What is the chain of cause(s) and effect(s) which does a butterfly's wing-flapping cause a hurricane 1000's of kilometers further away?

Question

I once asked if the butterfly effect in the weather system is a real effect: a butterfly flaps her wings, which can cause a hurricane thousands of kilometers further.

But how will this happen? To be more precise, how does a tiny change in the movement of air (of a tiny part of the WHOLE weather system) cause a hurricane to appear 1000's of kilometers further away?

A hurricane is a huge domain of energy, which can damage big areas on land. Can this really be caused by the flapping wings of ONE butterfly 1000's of kilometers away from the area where this hurricane is supposed to appear after this flapping?

I'm asking for the real chain of events following the flapping of her wings of which it is alleged that it can lead to the hurricane. To avoid more complications let's assume (the highly hypothetical) case that the Earth's is perfectly flat including the water. So what would be the chain of (forked) cause(s) and (forked) effect(s) leading from the flapping to a hurricane far, far away in this case?

It is said that it's just a metaphor, but why use it as a metaphor if it can't be related to chaos theory?

See also this question.

Answer 1

It's a mistake to attribute the hurricane to the butterfly. Rather, hurricanes are going to happen at various different times and in various different places, but the precise time and location are sensitive to small effects. Not just the butterfly, but also myriad other processes contributed to determining the time and place and size, most of them much more significantly.

Let's compare this to something a little simpler: an avalanche of snow. Someone might say that a small noise 'caused' the avalanche, in the sense that if the small noise had not happened just then, then the snow would not have started to slip just then. But of course, the main causes of an avalanche are first the build-up of snow over a long period, and then the particular properties of snow which enable it both to bind to itself and also slip past itself, and finally the influence of the gravitational field of the Earth and the shape of the mountain and so on. And if one particular noise had not set it off, then something else would have done, very likely, soon after.

This illustration does not capture the more interesting dynamics of a chaotic system, but it serves at least to show that the word 'cause' is being used in a rather misleading way when someone says an avalanche is caused merely by a cough, or when they say a hurricane is caused merely by a butterfly. Such a statement fails to mention almost all the actual causes!

I guess the point of the butterfly analogy is this. A chaotic system such as Earth's atmosphere has an exponential sensitivity to initial conditions. This sensitivity is owing to the combination of gravity, heating from the sun, the rotation of the Earth, the physics of gas and water, etc. The location, timing, and strength of a hurricane are influenced mainly by these large-scale facts, but owing to the sensitivity, the details of the location are also influenced by smaller things such as the shape of the landscape, and even the particular shapes of waves on the ocean, and things like that. Even after one has taken into account the effects of landscape and waves, one would still find that even smaller things have a non-negligible impact on the time and location, right down to tiny things like air currents disturbed by butterflies. However, I do not know what are the limits on such effects: see the note added below.

Added note

With thanks to Niels Nielsen, I will add that it is entirely possible that the local damping that is present in air is enough to make butterflies in fact irrelevant to weather systems. In order to model a portion of atmosphere, one might use some simple model such as the one introduced by Edward Lorenz. He was simply interested in exploring what sorts of things can happen; he was not attempting a thorough model. In that model one finds deterministic chaos with all its interesting features, especially the exponential sensitivity to initial conditions. In such a model even the effect of a butterfly will be inexorably amplified up and up. However, a more realistic model will involve further features, including local damping. In such a model I think there can be damping of very small perturbations and exponential growth of somewhat larger ones. Having said all that, I am not working directly in this area so I hope an expert who is will chip in to confirm this last statement.

Answer 2

The butterfly and hurricane is just a metaphor. It should not be interpreted as: A butterfly can trigger a hurricane. It's more like: the weather dynamics are so sensitive that you can not ignore any small perturbation, since that a small eddie somewhere can change the general behavior.

Just to take the easy well known Lorenz attractor

$$\dot{x}=\sigma (y-x)$$ $$\dot{y}=x(\rho-z)-y$$ $$\dot{z}=xy-\beta z$$

with $x$, $y$ and $z$ the convection, horizontal temperature and vertical temperature rates respectively and $\sigma$, $\rho$ and $\beta$ are the Prandtl number, Rayleigh number and a characteristic number to describe the fluid layer.

If you set $\sigma=10$ and $\beta=8/3$ and set $\rho$ as a bifurcation parameter from 20 to 30 or so, you can pass from a very predictable system to a hard to make long term prediction given a slight difference in the initial conditions system. That is the butterfly is a perturbation to the convection rate or temperature change and the hurricane one of the possible solutions, that will change due to the flap of the butterfly

Answer 3

The problem is that the butterfly effect has been poorly explained and poorly represented in a lot of popular sources. The cause-effect relationship is reversed. The flapping of the wings does not cause a hurricane, but omitting the flapping of the wings causes the mistake in the prediction.

The Chaos theory deals only with error propagation in mathematical calculations. It only wants to give an estimate of the possible difference between the estimated outcome and the actual outcome taking into account the complexity of the calculations and eventual defects by imprecisions in the measuring instrument or missing data.

EDIT:

The answer by niels nielsen marked as correct is NOT correct. I'm surprised to see how difficult it is to get such a simple concept. I'll try and rewrite my answer with other words:

The air moved by the wings of a butterfly is one of the billions of data to take into account by the calculations. By itself it does not causes the hurricane, but it contributes together with all the other data to the cause of the hurricane. It can be said that its contribution is so small that it is negligible. But the above blunt oversimplyfied statement is as much stupid as the oversimplified statement in Jurassic park style that created the misconception in the beginning.

All the Chaos theory says is that if your calculation if far too complex you cannot afford to miss even that tiny data that gives such a small contribution to the picture because that small error may increase exponentially.

Answer 4

It is not true that the flap of a butterfly's wings can give rise to a hurricane thousands of kilometers away. The atmosphere is sufficiently well-damped that butterfly wing flaps do not propagate more than ~1 meter, and only if there are no larger perturbations (like the flow of wind through a tree) in the vicinity.