Absolutely false. Chaos theory was discovered by scientists doing atmospheric modeling. Weather is the canonical example of a real-world chaotic system.
A small disturbance can globally perturb a system without supplying the energy that drives the system. The fact that there are storms at all is the result of the sun pumping solar energy into the atmosphere. But *where *and *when *the storms appear is the result of a chaotic process that is driven by tiny events all across the globe. The minuscule eddies that I create every time I exhale will, over weeks or months, change the course of the weather in China.
I also am very interested to hear what someone who has experience modeling the atmosphere says. Baring that, maybe you can elaborate on how you came to your understanding of the atmosphere’s level of “chaoticity”. Because to me at least, it seems like a practically unanswerable question. There are so many variables and external interactions that I don’t see how you can say that a tiny local event will make any sort of measurable impact on a large scale, all else being equal. This is doubly true when you consider the tendency for any dynamic system is to try and reach an equilibrium.
I guess I’d be satisfied if you agree with me on this hypothetical: Say you lived on a planet where for someone reason the atmosphere started at a uniform temperature, pressure, and density, and received no heat from an external object or the planet itself. You would agree the steady state behavior would just be to sit there in hydrostatic equilbrium, right? Would you also agree that if you were on that planet, nothing you could do by jumping around, waving your hands, etc. could create any macro change any time in the future? Seems to me that if you disagree with that, you’re disagreeing with the second law of thermodynamics.
Alright, I’m going to need a cite then that global weather states over a period of months are sensitive down to the level of perturbations over a few feet. I’ve been trying to explain why that seems like a lot and responses just seem to be “don’t you get it? it’s chaotic”.
As Sam said, it’s all a matter of scale. Do you think changing a single molecule’s velocity will have a macro impact? Over what time scale? And finally, how would you calculate that?
I had thought that the butterfly effect was about the possibility that a small event could make a difference in a larger one. For instance a system that might become a hurricane, or just a tropical depression, could at some point have the small change from a butterfly’s wings tip it in one direction or the other. Now this thread is telling me there is a lot more to know about chaotic systems than I imagined. Who knew chaos could be so complicated?:eek:
There is not really such a thing as “chaoticity” (there are lots of interesting differences between different chaotic systems, but this is not one of them). Either something is chaotic or it is not. Weather is chaotic. Now, that doesn’t mean that a butterfly can cause the earth to explode :); it doesn’t mean a small perturbation can cause anything to happen; all it means it that weather patterns, such as clouds, winds, rain, are chaotically sensitive to initial conditions. The definition of chaotic implies that they are arbitrarily sensitive to initial conditions. What that means is that even the tiniest movement of a single air molecule is enough to cause wildly different weather patterns down the line.
It depends what you mean by “macro change any time in the future”. If you mean “the locations of all of the individual air molecules”, then I disagree with you: the system is still chaotic. If you are looking for some more visible sign of chaos, such as a weather pattern, then by suggesting a system without enough energy being added to the environment to create weather patterns in the first place, then what you are doing is begging the question. If you were to suggest a system with enough energy being added to generate weather patterns, I would agree that there are certainly a few very artificial scenarios in which the macroscopic behavior would not be chaotic.
Technically such answers (“don’t you get it? it’s chaotic”) are correct. That is why I asked you whether you really understood what “chaotic” means. The entire point of chaos is that you can’t say “let’s make the perturbation small enough to not be noticeable”. That is why the term was invented: even the tiniest perturbation causes large changes down the line. No matter how small the perturbation. This is a re-statement (in laymen’s terms but nonetheless correct) of the technical definition of “chaos”.
At the small end of the scale we can look at attempts to model turbulence. There’s been a huge amount of work in this area because understanding turbulence is important for a number of practical problems: airplane design, nuclear weapon design, oceanography, etc. All efforts to *precisely *model turbulent flow have failed. For a given situation we can determine the average magnitude of the eddies that will form, but we can’t predict the specifics of each individual eddy. This behavior seems to hold even at very small scales.
A butterfly’s wing produces tiny eddies. The tiny eddies affect small eddies nearby. The small eddies effect medium eddies further away. The medium eddies affect large eddies. And so on. The behavior of the air mass at each scale affects the behavior of the nearby air masses at the next larger scale in a continuous chain from microscopic to global.
When I said it’s a matter of scale, I didn’t mean scale between the input and the size of the output. I meant that at macro scales the chaotic effects can statistically average out. We don’t know exactly where a particular cloud may form or what shape it might take, because cloud formation is sensitive to arbitrarily small input effects. But we still might be able to predict the total expected percentage of cloud cover over a large area, because the smaller chaotic patterns average out just like collisions of random particles in a pressure vessel can result in extremely predictable pressures at the macro level.
Sorry I didn’t get a chance to post this weekend. Hopefully this bump will bring about more discussion.
I don’t disagree technically with anything you’ve said, but all it really is is a restatement of the definition of a chaotic system. What I meant when I complained about the responses I was getting was not that people were just glossing over the definitions, but that they were just asserting that weather is a chaotic system without qualification and without backing it up. Notice you did this too.
I’ve no doubt that the phrase “weather is chaotic” is true in the most general sense but that leaves a lot of details that need to be filled in. There’s the question of what size of initial perturbation is required to create a trajectory divergence. You may find it intuitive to think that even just a single molecule being changed is enough to alter global weather sometime down the road but I don’t. I mean yes, it will be altered in the sense that eventually no atoms will be in the same state, but that’s not what we’re talking about. Please fight my ignorance and show me how we know that weather phenomena are sensitive down to the level of a single molecule or breath puff.
There’s also the question of time scale. A system may take millions of years to fully mix topologically and yet it would still be chaotic. I’ve seen claims of weeks and months for the difference of my breath to propagate to the global level. Where are these numbers coming from?
This may be true, but it doesn’t indicate that the actual weather phenomena will be different. An air mass containing a butterfly flap may have a different swirling pattern of atoms that its insect-less counterpart, but the two will be thermodynamically indistinguishable. Why should one go on to affect the global weather state any differently than the other?
You may want to go back and readdress my post as you totally missed what I was saying. My apologies if I was unclear though; I’ll try to be more explicit here.
The issue is NOT applying the definition of a chaotic system to weather, but rather supporting the assertion (made many times) that the evolution of weather phenomena IS a chaotic system. To be extra-specific, it is supporting the claim that given all other conditions being equal, a “micro-disturbance” will eventually lead to a completely different weather state on the global level within, say, one year. My point is that while you can certainly say (1) weather is chaotic and (2) by definition, any initial condition difference means divergence, you’re begging the question without showing (1).
I can go on for a bit more about details about what you would need to show to verify such a specific claim, but for now I just want to see your response to my actual position.
Yes, destiny. I recall an “It’s a Wonderful Life” type story in which the guide character definitely stated such. Persons with strong bonds of love,friendship, or hate in one universe would always be drawn together, though their interactions would differ. That is, you might find a world in which Rhett stayed with Scarlett rather than leaving her; you might find a world in which they were always deadly enemies and one ended up murdering the other. You’d never find a world in which they were indifferent to one another or didn’t know one another.
If it’s a cite you want. Haveatit. I thought that it was understood that weather is the prototypical example of a chaotic system. Not only prototypical, but emblematic. Pretty much any online resource should point out that the study of weather is what first led to the notion of chaos in the first place.
Now, mathematically it is trivially a chaotic system (any N body system where N >= 3). That is, if you follow the trajectories of the individual particles. But I understand your point that this is a necessary but not clearly (to you) a sufficient condition to understand that weather is chaotic. In response to that I refer you (see cites, but #3 specifically) to computational atmospheric modeling, where it has been repeatedly shown that if you input an arbitrarily small error into your system, your simulations become locally non-predictive after a few weeks. But this should be clear, for reasons that a number of previous posters made explained. For example the butterfly effect is not really that abstract, it is actually fairly intuitive once you grasp how it works:
and
I will also quote from the third link I provided above:
Yeah, I doubt we’ll be able to progress past here, just because there’s no way any modeling has been done which shows atmospheric dynamics in the range of continents all the way down to centimeters. I only looked through the third cite in any detail (which was still not much) and it looks like their “micro” scale is the tiny size of two kilometers. AND they don’t even include the dynamics of the micro and meso scales (up to 2000km) in prediction models.
So that’s why I said certainly weather phenomena are chaotic in general. When you are looking at tiny perturbations that are on that scale, it’s a no-brainer to think you’ll get chaotic behavior. But again, I fail to see why I should automatically accept the same behavior occurs as you arbitrarily scale down the perturbation.
Your selection of quotes are certainly intuitive explanations of the smallest disturbances propagating into massive transients, but I might find it intuitive to think such a small variation will almost instantly become lost in the thermal noise of the system and then we’ll have two states that produce identical macro-behavior. How do we prove whose intuition is right?
I’ll add a cite of my own, which is actually one of yours. Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?. I think Lorenz would answer the question in the affirmative but he concedes that we simply do not know and gives a few reasons why it may actually be no. So I guess the only takeaway I have to suggest is to perhaps be a little less sure when telling people about how the butterfly effect works in reality.
Christ, Saganist, there are literally hundreds of cites I could refer you to describing weather as chaotic. How many would be sufficient. You have admitted that you understand what chaos means… the only thing left is whether or not you trust in the academic establishment. If you don’t, then I suggest you start a different thread…
But to explore the cites I’ve already provided a little more, this is from the first cite:
[bold mine]
That cite also goes into detail about the non-linearity of the climate modeling equations. It has long been understood the the equations themselves describe chaotic behavior. So technically there is no need to show anything more.
Now, since you don’t seem to understand or accept that, here is more from the last cite:
Now, the ability to do these integrations for sub-km resolutions is dependent on available computing resources, but as computing power has increased, resolutions have increased and studies have continued to show (as per the non-linear equations used by the simulations have long been known to describe) chaotic behavior. But this is generally not emphasized in publications because such information is redundant (and because in practice predictive modeling cannot rely on sub-km precision of initial conditions – ie weather sensors are not everywhere).
This would have been a more appropriate response to post #49, not my most recent one. Now I have to repeat myself because once again, it appears you didn’t understand what I said.
Chaos is a mathematical property. So when you say there are plenty of cites showing weather is chaotic, what you mean is that there are plenty of cites showing certain mathematical models that correlate to weather behavior are chaotic. And, as I think is pretty obvious, current models operate at large scales. Scales that are on par with the size of the weather phenomena being observed. So tell me again why you can extrapolate these results without controversy to scales far smaller, which are not representable in these models.
Uh yeah, it talks about the Lorenz equations. You think the fact that the Lorenz equations show chaotic behavior implies the same for weather phenomena under any level of initial condition variation?
Then we may not be too far away from showing the butterfly effect to be real. I’m still not sure why you’re insisting that we know it already though.
Such models do not “operate at large scales.” In practice they are applied at large scales. The models themselves are applicable to any scale. Incidentally in the cite I quoted they used the phrase at all scales for this very reason. The models consist of non-linear equations whose behavior is chaotic. The errors on initial conditions used in numerical integration of those equations is irrelevant to the applicability of the equations themselves.
While the Lorenz cites are a great place to start reading about this stuff, certainly a lot of progress has been made since the 1970’s. The so-called Lorenz equations represent a simplified model of convection, and in no way represent more realistic atmospheric simulations. But yes, I do in fact actually think that the chaotic behavior of the Lorenz equations implies the same for weather phenomena. Perhaps the point requires mathematically “trained ears” but it would be implausibly miraculous if the equations of real atmospheric convection were not chaotic given that the Lorenz equations are. It would be like modeling a 6-body-system with a 3-body system, and arguing that though the 3-body system is chaotic, we don’t know whether the 6-body system is chaotic. Now, it is conceivable that some sort of miraculous cancellation and cooperation of dynamics comes into play when you add 3 more bodies to a 3-body system, but there is no hint of evidence for that, and such cancellations are only possible in extremely contrived examples. In general, if a simplified system is chaotic, the more complex model of the system is also chaotic.
I am not a climate scientist, but I trust climate scientists when they describe weather as chaotic. I know the definition of chaotic, and I assume that they do also. I don’t think that this is unreasonable, unless you want to get into a debate about the competency of climate scientists.
