit energy from Earth's surface. The
temperatures on Earth are not uniform, and there are large active energy
transfer mechanisms [3]. Some of the features add or subtract energy emission;
those can be estimated quite reliably.

There are threshold effects, like possible methane release from hydrates on
oceanic floor, or large areas of vegetation disappearing because of
temperatures dropping too low. Those are harder to estimate reliably, as very
small variation in temperature will change wide areas of Earth's surface or
atmosphere. We probably don't know all of them yet.

There are some negative and positive feedback loops [4], like desert
reflecting more than vegetation, and also icecaps reflecting more than
vegetation; deserts grow with temperature increase, icecaps grow with
temperature decrease. Positive feedback are exceptionally hard to estimate
reliably.

Some of the features multiply (!) effects of other features; take clouds for
example; they block a lot of solar energy from reaching whatever feature is
below them; yet they depends on several climatic factors -- some of them
increase, othres decrease formation. Now that's got power to throw simulations
off-track even further.

To wrap it up: complex systems with many intermingled nonlinear factors; in
effect very sensitive to input parameters. And there are countless input
parameters...

[1] <http://en.wikipedia.org/wiki/Stefan–Boltzmann_law>

[2] <http://en.wikipedia.org/wiki/Planck_law>

[3] like <http://en.wikipedia.org/wiki/Gulf_Stream> that causes Warsaw to have
pretty much same temperatures Toronto has, only that it's 8.5 degrees further
to the north.

[4] there's speculation they may lead to runaway effects like
<http://en.wikipedia.org/wiki/Snowball_earth>

\----

And a great animation of a somewhat chaotic system:
<http://en.wikipedia.org/wiki/Swinging_Atwood_Machine>

~~~
scarmig
Thanks for the thoughtful writeup. But I don't get specifically how this
doesn't prove too much--even though climate is a complex system, we know
(right?) that throwing it into the center of the sun would increase mean
global temperatures. But if I said that chaos and information theoretic
arguments prevent us from making any predictions, I'd rightly be laughed off
the stage.

What is the differentiating factor that makes those arguments apply to models
looking at CO2 concentrations as exogenous forcers but not being-in-the-
center-of-the-sun?

~~~
jerf
"But if I said that chaos and information theoretic arguments prevent us from
making any predictions, I'd rightly be laughed off the stage."

"Information theory bounds our ability to predict" is _not_ the same as
"Information theory prevents us from giving any predictions". You could
predict the temperature of "the Earth" (for some suitable definition) within
some bounds, but if you tried to do it to within a tenth of a degree Celsius
for a hundred years you would continue to encounter difficulty... rather more
than you have right now, actually, I'm quite confident the Sun's Lyapunov
constant will strip bits from you much faster than Earth's current climate
does.

Also, you're engaging in a bit of tautology without realizing it. The entire
concept of "forcing" is one divorced from a chaos-informed concept of weather,
it's part of the reason I really don't think they understand chaos. The
"forcings" are themselves subject to the prediction problem, so you may be
able to predict what the system will do immediately after the application of
them, but you don't know what will happen later. In Earth's specific example,
what if the hotter climate ends up triggering Yellowstone? Or that methane
clathate catastrophe? Or something entirely unexpected? Much more
realistically, what if the "forcings" tend to cancel themselves out over
longer times? There's been a couple of good papers that suggest that all the
water on Earth (oceans, etc) are very, very strongly effecting the climate by
virtue of cloud production in the tropics such that it is _extremely_ strongly
forcing the climate to look the way it does now, and all your "forcings" may
be nothing next to it. Or that may be totally wrong. By the time you're done
caveating "forcings" in light of what a chaotic system can do, they aren't a
very useful approach to the problem.

