There is a popular notion that the “butterfly effect” describes a climate system that is exceptionally sensitive to very small perturbations. As just one example, a google search turned up the following description of it,
“In the arcane field of chaos theory, there is what scientists call “the butterfly effect,” the popular notion that when a butterfly flaps its wings in Asia the action may eventually alter the course of a tornado in Kansas.” (http://www.roanoke.com/editorials/commentary/wb/xp-32390).
Even the scientific community presents this perspective, e.g., see http://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.html, from which the following text is extracted,
“The ‘Butterfly Effect’, or more technically the “sensitive dependence on initial conditions”, is the essence of chaos……..The “Butterfly Effect” is often ascribed to Lorenz. In a paper in 1963 given to the New York Academy of Sciences he remarks: ‘One meteorologist remarked that if the theory were correct, one flap of a seagull’s wings would be enough to alter the course of the weather forever.’ By the time of his talk at the December 1972 meeting of the American Association for the Advancement of Science in Washington, D.C. the sea gull had evolved into the more poetic butterfly - the title of his talk was ‘Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?’ In the applet we also see a second incarnation of the Butterfly - the amazing geometric structure discovered by Lorenz in his numerical simulations of three very simple equations that now bear his name.”
The solution of the Lorenz equations from this informative website illustrates the “butterfly” looking field that results. Figure 6 from Tsonis and Elsner (1989) has a particularly clear illustration of the butterfly solution.
This second definition of the butterfly effect is the correct use of the term “butterfly effect.” However, the first usage of the “butterfly effect” in that “when a butterfly flaps its wings in Asia the action may eventually alter the course of a tornado in Kansas” is incorrect. The information from the butterfly is quickly lost on scales close to the size of the butterfly, as the atmosphere is a dissipative system such that only particularly significant powerful forcings, or small climate forcings near a climate transition (but certainly not as small as a butterfly’s flapping winds) can upscale (i.e., teleconnect) to the global scale. Indeed, if we accepted the first definition of the “butterfly effect”, everything that an individual human does on any scale (e.g., brushing your teeth) would be a climate forcing that would influence weather thousands of kilometers away. Of course, that is preposterous.
To communicate accurately in climate science, we need to make sure we properly present the significance of Lorenz’s seminal research. There certainly are thresholds (i.e., “tipping points”) that can result in sudden and large changes in climate regimes. This is a characteristic of chaotic nonlinear systems which we discuss for example, in
Pielke, R.A., 1998: Climate prediction as an initial value problem. Bull. Amer. Meteor. Soc., 79, 2743-2746.644.
Zeng, X., R.A. Pielke, and R. Eykholt, 1993: Chaos theory and its applications to the atmosphere. Bull. Amer. Meteor. Soc., 74, 631-644.
Pielke, R.A. and X. Zeng, 1994: Long-term variability of climate. J. Atmos. Sci., 51, 155-159.
Zeng., X. and R.A. Pielke, 1993: What does a low-dimensional weather attractor mean? Phys. Lett. A., 175, 299-304.
The consequences of this chaotic nonlinearity is the ocureence of the concept of critical thresholds which we discuss in Rial, J., R.A. Pielke Sr., M. Beniston, M. Claussen, J. Canadell, P. Cox, H. Held, N. de Noblet-Ducoudre, R. Prinn, J. Reynolds, and J.D. Salas, 2004: Nonlinearities, feedbacks and critical thresholds within the Earth’s climate system. Climatic Change, 65, 11-38.
The flap of a butterfly’s wings, however, while it seeks to capture the concept of the “sensitivity of the climate system to small perturbations of the initial conditions” overstates the true characteristic of the Earth’s climate system.
Um, that was pretty weird. “The information from the butterfly is quickly lost on scales close to the size of the butterfly, as the atmosphere is a dissipative system such that only particularly significant powerful forcings, or small climate forcings near a climate transition (but certainly not as small as a butterfly’s flapping winds) can upscale (i.e., teleconnect) to the global scale.” is wrong. Its provably wrong in climate models, and generally believed to be wrong in reality.
I recommend my post instead
Also upscale is not at all the same thing as teleconnect.
Comment by William Connolley — October 6, 2005 @ 1:03 pm
Actually, the second definition of the butterfly effect is correct. What’s left out of the standard description is that the tornado will only appear if the perturbation occurs under the correct circumstances. In the case of the Lorenz attractor, if the state of the system is out in the wings, perturbations do nothing of interest and the evolution is essentially unperturbed. But, if it is in the fold between the wings, small perturbations send the system into one or the other wing unpredictably.
The weather system is far more complicated but there are undoubtedly similar circumstances which cause unpredictable evolution. We must observe something like that with the weather because at certain times of year 5 and 10 day forecasts are quite good, but at other times even 3 day forecasts become uncertain.
Comment by Paul — October 6, 2005 @ 1:41 pm
William-I see your point about including too much with “teleconnection”. Perhaps it is clearer if it would say “upscale and teleconnect’”. On your broader point, I am unclear on what you disagree with on the lack of a significant influence on weather and climate associated with the butterfly’s flapping?
Comment by Roger Pielke Sr. — October 6, 2005 @ 1:51 pm
Ummmm… because in a non-linear dynamical system like the weather, even the smallest perturbation will amplify. It won’t die away.
Its perfectly clear that a butterflys wing will have no impact on climate (unless one is very near some mythical tipping point) but also clear that it will completely change the weather, globally. You have confused “climate forcing” with “weather forcing”, effectively.
The atmosphere being dissipative is irrelevant.
Comment by William Connolley — October 6, 2005 @ 2:31 pm
William-the energy associated with such a small perturbation will dissipate into heat (the velocity components of the disturbance, will “die away”). Please see my paper Pielke, R.A., 1998: Climate prediction as an initial value problem. Bull. Amer. Meteor. Soc., 79, 2743-2746.644 http://blue.atmos.colostate.edu/publications/pdf/R-210.pdf which discusses why “weather forcings” are just a subset of climate forcings.
Comment by Roger Pielke Sr. — October 6, 2005 @ 3:26 pm
Roger - of course the perturbations, insofar as they can be identified, will dissipate into heat. But since its a non-linear system they interact with all the other various waves and energy floating around and the influence doesn’t decay to nothing, it grows. This is provably so in climate models (you know this: you use them), which include dissipation. Its almost undoubtedly true in the real world too.
Your letter to the editor (not paper) is interesting but irrelevant to the butterfly discussion, and its best not to get distracted from the main point.
Comment by William Connolley — October 6, 2005 @ 3:56 pm
I have to agree to William. The butterfly doesn’t make any (noticeable) difference to the overall energy or momentum budgets, but the perturbations will not ‘die away’. In a model (any chaotic model, not just a GCM), any tiny perturbation to the last digit of any prognostic variable will lead to a divergence of the solution after a number of Lyapunov timescales. For GCMs the correlation between two perturbed simulations goes to zero after around 3 weeks (while the long term climate statistics remain very similar). To the extent that we all think that the real world is chaotic, this will be true there as well. A statement to the contrary implies that you don’t think that real weather is chaotic, which would surprise me.
Comment by Gavin — October 6, 2005 @ 4:38 pm
William and Gavin-Thanks for the quick feedback. All perturbations do not grow; indeed, as with a butterfly’s flapping wings, all of the information of such small scale perturbations can be quickly smeared by turbulent processes into heat. The perturbations “die away”. The system is changed slighly by the heat, but there is no way the real climate, long distances away, responds differently to adjacent very small perturbation.
With respect to the Lorenz equations, I agree with you. However, the real climate system is much more complex and what would be amplifying modes can be smeared into heat on these small scales by a diversity of climate (including weather) processes.
Both the weather and climate systems are, of course, chaotic (which I discuss in the BAMS paper cited in my earlier response). Since weather, on all time scales, is a subset of the climate system, it should not be a surprise that the climate system must be chaotic. The nonlinear coupling of the climate system with the oceans, land surface processes, and other components of the climate system make chaotic responses even more likely. This does not mean, however, that all perturbations amplify. The very small scale unstable modes are mixed out before they can continue to amplify.
This topic is of considerable importance beyond the butterfly effect, and I plan to write a weblog on this subject next week. Thanks for engaging in a constructive discussion of this subject.
Comment by Roger Pielke Sr. — October 6, 2005 @ 6:20 pm
All perturbations do not grow initially - only the projection onto the growing lyapunov vector(s) will grow, the projections onto lv with growth rates of less than 1 will decay. However, a random perturbation _will_ have a non-zero projection onto growing lv, and although the shrinking components mean that it will initially decay in magnitude overall, the growing component will eventually dominate and cause the model trajectories to diverge. This might well take longer than a particular forecast duration for it to be noticeable, which is one reason why people search for the most rapidly growing perturbations (eg via bred vectors and SV methods) to generate ensemble forecasts. But on a long enough time scale, the runs will eventually diverge.
I must admit, I’m more than a little surprised to see a State Climatologist apparently misunderstand this. You only have to run a NWP model with slightly perturbed initial conditions to see it for yourself (note that if the perturbation is small enough, it could vanish due to rounding errors, but this is entirely due to limited digital precision).
Comment by James Annan — October 6, 2005 @ 7:42 pm
The big point regarding chaos theory and climate science is that chaos theory invalidates GCM’s. GCM’s create ideal world simulations as opposed to real world simulations.
The core implication of chaos theory is that the probability of the GCM’s producing real world accurate results quickly approachs zero as time progresses in the model. I believe the results of every GCM every created verify this.
Chaos theory also means that no amount of tweaking the GCM’s will change them into real world accurate models. What is needed is a new form of mathematics to replace the differential equations that comprise the GCM’s. That new math may not exist for it would be the equivalent of pre-determining the future.
Comment by Reid B — October 7, 2005 @ 4:03 am
A very interesting discussion, but I am not sure the two sides are arguing the same thing. Roger appears to be making a point about the discernability of of the flap of the butterfly’s wings; a point which is most assuredly true. Gavin and William are arguing about the growing effects of even very minor disturbances in modeled chaotic systems; also a point that is difficult to refute.
The problem seems to be the attempt to equate a growing effect with a growing discernability. In a system as complex as weather and/or climate, a tiny, but growing disturbance will rapidly lose any discenability in the larger system. Even if this tiny effect eventually triggers a ‘tipping point’, it remains impossible to predict such an outcome or discern it when it happens.
Both sides seem to be correct, but Roger’s take is far more practical in the study of these chaotic systems. To embrace the other argument leads to the paranoa of the Precautionary Principle, inwhich any action could theoritically lead to some catyclismic shift in the future of the system, prompting the notion that no action should be taken! Ironically, ‘no action’ is a form of action that could also lead to a ‘tipping point’, revealing the Precautionary Principle as completely unworkable, if applied literally.
The main point here is that the models do not and can not accurately ‘model’ the reality of weather and climate. The most interesting sentence of the debate was when William stated “Its provably wrong in climate models, and generally believed to be wrong in reality.”
Comment by Jim Clarke — October 7, 2005 @ 6:57 am
Nice discussion. Roger is right that not all pertubations grow. With respect to sensitivity to initial conditions, that is the principle behind the creation of ensemble initial conditions using the “breeding of growing modes” technique. Some perturbations grow, others don’t.
William and Gavin are also right in that even extremely small perturbations in the right place at the right time can have a tremendous impact on the solution of a weather prediction model. Again regarding sensitivity to initial conditions, that is the principle behing “targeted observations.” I like to think of sensitivity in this way: If a parcel of air is in the broad confluent entrance region of a jet, it would take a massive perturbation to keep the parcel from entering the jet core. On the other hand, if a parcel is in the jet core prior to exiting into the broad diffluent region, a very small perturbation may influence the parcel to head southward rather than northward.
Placing that again in the context of initial conditions, we can add all the observations we want in the entrance region and the model solution won’t change much. But observations located near the exit region will have a big influence on the model solution.
Comment by Dan Pawlak — October 7, 2005 @ 9:14 am
Fascinating discussion, gentlemen.
And an excellent use of the internet to boot. Thanks for making us privy to the debate.
Comment by Kerry — October 7, 2005 @ 12:43 pm
Gavin observes that in a gcm, any change to any prognostic variable, no matter how small, leads to divergence.
Roger observes that all perturbations do not amplify, either because of their location, timing, or nature.
Paul observes that perturbations in the “wings” of the butterfly do not amplify, because of their location, but perturbations in the fold can amplify greatly.
I suggest that they are all correct, and further, that this difference between models and reality is one of the reasons that computers models are not particularly successful in modeling either the weather or the climate.
I also suggest that a look at information theory shows that it is not theoretically possible for all perturbations in a system to amplify. There is a finite amount of energy flowing through the climate system, which means that there is a finite amount of information flowing through the system as well. The amplification of one signal, therefore, can only come at the expense of another signal.
Part of the information, of course, is constantly being converted irreversibly into random heat. And part of the information, of course, is being lost to space, leaving the system entirely. At the same time, new information is constantly arriving in the system, in the patterns of light and shade of the energy of the sun.
In this insanely complex system, at some points in time and space a transient forcing change of a few watts/m2 will make no lasting impression, no measurable difference whatsoever in either the local weather or the local climate.
In another point in time and space, the same transient forcing change of a few watts/m2 could be the butterfly that tips the Pacific Decadal Oscillation from its decades of warm phase to its decades of cool phase.
The difficulty, of course, is telling the sheep from the goats — which difference makes a difference?
w.
Comment by willis eschenbach — October 7, 2005 @ 5:46 pm
Thanks all for a very useful discussion (and keep the comments coming!). I have response and a question. First, all perturbations applied in NWP and climate predicion models, as part of creating ensembles, involve much larger variations in kinetic energy than is involved with a butterfly’s flapping of its wings. Is it there anyone who actually concludes there is any situation where the flapping wings of a butterfly can alter even local weather patterns, much less a tornado many kilometers away?
Comment by Roger Pielke Sr. — October 7, 2005 @ 7:54 pm
Folks, as a scientifically and mathematically literate outsider, I’ve got to say this whole argument seems a little silly. I find it difficult to imagine that anyone sensible is seriously saying that a butterfly, a knowable buttrfly in a knowable place, is flapping its wings in such a way that it knowably leads to a hurricane later. People who take it as that don’t need to know about climate or weather forcing, not at least before some more general points are explained.
The whole “butterfly effect” notion is a metaphor that clarifies what “sensitive dependence on initial conditions” means — that small and unmeasurable differences in the initial state of a chaotic system can lead to widely divergent future trajectories. Whether a real physical butterfly can introduce sufficient information to knowably influence a hurricane later is as misguided as pointing out that “Shall I compare you to a summer’s day” is foolish because one person isn’t at all like the state of weather in one hemisphere.
Comment by Charlie (Colorado) — October 8, 2005 @ 12:39 am
Charlie-I agree that the “butterfly effect” should be a “metaphor.” Unfortunatley, as documented in my original comment on this subject, the concept has been taken literally, even by some in the scientific comuunity. We need to emphasize that the real “butterfly effect” is described by the appearance of the solution space that result from solving the Lorenz equations.
Comment by Roger Pielke Sr. — October 8, 2005 @ 8:03 am
While the world is a strange & mysterious place, and our knowledge of things and their interrelations is , from one point of view, barely rudimentary, some things might just be what they seem. Along the lines of the famous Zen koan put to students by their teacher. “What is the sound of one hand clapping, grasshoppers?”, he says ever so seriously. All the serious students contemplate this , some wave their hand to further explore the possiblities of the cryptic puzzle. Only the illiterate farm boy can state immediately and without equivocation , ” None, Master. It takes two hands to clap.” It can also be proved that Alexander can never reach home if he makes successsive trips of half the journey. Mathmatically, He will always be some small fraction of a distance from home.
Comment by Nick — October 8, 2005 @ 3:00 pm
Re: 15, Roger, thank you for asking interesting questions.
Can a butterfly influence a tornado? Yes, but only in the sense that a straw can break a camel’s back … it can, but it has to be the last straw.
Consider an eddy downstream of a rock in a river. The eddy is semi-stable, in that there is almost always an eddy in that location. However, some of the time it spins clockwise, and sometimes it spins counterclockwise, and in the short periods in between, there is no eddy. Common behaviour, seen in any river.
Now this is a bi-stable, semi-chaotic system that flips from one temporary stable state another. Can a butterfly flip this system over? (We can imagine that this butterfly is swimming downstream.)
Yes, it can, but only once in a great while. Most of the time a butterfly will just get swept through the eddy and downriver, with no sign of its passing. But if the butterfly is swept around the rock at just the right time, just when the transition is occurring, it could either prevent, or instigate, the transition of state, with large consequences.
The problem with creating a model of the climate is that the climate system is neither 100% linear nor 100% chaotic. You could say that it is all chaotic, but only part of the time. Or you could say it is part chaotic, but all of the time. This behaviour is very challenging to model.
The modelling problem is made worse by the tendency of natural fluid systems (of which the climate is one) to run at the edge of turbulence, in that gray zone between fully laminar flow (linear) and fully developed turbulence (chaotic). While there are tractable mathematical approximations of both laminar and fully turbulent flow, the space between is generally a mathematical no-man’s-land. I am reminded of the engineer’s prayer … “Dear God … please make the world linear.”
w.
Comment by willis eschenbach — October 8, 2005 @ 3:20 pm
The “Butterfly Effect”
Anyway, this is all pretty well-known stuff - or so I thought, before reading Roger Pielke Sr’s blog. He seems to deny that this “butterfly effect” exists at all. Myself, Gavin and William have all pulled him up on it, so it will be interesting to s…
Trackback by James' Empty Blog — October 9, 2005 @ 5:56 pm
The response #20 shows further how the “butterfly effect” continues to be misunderstood. I am still waiting for someone to claim that under any circumstance, the flap of a butterfly’s wings can cause a tornado that otherwise would not occur.
With respect to the Lorenz equations, one certainly can show that even the slightest changes will cause the solution to change from what it otherwise would be. However, this is not true of the actual climate system due, for example, to turbulent mixing. A clear example of this is baroclinic synoptic weather development. There is a prefered spatial scale of cyclogenesis because mixing processes prevents development on a smaller scale.
Hopefully, we will soon all agree that the butterfly effect is specifically applicable to the solution space of the Lorenz equations, but not a literal statement of the effect of a butterfly.
Comment by Roger Pielke Sr. — October 10, 2005 @ 3:09 am
Statistical thermodynamic theory confirms the chaos theory notion that a flapping butterfly wing can cause a tornado. But the odds of that happening are infinitessimally small and scientists would have no way of knowing the cascading event took place if it occurred.
Comment by Reid B — October 10, 2005 @ 5:15 am
In order to provide a further perspective on this subject, lets do a thought experiment. Consider two butterflies that are flapping their wings 10m apart. Then they stop flapping. The turbulence that the flapping caused cascades to smaller scales and dissipates into heat. At a distance of 1 km, how would the atmosphere distinguish between the effect of the two butterflies? It cannot. Only the heat remains.
The fundamental misunderstanding here is that the sensitivity to initial conditions does not guarantee that all perturbations result in the solutions from all nonlinear systems deviating over time. In the Lorenz system of equations they do.
However, in the real climate system, the multitude of forcings and interactions results in the damping of many perturbations (such as a butterfly flapping its wings)such that their influence does not upscale nor teleconnect. The odds of a butterflies flapping causing a tornado is zero.
Comment by Roger Pielke Sr. — October 10, 2005 @ 10:18 am
Good grief, Charlie Brown
Let’s accept the inescapable conclusion that the weather will destroy this planet because there are just too many butterflies.
Comment by Jeff Cain — October 10, 2005 @ 12:50 pm
Re #21 - Roger, we’ve all been saying that. I’ve said so explicitly: this comment.
Comment by William Connolley — October 10, 2005 @ 2:09 pm
This is a fascinating thread, and surely an enormous source of confusion among environmentalists and the public at large is dispelled here.
The notion of the fragility of nature to causal disaster is widely accepted, even wildly embraced by todays pagan worhippers. But reality is more confounding than religious truth. As Dr. Peilke makes clear, this is rooted in a confusion of relevant with irrelevant variables. Strictly speaking, abuse of a metaphor isn’t the problem; confusion about the salience of natural causation is.
Comment by T J Olson — October 18, 2005 @ 3:04 am
O.k., I’ll go out on the limb - the butterfly CAN cause the tornado. However, let’s examine the context of this statement. Begin with two worlds that are identical, except that in A the butterfly does not flap its wings, while in B the butterfly does. Not all of the butterfly’s energy in B cascades down - a small amount will cascade up. This effect will be slow (the Lyapunov exponent) but inexorable. Eventually, the weather on A and B will be entirely different from each other, everywhere. As one small part of this difference, B will certainly have a tornado at some time and place where A does not; but some of the tornados on A will occur at times and places where B does not. If you wish to say that B’s butterfly caused B’s tornado, you would also have to admit that A’s butterfly caused A’s tornado by NOT flapping its wings. I don’t want to get into the philosphical analysis of the word “cause”, but it is arguably applicable to BOTH A and B, or else it is applicable to neither. In either case, there WILL BE tornados at different times and places on A and B.
Comment by Eric S. Posmentier — November 10, 2005 @ 1:42 pm
Eric- thank you for your further discussion of the butterfly effect. With respect to butterfly A and B, the entire energy produced by both insects in the real world will dissipate into heat over a very short distance. Imagine a thought experiment where the butterflies are in a closed room. The walls of the room would absorb any velocity perturbations and convert into heat. No coherent signal from these butterflies would exit the room. We should not expect a different behavior when the atmopshere itself absorbs the velocity perturbations over a very short distance and coverts into heat.
Comment by Roger Pielke Sr. — November 11, 2005 @ 9:18 pm
Hello anyone,
I fear I’m joining the debate way too late, but it’s still interesting even months later…
Anyway, a simple thought experiment:
Our good ol’ buddy GW figures out the “intahnet”
Also, he learns to read good.
He eventually reads this discussion.
Based on a dream he has with Jesus and Blair, he decides to set up a scenario where he puts a butterfly in a black box with a flash camera.
He sets the rules that if he snaps a photo and sees that the butterfly was indeed flapping his wings at the time of the flash, he will authorize the pushing of a BIG RED BUTTON (or multiple buttons).
This event triggers a major climate change (eventually tornados), and really angers the butterfly, who just wanted out of the box.
I know it’s a random, painfully stupid scenario, but humans are a part of the natural world and any possible poorly conceived chain of events with humans in the driver seat is also “natural,” so it’s rather simple to imagine some “insignificant” butterfly flutter (dissipating energy and all) could lead to a major climate change if foolish humans exist. We can make the world system as non-linear as we want, and correlate most phenomena with our own observations and reactions.
Comment by Dave — August 17, 2006 @ 5:04 pm