Follow Up (April 27 2008)
Ray – In searching for what Professor Lorenz has said on this issue, please see Chaos Avant-Garde: Memories of the Early Days of Chaos Theory
In this essay he writes,
“Returning now to the question as originally posed, we notice some additional points not yet considered. First of all, the influence of a single butterfly is not only a fine detal – it is confined to a small volume. Some of the numerical methods which seem to be well adapted for examining the intensification of errors are not suitable for studying the dispersion of errors from restricted to unrestricted regions. One hypothesis, unconfirmed, is that the influence of a butterfly’s wings will spread in turbulent air, but not in calm air”
This certainly would rule out the butterfly in the jar! More importantly, he recognized that there remain questions about the “butterfly effect”, one of which is when small pertubations result in altering larger scale atmospheric flow, and when they do not.
Sixth Update (April 27 2008)
A Further Reply By Ray Pierrehumbett
[Response:Roger, I can’t make sense of what you’re trying to say here. For those picokelvins of temperature to be lost to space, first they have to appear in the atmosphere as an increase of temperature, right? So there you have your change of one digit in the initial conditions, just like in Lorenz’s example. And your statement is just flatly inconsistent with thermodynamics. The butterfly dissipates heat locally, and that heat will be gradually diluted over a larger and large area. So just divide by Cp and there’s your answer. Do you think there’s some way to magically teleport the heat away, leaving the fluid to heal back to exactly the same condition it would have had without the flap? That’s really a stretch. Your remarks about simple models and GCM’s don’t make much sense to me either. The GCM doesn’t resolve butterfly-scale motions, but once you have influenced a dynamic variable (e.g. temperature) at a resolved scale, any number of actual twin experiments in GCM’s confirm the divergence. If you are claiming there’s some fundamental difference between sensitive dependence to large scale changes in a GCM and sensitive dependence in the atmosphere, I’d like to see some evidence to back up that claim. The success of GCM’s in short term weather forecasting would be pretty much impossible to reconcile with such a claim. –raypierre]
You are correct in that you and I probably agree on most issues in chaos and nonlinear dynamics. All NWP and climate models show the sensitivity of large scale circulation features to initial conditions when perturbations are inserted in their initial state or in their parameterizations (these are all much larger effects than the energy that a butterfly places in the system). We also agree that the added heat from a butterflies flapping wings results in a slightly different system than if this flapping did not occur. However, the issue is whether the heat (the “information”) from this effect can translate (teleconnect) to larger scale so as to result in alterations in large scale features.
Even Issac Held seemed to indicate that there is a lower limit to when this upscale effect can occur (i.e. this ability disappears when the flow becomes laminar); he said in this thread
“the scale of the perturbation has to be larger than what is often referred to as the Kolmogorov microscale, the scale below which the flow is effectively laminar, to avoid being damped out immediately. This scale is typically a few millimeters in the atmosphere….”
I agree with this, but maintain that the smallest turbulent scales also are damped out due to the physics of non-motion transfers (i.e. radiative transfers) of energy. I have been in communication with Professor Ekyholt on this question, and he and I agree that you are misinterpreting the butterfly effect for very small scale perturbations. We will be preparing a paper on this to demonstrate that there is lower limit to which the “butterfly effect” applies.
On a separate note, I see commenters on this thread are somehow skewing this discussion to be on climate change. It is not. This issue of the scale at which the “butterfly effect” occurs is a pure discussion of the science such as we all used to have as graduate students and need more of!
Also, you questioned as to why Roy Spencer posted a guest weblog. The answer is that he has introduced a novel and important new perspective into how variations in atmospheric/ocean circulations can result in alterations in the global average radiative balance. Disagreements with his results and conclusions should be on his science. I invite others (including any interested Real Climate climate scientist) to post unedited guest weblogs on Climate Science.
Additional Response From Ray Pierrehumbert
[Response:Regarding the butterfly in the room — even in a jar in the room — sure I think it’s likely that it would ultimately affect the large scale weather. Look at it this way: Temperature has a dynamic influence through buoyancy. The heat dissipated by the butterfly might warm the room by a few tens of microkelvins, say. That increased temperature will change the heat flow between the house and the environment, which will ultimately change the temperature of some parcel of air by a few nanokelvins. Then before you know it, some parcel of air the size of the state of Illinois has a temperature different by maybe a few picokelvins. I guarantee that if you take a GCM and change the temperature of the air over Illinois by a few picokelvins (given sufficient arithmetic precision) that that will lead to divergence of the large scale forecast given infinite time. I have seen no indication either in dynamical systems theorems or in numerical experiment to suggest that anything else would be the case. –raypierre]
Ray- We certainly disagree with respect to the butterfly in the room in a jar. :-). Other readers of Real Climate (and Climate Science) can make up their own minds on this.
You are, however, taking the concept of chaos too narrowly and are focusing on idealizations (simple illustrative models and GCMs) of how the real atmosphere (and climate system) works. You are ignoring the consequences of the dissipation of kinetic energy into heat within a open system. The “picokelvins” of heat, even if they could cause such a temperature perturbation over the state of Illinois (which it would not), would be lost to space long before an “infinite” time were reached.
Fourth Update (April 26 2008)
Additional Response From Ray Pierrehumbert
[Response:Have a look at Isaac’s remark above. I think what you probably have in mind is the possibility that if a perturbation is at a scale where you have primarily downscale energy cascade to the dissipation range, it might never project on the large scale quantities whose behavior determines large scale predictability loss. Given the nature of turbulence, it is hard to absolutely exclude this possibility a priori, but for this to happen, there would have to be ZERO leakage to large scales. Not just small but ZERO. That is exceedingly unlikely, and would be contrary to most of what is know about turbulent cascades. As a practical matter, I do agree that if the initial perturbation is at sufficiently small scales, the projection on large scales would be small enough that it could take an exceedingly long time before it affected the evolution of the large scales. –raypierre]
My Reply [posted on Real Climate]
Ray – Thank you for getting involved in this discussion. The question of the leakage time scale is, of course needed, in order to determine when the exceedingly long time scale becomes infinite (in terms of where the heat goes). If we both agree that ALL of the turbulence quickly dissipates into heat when the flapping stops, then what is your estimate of the residence time of this heat within the atmosphere before it is lost to space?
Also, as another thought example, if a butterfly flaps its wings inside a room with the doors shut, would you still maintain that this has an influence on atmospheric circulation at large distances? All of the heat generated would be absorbed by the walls of the room, and subsequent heat conduction is, of course, laminar. An analogous behavior will occur in a very stable boundary layer (and any region of the atmosphere for such small perturbations), and if we can agree on this “exception” than we have made progress in understanding this issue. My point here is that if there is an part of the process which results in complete loss of the turbulent flow, then it is not communicated over large distances.
Issac’s Held’s answer also actually contains part of the answer on this issue. If the turbulence dissipates into heat, as illustrated in the above example, than its further behavior can be described by non-turbulent behavior. As he explained, he was “was thinking that the scale of the perturbation has to be larger than what is often referred to as the Kolmogorov microscale, the scale below which the flow is effectively laminar, to avoid being damped out immediately. This scale is typically a few millimeters in the atmosphere “. This is what occurs with the flapping of the wings of a butterfly; all of its energy dissipates into heat and the spatial structure of this heated air is less than a few mm. To disprove this total transfer downscale, one would have to show that a coherent turbulent structure remains and becomes progressively larger in scale and/or is monitored propagating away from the location of the flapping wings as a coherent disturbance of the air flow; in both cases, while still retaining the conservation of total energy. Since the total energy of the flaps of the butterfly’s wings must be accounted for (as kinetic energy in the turbulence, heat) what is your estimate of the magnitude of this energy that reaches thousands of kilometers away, as well as the path this energy would take to get there?
Third Update (April 25 2008)
Further Response From Gavin Schmidt
Response: As we said above, this is what you believe. Why you accused us of misrepresenting you is a mystery. However, your claim about Ekykholt’s belief is contradicted by his quote above. He states very specifically that exponential growth saturates at the time the perturbation reaches the size of the attractor. That, for the atmosphere, is very large indeed and is certainly large scale enough to encompass storms thousands of miles away. Isaac can certainly speak for himself, but as far as I know there is no demonstration that there is a minimum scale below which perturbations do not grow. Such a thing may exist, but your certainty on the matter seems a little overconfident. Perhaps you’d care to point out a reference on the subject? – gavin]
My Reply [posted on Real Climate] Gavin – I am glad this discussion is continuing. I will be having more to say on this next week in a weblog on Climate Science, however, you are failing to distinguish between an open and closed system, and between the real world and models. With nonlinear atmospheric models such as analyzed by Professor Lorenz, the results for large scale features are sensitive to the initial conditions regardless of how small they are. This is because the system is closed. The real world climate system, however, is not closed, such that energy (i.e. in the form of heat) can leak out of the system. In the case of such a small perturbation as the flap of a butterfly wing, the kinetic energy of the small amount of turbulent air that it generates will quickly dissipate into heat, once the flapping stops. Radiative loss of this heat to space will prevent the flapping to have any effect at large distances.
This is one of the reasons that you are mistaken in stating that “there is no demonstration that there is a minimum scale below which perturbations do not grow.” If a perturbation in the system (i.e. the atmosphere) dissipates into heat, it can be lost to the system before affecting atmospheric features at large distances. I will have more on this topic on my weblog next week, and will post a comment on Real Climate when it appears.
Second Update (April 24 2008)
Gavin Schmidt has replied
Response:You misinterpreted this back on the original thread and you are misinterpreting it here again. However, just repeating the same argument is pointless. Since I agree with Dr. Eykholt’s statement, and so do you, let’s just leave it at that. (if other readers are interested in what this is about, please go to the original thread. The clue is that ‘larger scales’ in the Eykholt quote means the attractor itself (i.e. climate), while RP thinks he means the large scale flow (i.e. the specific position on the attractor)). – gavin]
My Response is
Gavin- I agree readers can go through the thread to see the discussion. However, you are misrepresenting my views. Rich Eykholt and I are in 100% agreement on this subject. The question that was being discussed is whether an atmospheric perturbation as small as a real world butterfly could actually affect large scale weather features thousands of kilometers away. The answer, as given by Professor Eykholt, is NO under any circumstance. The perturbation has to be much larger (Issac Held, as I recall said meters in his NPR interview; I suspect it is a few kilometers or more) for a perturbation to affect an atmospheric feature thousands of kilometers away.
This issue, based on our disagreement, would benefit from further quantitative evaluation with both analytic and numerical models. We do have papers on the use of analytic models to examine chaos and nonlinear dynamics which document that we are quite familiar with the subject of sensitivity of the climate system to initial conditions; e.g. see
Pielke, R.A. and X. Zeng, 1994: Long-term variability of climate. J. Atmos. Sci., 51, 155-159.
Update (April 24 2008) : Following is my comment, Gavin Schmidt’s reply, and my response on Real Climate
[Response:In the linked piece, you very clearly state that you do not believe that the real world is sensitive to initial condition variations like butterflies. That is all we are discussing here. If you now think that it is, feel free to expound on your viewpoint. We were just trying to make sure that a diversity of points was presented. – gavin]
Gavin – Thank you for posting my Climate Science link. In terms of actual butterlies, this is clearly explained by an expert in the physics and mathematics of nonlinear dynamics and chaos in geophysical flows, Professor Richard Eykholt (see http://climatesci.org/2005/10/12/more-on-the-butterfly-effect/), where he writes “
Roger: I think that you captured the key features and misconceptions pretty well. The butterfly effect refers to the exponential growth of any small perturbation. However, this exponential growth continues only so long as the disturbance remains very small compared to the size of the attractor. It then folds back onto the attractor. Unfortunately, most people miss this latter part and think that the small perturbation continues to grow until it is huge and has some large effect. The point of the effect is that it prevents us from making very detailed predictions at very small scales, but it does not have a significant effect at larger scales.
Real Climate has published a well written summary of the seminal accomplishments of Professor Ed Lorenz in the field of deterministic chaos and nonlinear dynamics (see). Professor Lorenz’s contribution to the understanding of the mathematics and physics of geophysical flows (and other dynamic systems) has altered how the science community investigates these processes. I had the opportunity to sit and talk with Professor Lorenz during one of his trips to Colorado State University, and enjoyed and learned from his perspective on the nonlinear aspects of the climate system including its behavior, as with any other nonlinear system with strong feedbacks, as being sensitive to initial conditions.
At the end of the well deserved recognition to Professor Lorenz, Real Climate writes
“So what does this have to do with the IPCC?”
Real Climate then writes
“Even though the model used by Lorenz was very simple (just three variables and three equations), the same sensitivity to initial conditions is seen in all weather and climate models and is a ubiquitous phenomenon in many complex non-linear flows. It is therefore usually assumed that the real atmosphere also has this property. However, as Lorenz himself acknowledged in 1972, this is not directly provable (and indeed, at least one meteorologist doesn’t think it does even though most everyone else does). Its existence in climate models is nonetheless easily demonstratable. “
I am the “one meteorologist”. Real Climate refers to one of the Climate Science weblogs on this issue that was published (see).
However, Real Climate is wrong in its statement on my research conclusions! I have written several papers on climate as an initial value problem: e.g. see
Pielke, R.A., 1998: Climate prediction as an initial value problem. Bull. Amer. Meteor. Soc., 79, 2743-2746.
Pielke Sr., R.A., G.E. Liston, J.L. Eastman, L. Lu, and M. Coughenour, 1999: Seasonal weather prediction as an initial value problem. J. Geophys. Res., 104, 19463-19479.
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.
Real Climate should report accurately on the research of others.
What we disagree on is whether the multi-decadal global climate model predictions can be used to accurately quantify the degree of nonlinearity and predictability of the real world climate system (the nonlinearity of the climate system is shown, for example, in the Rial et al paper).
Real Climate, however, reports on the use of a model to investigate this issue. This is a typical mistake they are making; a model is itself a hypothesis and cannot be used to prove anything! The multi-decadal global model simulations only provide insight into processes and interactions, but we must use real world data to test the models. So far, the models have failed, for example, in their ability to accurately predict the regional weather and climate features we discuss in the Rial et al paper. Lets have more accurate reporting on Real Climate.