In response to the variety of comments on the weblog of October 6, 2005 entitled “What is the Butterfly Effect”, I asked Associate Professor Richard Eykholt of the Department of Physics at Colorado State University to provide his perspective on the discussion. Professor Eykholt is an internationally respected expert on chaos and nonlinear dynamical systems. His website provides information on his excellent professional and academic credentials.
His response to my request (dated October 11, 2005) is reproduced, with his permission;
“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.
Richard Eykholt”
This summary should put to rest the misconception about the “butterfly effect.” In answer to the question presented in the original weblog on this subject, “Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?”, the answer is absolutely no.
Roger A. Pielke Sr.
Hi Roger. You’re still wrong, and RE seems to lack weather expertise. The idea that “it does not have a significant effect at larger scales” is painfully wrong. But its good to see you nailing your colours to the mast!
Comment by William Connolley — October 12, 2005 @ 12:20 pm
Well, I can’t make head or tail of this - it seems to be even more wrong than you were in your original post. The idea that a small pertubation grows off the attractor, eventually folding back onto it, is contradicted by everything I have ever learnt on the subject. Furthermore, although in reality any perturbation actually rapidly collapses back onto the attractor, it collapses onto a slightly different point from the unpertubed state, and as both points move around the attractor the pertubation continues to grow until it is bound by the size of the attractor itself. At this point, the “weather” in both the original and perturbed models on global scales is completely unrelated.
As I said, this is easy to demonstrate with models, so long as you do not lose the pertubation to rounding error.
James
Comment by James Annan — October 12, 2005 @ 3:59 pm
It is obvious that the perturbation cannot grow exponentially for ever. However the correlation between (say) the sea level pressure field anomaly in the two simulations will tend to zero in time, while the RMS difference will increase and then plateau. The large scale features will be different in the two cases, and the position and timing of storms will have been affected. I guarantee I can find a time when the first case did not have a storm in Texas, and when the other case did. Clearly, that isn’t the same as being able to predict a storm in Texas from the various butterfly flaps, but no-one has claimed that.
Comment by Gavin — October 12, 2005 @ 4:21 pm
James and Gavin- thanks for your further comments. As Professor Eykholt, a recognized expert on nonlinear dynamics, has clarified, very small perturbations do not have a significant effect at larger scales in weather and in the climate system.
In a numerical weather prediction model, the perturbations that are applied are substantial in that they are always grid volume averaged information, as I discuss in detail in Chapter 4 in Pielke, R.A., Sr., 2002: Mesoscale meteorological modeling. 2nd Edition, Academic Press, San Diego, CA, 676 pp. Their magnitude is much larger than the flap of a butterfly (or other very small perturbation).
There have been claims that a butterfly’s flapping will affect the weather. I presented examples of these claims in the weblog. It was because of this misconception that I posted the weblog.
Comment by Roger Pielke Sr. — October 12, 2005 @ 4:45 pm
The real meaning of the effect is to make long term prediction impossible. A small perturbation puts you on a slightly different trajectory of the attractor [if such a thing even exists for the weather] which in the long run has a very different time evolution than the initial unperturbed state.
Comment by Paul — October 12, 2005 @ 11:22 pm
Prof. Pielke,
It seems to me you are trying to average your way out both in your question and your answers. Weather and climate have two dimensions, time and space (well, true, space has three…except if you are a string fantasist). What everyone is telling you is that if you look at a fixed time at a particular point, yes, the butterfly has an effect on the weather and if, as most people, you define weather as what their nose encounters at a particular moment.
This is all quite consistent with what Prof. Eykholt and everyone else is saying to you. However, mathematicians, because of their rather particular precision are prone to telling the absolute truth in useless ways as the bad joke about Zeno’s paradox (about 2/3 of the way down http://www.phy.ilstu.edu/~rfm/EPMjokes.html) shows.
On the otherhand, once you start averaging over time and space this all averages out, so that at the rather long end you get climate, which is not significantly effected by the butterfly’s wings. Everyone agrees that if you are choosing to define weather as an average over time and space, which again, most people also do, well, that butterfly will have to work very hard, and as the monarch queen has left for her Mexican winter home I cannot guarantee enough day laborers.
OTOH, your query itself almost assumes the answer. Perhaps a more useful question might be: What is the maximum path deviation for a tornado in Texas caused by a butterfly in Brazil?
Comment by Eli Rabett — October 13, 2005 @ 6:00 am
Prof Pielke,
Professor Eykholt’s comment is certainly apropos and correct until his bizarre non-sequitur of a final sentence. The attractor in question here is strange, and as Lorentz showed all those years ago, the region of phase space it explores is large. It is obvious that the attractors of the climate and weather systems include tornadoes, hurricanes, and mesoscale cyclones (because we observe all these things) and hence they are small compared to the attractor. His reasoning is correct - but his (and your) conclusion doesn’t follow.
Comment by emeasure — October 13, 2005 @ 7:22 am
Thank you for the follow-up comments. Several of you are using the Lorenz model, and NWP and climate models, as your exact analogues to the weather and climate system. However, the real weather and climate system involve more physics, including the second “law” of thermodynamics. In the real world, the butterfly’s flapping wings will never result in significant weather effects.
Comment by Roger Pielke Sr. — October 13, 2005 @ 8:34 am
Hi All- As an additional illustration of the inability for a very small perturbation from making a large scale change, consider the following example. You lay a rope from your location to a town 10 km away. Jiggle the rope (or even vigorously shake it!). No matter now many times that you shake the rope, the rope will not move in the town. The energy that you impose will be dissipated into heat long before the motions influence the rope at that distance.
Comment by Roger Pielke Sr. — October 13, 2005 @ 10:59 am
Re 9: there are any number of examples of situations where small perturbations produce small or negligible effects. Consider a mountain: I hit it: nothing happens. So what… this proves nothing about weather, which is a totally different dynamic; and your rope illustration is the same. The rope is a case where perturbations produce a linear decaying response and are dominated by damping. The atmosphere isn’t like that. You’re in a hole. Stop digging!
Comment by William Connolley — October 13, 2005 @ 1:43 pm
Re #9. The point being that the dynamics of ropes are not chaotic?
More relevantly, you appear to be asserting that despite the fact the NWP models paths are demonstrably sensitive to the smallest possible perturbation (say the last digit in a 32 bit surface temperature variable, around 10^-16 degrees), there is a smaller scale at which (should they be capable of resolving it) they would not be. So, my questions to you are, what is this scale? and what evidence do you have for that? My feeling is (and everyone else appears to agree with this) is that there is no such limiting scale. Weather is chaotic all the way down. “Weather” you like it or not!
Comment by Gavin — October 13, 2005 @ 3:04 pm
William and Gavin-Sorry but not everyone else agrees with your perspective (such as an expert on nonlinear dynamics). All of us agree on the sensitivity to initial conditions of the models to very small perturbations. However, the models do not include all of the essential physics of the real world, such as adherance to the second “law” of thermodynamics, which precludes very small perturbations from influencing weather and climate at larger scales.
I do agree that both weather and climate are chaotic. This is why the concept of sudden thresholds in response to human- and natuaral- forcings is of such critical societal importance.
We discuss this nonlinearity, for example, in Pielke, R.A., 1998: Climate prediction as an initial value problem. Bull. Amer. Meteor. Soc., 79, 2743-2746. http://blue.atmos.colostate.edu/publications/pdf/R-210.pdf and 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.http://blue.atmos.colostate.edu/publications/pdf/R-210.pdf
However, the forcings which result in a sudden regime transition must be energetic enough to upscale in order to alter the larger scales. A butterfly’s flapping wings will not do this under any circumstance.
My suggestion is to invite another specialist in nonlinear dynamics and chaos to provide a comment which refutes my (and Professor Eykholt’s) conclusion. This would help readers of this weblog make up their mind who is correct on this issue.
Comment by Roger Pielke Sr. — October 13, 2005 @ 4:13 pm
Roger, you are simply wrong here. Further email correspondence with Professor Eykholt has confirmed that he agrees with the well-understood position that small pertubations do in general affect the long-term evolution of the (any chaotic) system.
(The point he seemed to be trying to make is that small perturbtions of a similar magnitude to rounding error cannot materially affect the prediction skill of a model, since any practical model already has such small-scale errors. This point is fundamentally unrelated to your comments about the trajectory of the system itself.)
Rather than focussing on the purely thermal/turbulent aspect of a forcing, consider also the (very small) change to the momentum field. Does this also dissipate? And what relevance does a “sudden regime transition” have to any of this? This seems to be a straw man entirely of your own construction.
Comment by James Annan — October 13, 2005 @ 5:19 pm
James-I am unclear what futher correspondence with Professor Eykholt you are referring to. His message, as posted on my weblog, is that very small perturbations do not have a significant effect on larger scales. Any very small change to the momentum field (in the real world and not in the Lorenz equations or in an NWP model) will dissipate into heat. At a distance of even a few meters from a butterfly, it is impossible to distinguish its influence from another butterfly a short distance away. Do you disagree with this statement?
My point about “sudden transtions” is to emphasize that for large enough perturbations, the ability to produce very large effects (i.e. a regime shift) in the chaotic climate system is a real risk. However, very small pertubrations, such as the flap of a butterfly’s wings never can.
Thanks for your feedback.
Comment by Roger Pielke Sr. — October 13, 2005 @ 8:11 pm
Roger, I’m referring to email that I exchanged with Prof Eykholt. His comment above still seems a little oddly worded, but his clarification (regarding the relevance of minuscule perturbations to practical prediction) is uncontroversial enough. However I cannot see how it supports your claims in the slightest.
As for your suggestion that momentum is not conserved in the real world, I’ll have to file that under “interesting”.
Comment by James Annan — October 13, 2005 @ 9:44 pm
James- thanks for following up. Professor Eykholt states “Roger:I think that you captured the key features and misconceptions pretty well.” This certainly indicates that I am presenting the issue correctly.
On the conservation of momentum, in the presence of the molecular dissipation of motions into heat (as is well understood in turbulence theory), monmentum is not conserved! The real world, of course, includes this dissipation, but it is not present in the Lorenz model or in NWP and climate models (although in the later models, numerical diffusion which is present explicity or implicitly removes momnetum). Two delta features in such models, for example, are often completely filtered out.
Comment by Roger Pielke Sr. — October 14, 2005 @ 7:30 am
The following comment was posted on http://julesandjames.blogspot.com/
“Well, RP Snr’s promised followup post consisted of an odd quotation from Richard Eykholt who works in the physics department of Colorado State University. His quotation contains the comment that
“The point of the [exponential growth of minuscule perturbations] 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.”
which RPSnr takes as vindication of his claim that small perturbations do not affect the real world, only models.
The quotation seemed a bit unclear, and RPSnr’s position is clearly wrong, so I emailed Prof Eykholt to see what he really meant. He explained that he was referring to issues of practical prediction, in which case any additional perturbation of a similar size to the rounding error of a finite precision computer model cannot harm predictability over broad spatial scales, since the model already has lots of perturbations of equal size (not to mention more substantial errors) which limit predictability. Of course this is true (to the point of triviality), but it hardly seems relevant to RPSnr’s claims. Further, Prof Eykholt explicitly agreed that a small random perturbation in a chaotic system will grow to a point at which the original and unperturbed systems are completely decorrelated.
Meanwhile, back in the comments section of his post, RPSnr seems so determined to defend his untenable position that he is now claiming that not only turbulent energy but even momentum itself will dissipate in the real world…”
My response is reproduced below:
James-if you are going to refer to communications with Professor Ekyholt, it would be appropriate to post them. I also find it disappointing that you do not categorically state that a butterfly’s flapping wings can have a significant effect on the weather thousands of kilometers away, if that is your conclusion. Quite frankly, it is amazing that you are focusing on a physically incomplete modelling perspective, rather than how the real world works.
Comment by Roger Pielke Sr. — October 14, 2005 @ 1:40 pm
One further comment on the http://julesandjames.blogspot.com/ post. Turbulent energy is composed of momentum! Since turbulent energy dissipates into heat, so will momentum. Turn the Sun off, even in a climate model, and watch the momentum go to zero after a period of time.
Comment by Roger Pielke Sr. — October 14, 2005 @ 2:22 pm
I hope, Professor Pielke, that you are referring to transfer of momentum to the Earth through wind stress. If you really thinks you have found an example where momentum isn’t conserved you should talk to the Swedish Royal Academy. If you can convince them that Newton and Einstein are wrong, you should get a nice reward next October.
I notice you make the same claim (that momentum is dissipated)in your modelling book. This is seriously misleading, to say the least. Kinetic energy can be changed into internal energy (microscopic kinetic energy), but there is no analogous process for momentum.
Comment by emeasure — October 14, 2005 @ 6:01 pm
Roger,
Since your position appears to be that the basic equations of motion are fundamentally wrong in some (as yet not clearly specified) manner, any appeal to Prof Eykholt or chaos theory are unlikely to shed further light on the issue.
It seems to me that the onus is on you to propose some model that you believe describes reality better than existing atmospheric models. Perhaps you could start by identifying the threshold below which you believe perturbations will simply vanish? Can you suggest any way in which this theory can be tested, even in principle? In the absence of such, your statement appears to be a fundamentally unfalsifiable belief that sits oddly on a “science” blog.
I don’t think blog ping-pong is a good way of continuing 2-way or multi-way conversations. I have posted on usenet here and look forward to your or anyone else’s further comments. Of course you can have the last word here.
(Oh yes - in response to your question, of course I think a small perturbation will ultimately affect the trajectory of reality on large scales, as it does in all realistic models. However it will take a very long time to grow to a significant size, and so has no practical impact on forecasting)
Comment by James Annan — October 14, 2005 @ 7:12 pm
RP - “Turbulent energy is composed of momentum! Since turbulent energy dissipates into heat, so will momentum.”
I think you need to consult an elementary physics book on the difference between energy and momentum, and on what happens in an inelastic collision (some kinetic energy changes form to internal energy, sound, light etc.) but momentum is rigorously conserved - always and ever.
Comment by CapitalistImperialistPig — October 14, 2005 @ 9:44 pm
Dissipative forces exist on a macroscopic level, but then again, so do infinite heat baths. On the other hand, they result not only in the loss of conservation of momentum, but also of conservation of energy. However, as soon as you start looking at the bath in microscopic detail, dissipative forces disappear and you regain the conservation laws.
To the extent that a model treats part of the system as an infinite bath for energy or momentum, you can beat the conservation laws but you have to be careful in the analysis lest you push the assumption too far. It appears to me that the surface of the earth is such an energy bath for atmospheric dynamics models, as the surface of a pipe is for fluid flow. However, as long as the perturbation stays away from the wall it will not dissapate. Since the surface is in all climate models, and since the models show that small perturbations in the atmosphere propagate, for the purposes of this discussion, the butterfly flaps.
Comment by Eli Rabett — October 15, 2005 @ 12:01 am
Dr. Annan, after a visit to your usenet post I can hardly see how it will further the discourse you gentlemen have on this topic.
It has already been hijacked by a polemical internet Torquemada invoking “scientific death penalties”, others rambling on about Hurricane Katrina, male genitalia and other extraneous issues, and a distinct lack of focus on the topic at hand. One can only wonder at how bizarre the thread will become in the next 24 hours.
Perhaps sharing with Dr. Peilke the contents of Dr. Eykholt’s e-mail would help clarify RPsr.’s error?
Your last post on the topic at your blog, although cryptic, seems to indicate that Dr. Eykholt does not “agree(s) with the well-understood position that small pertubation” as wholeheartedly as you originally indicated.
Comment by Kerry — October 15, 2005 @ 6:00 am
Following are two e-mails from Richard Eykholt (published with his permission) in response to the original email from James Annan. The first in response to the request for clarification from James Annan and the second in response to how he used the first answer on a subsequent post.
Subject: Re: Your comments on chaos
Date: Wednesday 12 October 2005 09:35 pm
From: Richard Eykholt
To: jdannan@jamstec.go.jp
James:
You have asked me to point out the flaw in your reasoning, but I don’t really
understand your reasoning. Are you saying that, if you run a numerical
simulation, and then run it again with a perturbation, the two trajectories
become decorrelated? If so, I agree with you. However, you need to realize
that your numerical trajectory is not a true trajectory. At every stage,
your computer program is rounding off the result, so a new perturbation is
being introduced at every step in the calculation. This series of a large
number of perturbations produces a rapid loss of correlation. On the other
hand, the shadowing lemma guarantees that there is a small tube around your
numerically generated trajectory for which a true trajectory lies inside of
this tube. However, your numerically generated trajectory will not have the
same correlation properties as this true trajectory. Basically, the size of
this tube tells you the scale on which perturbations disrupt trajectories.
Since the tube is small, your numerical trajectory is close to a true
trajectory, but small perturbations do not disrupt behavior beyond a small
scale. If small perturbations did disrupt trajectories on a large scale,
then the tube would be large, which would mean that your numerical trajectory
would be meaningless.
If this does not address your question, please try to state the question more
clearly for me.
Richard Eykholt
On Wednesday 12 October 2005 07:40 pm, you wrote:
> Dear Professor Eykholt,
>
> I saw your comments on chaos, published on Roger Pielke Snr’s blog
> http://climatesci.atmos.colostate.edu/?p=70
>
> I am puzzled by your claims about the effects of a small perturbation.
> Although of course a perturbed model state will return to the attractor
> (if it ever leaves it - a large part of numerical weather prediction
> research is focussed on generating “balanced” perturbations which do not
> leave the surface of the attractor in the first place), it will be in a
> different location from the unperturbed model, and their future
> evolution will decorrelate on all scales. This behaviour is readily
> observable in NWP models (although “minuscule” is limited by numerical
> precision).
>
> Of course a minuscule perturbation takes time to grow sufficiently large
> to have a significant effect, and there are other issues such as model
> inadequacy which may provide a more severe limit forecasting ability in
> practice, but it never occurred to me that the basic theory could be
> disputed in the way that Professor Pielke and yourself appear to be doing.
>
> If you have the time, I’d be grateful if you could explain where you
> think I got it wrong at
>
> http://julesandjames.blogspot.com/2005/10/butterfly-effect.html
>
> Thanks,
>
> James
From Oct 14, 2005
“James:
You emailed me with a question related to Roger Pielke’s weblog. It was a
very specific question regarding why numerical simulations showed a
decorrelation that seemed to you to be inconsistent with what Roger and I had
stated. You asked for the flaw in your reasoning. I did my best to explain
where your reasoning went wrong. You have now claimed that I agree with you
about something we never discussed. This is totally unethical. Thus, this
will be my last communication with you. Let me state this as clearly and
succinctly as I can. The shadowing lemma gives you a scale beyond which
small perturbations cease to have any important effects. You don’t see this
in your numerical simulations for the reasons that I tried to explain before.
However, the fact that numerical simulations have their own problems with
simulating chaotic behavior does not invalidate what Roger Pielke is trying
to say. You are simply wrong in your understanding of the butterfly effect.
Richard Eykholt
Comment by Roger Pielke Sr. — October 15, 2005 @ 7:26 am
Professor Eykholt - Perhaps I am mistaken, but doesn’t the Shadow Lemma just say that for any small perturbation of the system, the resulting orbit will shadow “some orbit” of the unperturbed system? I don’t think it says that it will shadow the original unperturbed orbit.
Since we know that the unperturbed system has orbits that include tornadoes, etc., isn’t it possible that the perturbed orbit will shadow an orbit of the system that contains a tornado not in the original, unperturbed orbit.
Also, the Shadow Lemma has assumptions which may not apply in real climate systems. In particular, system near bifurcation points may be sensitive to noise.
Comment by CapitalistImperialistPig — October 15, 2005 @ 8:46 am
Possibly we are making progress here. Prof Eykholt (and presumably Roger Pielke) agree with myself, James and William that models do indeed exhibit sensitive dependency to initial conditions (i.e. model butterfly flaps do affect weather). Maybe that was obvious all along, but it’s worth stating clearly. Secondly, I (and I presume James and William) agree that if you start close enough to the original trajectory you can stay arbitrarily close to that trajectory. So far so good.
But… while that is a neat mathematical statement (and since I started off as a mathematician, I appreciate that), there are some corollaries which are relevant. In particular, how do you define ‘close’? For a fixed time period, the lemma says that I can find a ‘tube’ of radius ‘delta’ (say) for which all trajectories will remain within ‘epsilon’ of the original trajectory after time ‘t’. As ‘t’ increases (and epsilon is fixed), the size of ‘delta’ decreases exponentially. In the real world, the time periods we are talking about are long compared to the Lyapunov timescales, and so for time periods of a month say or a year, the ‘delta’ would quickly get be way below butterfly scales (and I can simply increase the time period arbitrarily to force that).
What the shadowing lemma does not state is that there is a a tube of radius delta that always stays within epsilon of the original trajectory (which would be only the case in a non-chaotic system). Prof. Pielke’s statement about dissipation would correspond to stating that for a small enough delta, epsilon can be made arbitraily small for a long enough t. This is clearly not the same thing as the lemma.
So it still appears to me to that in any chaotic system the butterfly flap will affect the weather given a long enough time.
One minor point of clarification on Prof Eykholt’s statement about models is needed. He appears to claim that the decorrelation seen in models is due to round off errors adding new ‘perturbations’ to the system and therefore continually moving the model’s trajectory away from the true and initial perturbed trajectories. This is not correct. The model’s mathematics is completely deterministic, thus round off error is completely predictable and part of the model equations. That is to say, the model does follow the perturbed trajectory exactly.
Comment by Gavin — October 15, 2005 @ 9:56 am
Hi All- There has been a remarkable amount of attention on this post. Readers have quite a range of views to digest. The reason for concluding that the flapping wings of a butterfly cannot under any circumstances alter the weather long distances can be further illustrated by a simple question.
Does clapping your hands in one place, as opposed to having someone 10 meters away clapping their hands, have any possibility of affecting the air motions 1 km away? Assume the heat generated at the very small scale turbulence dissipates is the same for each. Non-zero turbulence spatial-temporal correlations from this clapping will quickly go to zero. There is never a long distance effect for this situation.
Comment by Roger Pielke Sr. — October 15, 2005 @ 1:03 pm
Gavin - Your understanding of what has been agreed differs sharply from mine. You and James believe in the butterfly effect, Pielke and Eykholt don’t. Nobody ever admitted to doubting sensitive dependence on initial conditions.
I don’t believe that the shadowing lemma says that the perturbed orbit closely matches the original - it says that the perturbed orbit closely matches “some orbit” of the unperturbed system.
It’s not clear that the classical shadowing lemma applies to weather or climate systems. Much weaker forms, characterized by some “shadowing time” may.
Comment by CapitalistImperialistPig — October 15, 2005 @ 3:32 pm
Gavin et al,
As you can see, Roger actually does dispute that a small perturbation will affect the trajectory of the real world. This is the entire basis of the disagreement.
Perhaps a different tack will help. There are two relevant aspects to the perturbation: its magnitude, and its direction when considered as a vector in model space. I wonder if Roger will be able to clarify whether he claims that a very small pertubation even in the direction of the leading Lyapunov vector (or singular vector for a finite forecast interval) will inevitably decay? Typically, such a perturbation will have a coherent large-scale structure, but one can take the magnitude to be arbitrarily small. A few thousand butterflies twitching their antennae, perhaps…
As far as I can tell, Roger’s reasoning that a small perturbation should simply die away, could equally be applied a priori to a model (which after all contains disippative terms). Prior to Lorenz, such intuition was presumably widespread. Of course, in the model world, such intuition can be directly proven wrong.
Comment by James Annan — October 15, 2005 @ 3:39 pm
Question for Prof Pielke - If sensitive dependence on initial conditions does not mean that a micro change (like a butterfly wing flap) can lead to a macro change later, and far away (like a tornado in Texas), what does it mean? An example would be helpful.
Comment by CapitalistImperialistPig — October 15, 2005 @ 3:46 pm
Thanks for the continued constructive dialog. The sensitivity to initial conditions does not mean that all perturbations upscale and teleconnect. I present an illustrative example under my comment #27 of when there is no such upscaling.
I feel the disagreement is that it is not being recognized that real world perturbations which produce velocity fluctuations cascade to smaller spatial scales through nonlinear interactions which are then dissipated into heat. This dissipation results in the inability for very small velocity perturbations to grow and upscale.
With respect to a global atmospheric model. The kinectic energy will decay to zero (the winds will eventually stop blowing everywhere) if you turn off all boundary energy inputs into the model, including the Sun’s insolation. We have tried this; it took about 2 weeks of simulated time. Although this does not represent the real world, of course, it shows that without the continued input of energy, all modes of motion in the model are eliminated (by numerical and explicit diffusion in this case).
Comment by Roger Pielke Sr. — October 15, 2005 @ 4:36 pm
Net momentum doesn’t change though. We know the net momentum of the atmosphere is zero - since it stays attached to the planet. Local momenta cancel with other local momenta to yield this result. When you turn off the Sun, the local momenta of large scale turbulent structures decay by exchange of momentum with other turbulent stuctures of opposite momentum, and total momentum stays conserved and zero with respect to the Earth.
Comment by CapitalistImperialistPig — October 15, 2005 @ 5:23 pm
It has been asked if analog weather prediction provides an evaluation of the sensitivity to initial conditions, as these forecasts always diverge from each other, no matter how closely one chooses the analogs.
This is a good demonstration of real world sensitivity to initial conditions. However, the deviations (i.e. the perturbations) between each case within an analog is actually quite large, no matter how much one attempts to find nearly identical cases.
These large perturbations do demonstrate the sensitivity of weather and climate to initial perturbations. This is why we need to assess the magnitude of the perturbations to determine if they will damp out before upscaling (as with a butterfly’s flapping wings) or result in a progressively divergent solution as can occur with large perturbations (such as with cyclogenesis).
Comment by Roger Pielke Sr. — October 15, 2005 @ 9:00 pm
Is it time to try a few more new tacks? Sorry about the length of this post. Hopefully it is simple enough to help locate exactly how such a difference of opinion arises.
Prof Pielke seems to accept that the models show chaotic sensitivities to initial conditions but claims the real world is not like that. This seems wrong to me – I can think of lots of processes that are not modelled that might introduce/increase the amount of chaos:
Scientists are used to thinking about lots of random motion of molecules all cancelling to get equal radial pressure. Does this mean that disturbing the path of one molecule leads to not only no change in the equal radial pressure but also no other effects on other systems? Consider Browning Motion, I think the same will apply to pollen suspended in the atmosphere rather than floating on water. The expected vector movement of a pollen grain over a long period of time is zero but there is an expected distance moved over a period which is not zero. The pollen grain movement is a larger effect than the disturbance of the path of one molecule which caused it so there is a growing effect.
I’ll quickly give a couple more examples of other growing effects: The pollen grain movement could make the difference between an animal sneezing and not sneezing. An animal sneeze could reveal the presence of that animal and change all animal interactions.
I have seen this referred to as a long string of unlikely events could lead to a change in weather. I think this is misleading/wrong: A particular sequence of event is extremely unlikely however disturbing the path of one molecule, it is certain that one molecule will hit another and those two will hit two others then 8 16 etc. So it becomes certain that the paths of all air molecules and all suspended pollen becomes affected. It therefore becomes certain that all sneezes caused by pollen will eventually be changed.
If a GCM shows chaos, I cannot begin to imagine how I could believe that adding all these un-modelled processes will somehow magically remove the chaos.
I really regard the chaos in weather systems as a numbers game: Let’s start by creating a mythical dynamic system where I make up the rules. Our perturbation affects surrounding locations so that there are 6 locations each with an effect that is half the size of the original. This has not created 3 times the energy it has merely redistributed existing energy a bit. From this you could claim the effect is dieing out because the perturbations are smaller (half the size). Alternatively you could claim there is a larger total effect (6*.5>1 so (6*.5)^n gets larger as n gets larger). Both are wrong, you need to see some further rules to see what happens. Suppose 2 of the 6 move away creating 12 changes that are a quarter of the size of the original. The other 4 interfere; one pair destructively and one pair constructively. At the end of this second time-slice, we have one perturbation of the same size and 12 perturbations of quarter size. I think it is now clear that this system is going to cause growth of effects (6*.5*.5>1).
Of course, this system proves nothing because you can create a different system (e.g. 3*.5*.5
Comment by crandles — October 16, 2005 @ 2:22 pm
sorry try again with the continuation….
(e.g. 3*.5*.5 less than 1) that does not grow. The problem is if you believe the real world is like that, how would you get any weather? Wouldn’t large perturbations die away into small perturbations that then die away completely? Maybe you could (or maybe it would be very difficult?) create a system with different rules for large and small perturbations. Is there good reason to think that applies to the real world? Or do equations that appear to govern the behaviour of motion in an atmosphere apply at a large range of scales?
My ‘numbers game’ is therefore a matter of what happens faster the initial reducing size of the perturbation or the rapidly growing number of locations at which effects occur.
Perhaps there is a straw for Prof Pielke to clutch at if he wants to claim that all he wanted to do was point out the initial reducing size of effects until they are trivially small. It appears to me he has not adequately considered that the rate of growth in the number of locations affected needs to be so much greater such that these effects interfere and not always destructively so that they win in the end (provided that the number of locations is not artificially restricted).
By ‘artificial restriction’ I mean that if you put a butterfly wing flap perturbation in a sealed room that did not have chaotic weather then the effects would die out. I wonder if it is possible for chaos theory to make some sort of (testable?) prediction of the smallest sealed room size that could have chaotic weather.
Comment by crandles — October 16, 2005 @ 2:25 pm
Conservation of Momentum
I think the law says that momentum is conserved in the absence of external forces. The classic examples that we’ve all seen usually involve collisions between two perfectly elastic spherical balls. The statement of conservation of momentum is then used to state that the total momentum before and after the collision is equal. The example is sometimes extended to the case of two perfectly inelastic spherical balls. None of the equations that I’ve seen for this classic example contain terms for external forces acting on the balls. The examples are of course presented to illustrate the conservation of momentum in the absence of external forces. Note, however, that the initial momentum assigned to the bodies most likely required the action of an external force
What can we say about momentum when external forces are acting on a body? It seems to me that gravity, an external force acting throughout a body, can act to change the momentum of bodies under motion in the presence of the force. Consider the case of an object dropped from a distance above the surface of the earth with no initial speed in any direction. I use the word “speed” and not “velocity” because I’m going to be looking at motions in only one direction and that direction is parallel to the gravity vector. The speed of the object increases along its path when the body is falling in the direction of the gravitational force. Taking the momentum to be the product of mass and speed, the momentum is changing and increasing. This description is correct even in the presence of friction between the body and the surrounding fluid. If the gravitational body force is not present, there would be no motion and the body remains at rest.
Likewise if the body is given an initial speed in a direction parallel to but directed opposite the gravity vector, its momentum first begins to decrease and at some point its momentum becomes zero and its direction of travel reverses and the momentum begins to increase. The friction between the body and the surrounding fluid can be considered or not and the results are the same.
Other examples are possible. Consider a fluid in motion as a slug in a one-dimensional flow channel that is perpendicular to the gravity vector. If the driving potential for the motion is shut off the fluid speed will decrease due to friction between the fluid and the flow-channel wall; the momentum goes to zero from an initially non-zero value. The change in momentum is due to the action of the friction at the fluid-wall interface.
So, it seems to me that in the presence of external forces, it is possible for momentum to not have a constant value for all times of interest. I do not know if this is considered to be “conservation” or “non-conservation” of momentum, within the framework of these discussions, but the momentum is not constant. In a sense, momentum is generated by the actions of external forces. This is in contrast to the case of mass and energy, for which generation is not possible in the absence of sub-atomic particle considerations.
I think someone also said that in the absence of external forces a body initially at rest will remain at rest and its momentum will be constant and have the value zero.
Thermal effects can act by way of external forces and give rise to fluid motions via the gravitational body force. Heating or cooling of a fluid introduces density gradients into the fluid and these gradients induce motion in the fluid through the gravity force. I think that a well-designed numerical experiment will show that in the absence of energy addition into the fluid through heating, fluid motion will cease. By well-designed I mean that all the initial and boundary conditions and interactions among the components of the models must be investigated to ensure that motion is not induced by these. Of course as is the case for all numerical experiments, the possibility that artifacts of the numerical methods themselves are dominating the fluid motion are critically important and must be eliminated. Grid independence of the solution is the minimum acceptable test. Ideally, a numerical solution method that is both non-dissipative and non-dispersive ( MOC, for example) is used. Realistically this is seldom the case.
Comment by Dan Hughes — October 17, 2005 @ 2:48 am
Physical Dissipation
Dissipation of kinetic energy and its transformation into thermal energy occurs in all fluid motions of real fluids. These processes are due to the viscosity of fluids and the associated shear stresses which arise when gradients in the velocity distribution in the fluid are present. The dissipation arises in the basic equations for fluid motion when the dot product of the velocity vector with the vector momentum equations is formed; the results is a scalar equation for the kinetic energy. The dissipation contribution appears as a positive-definite term on the right-hand side of the thermal forms (internal energy, temperature, enthalpy) of the energy equation for the fluid; dissipation always increases the thermal energy of the fluid. It makes no difference if the fluid motion is laminar or turbulent, forced or free. So long as there is a gradient in the velocity distribution for the fluid motion, dissipation is always present for real fluids.
For the case of turbulent single phase fluid flow the dissipation of the kinetic energy of the turbulent fluctuations is considered to be what causes the flow to be stable. Stable in the sense that the fluctuations do not grow in time. The dissipation generally occurs within the smallest scale of the turbulence. Algebraic friction factor and heat transfer correlations are so very accurate and effective for the case of turbulent single-phase flows because it is this single dominant scale that controls the flow. And, while these correlations are usually reported in terms of macro-scale properties of the flow, they can be equally developed and reported with micro-scale properties; scales associated with the micro-structure of the turbulence. Multi-scale dominated flows are very much more difficult relative to development of accurate and effective algebraic correlations.
Comment by Dan Hughes — October 17, 2005 @ 2:51 am
By changing the position of a boulder at the sea edge, does that affect the timing of high tide some time in the future? If not, why not?
Comment by stephan harrison — October 17, 2005 @ 2:51 am
It should be obvious to everyone that every living thing on the planet, and all the things that all these creatures have made, produce an essentially uncountable number of perturbations every second of every day. So how can the effect of a single perturbation by a single butterfly be the issue. The mechanical, chemical, and thermal perturbations introduced by human-made equipment and processes are especially enormous in magnitude, number, frequency of occurrence, and distribution around the planet. If the perturbations of a single butterfly are important, and this importance is captured in the fundamental equations used in the climate-change models, then how can all other perturbations be neglected.
Unbounded exponential growth of physical processes are not observed in nature. If they were, we probably would not be here. And I am not aware that the Lorenz equations exhibit unbounded exponential growth. So it seems to me that there are two separate issues involved here; the chaos in the Lorenz equations and the problems posed by ill-posed initial value problems (you might say).
Linear stability analysis applied to ill-posed initial value problems will indicate unbounded exponential growth. Generally, however, the application of linear stability analysis by its very nature is applicable to only the early stages of the physical processes that are being modeled. Maybe even to only the onset of the growth.
Many systems that exhibit initially unbounded growth are caused by discontinuities in the physical parameters of the system. Discontinuities in the distributions of fluid density, temperature, and velocity within the flow field of interest produce unstable physical phenomena. The atmosphere-ocean interface is a good example in which discontinuities in all three of these are possible. Mathematical representations of the physical discontinuities will give calculated results that exhibit unstable behavior similar to the expected physical instability.
Relative to initial instabilities, I think that linear stability analysis applied to the Helmholtz, Kevin, Taylor, Raleigh, etc., situations will indicate unbounded exponential growth for some wavelengths. However we know from experiments that these initially unbounded processes do not in fact grow without bounds. As another example, transition from laminar to turbulent flow does not results in unbounded growth of the perturbations.
Generally, the very shortest wavelengths do not ever grow in nature or in well-posed mathematical problems; these correspond to the region of highest physical dissipation. The shortest wavelength that can be approximated in numerical solution methods is two times the grid spacing. This is usually much larger than the spatial resolution required for accurate resolution of the real physical processes.
The numerical approximations and rough approximate solutions obtained in most models of complex physical processes introduce additional difficulties that deserve separate discussions.
Comment by Dan Hughes — October 17, 2005 @ 3:22 am
#38
The answer is yes. But the effects on the timing of the tide will never, as in for all time, be of theoretical or practical interest. So long as the boulder is small in size relative to (1) the size of the ocean basins, and (2) the distribution of the mass of the earth. And I assume that a local-instantaneous timing value is not the focus of the question.
Under these assumptions, modeling of the effects of the boulder is beyond our understanding of the totality of the physical phenomena and processes that govern the effects of small-scale changes on the timing of the tides. I will venture to say that there are no models that account for and include a mathematical description of the effects of scales of the order of those in the assumptions. I suspect tide timing is calculated based on the assumptions of homogeneous distribution of the mass of the earth within a non-spherical earth.
Additionally, the changes induced by moving the boulder will not change the chaotic and/or initial-condition sensitivity of the physical phenomena.
Changes to the boundaries of the ocean basins and distribution of the mass of the earth consistent with the scales in the assumptions have occurred for centuries. These changes have been both natural and human-made. I doubt that those you devise tide tables have included these effects. Primarily because the changes are not known to those people.
Comment by Dan Hughes — October 17, 2005 @ 8:12 am
Just a small point about the interaction between James Annan and Richard Eykholt: I sincerely hope that Eykholt does not consider every honest misunderstanding to be a “totally unethical” cause for ceasing all communication with the offending party. If so, he must be a member of a very narrow academic discipline indeed!
Comment by Steve Bloom — October 17, 2005 @ 7:39 pm
Dan Hughes - Suggest you get out that old freshman physics book. Momentum is conserved in all the situations you mention - and everywhere in the universe, so far as we know. External forces can change the momentum, but, according to newtons third law, only by changing the momentum of the external source. The Earth’s atmosphere can exchange momentum with the Earth, or even (in tiny amounts) with the external celestial bodies by tidal forces, but momentum is conserved.
Comment by CapitalistImperialistPig — October 18, 2005 @ 5:45 am
Steven Harrison - There is no reason to believe that tides are chaotic in their timing - unlike the weather. The tidal system is strongly dissipative, and the driving force is very stable. Moving the boulder makes a tiny difference in the tides but that difference does not grow or produce chatic changes in the pattern of tides.
Comment by CapitalistImperialistPig — October 18, 2005 @ 5:51 am
#42
CapitalistImperialistPig
Note that I explicitly stated in the post:
“So, it seems to me that in the presence of external forces, it is possible for momentum to not have a constant value for all times of interest. I do not know if this is considered to be “conservationâ€? or “non-conservationâ€? of momentum, within the framework of these discussions, but the momentum is not constant.”
Maybe I should have added that “the momentum OF THE BODY is not constant” in the last sentence above. My entire discussion was about a body of mass; not the totality of the momentum of the entire universe.
Can you point me to errors in this entry: http://en.wikipedia.org/wiki/Momentum
My discussions are completely consistent with that entry.
Further this entire thread is focused on the behavior isolated bodies of mass not the entire universe.
Finally can you explicitly point out an error in the post.
Thanks for your assistance.
Comment by Dan Hughes — October 18, 2005 @ 6:15 am
#42 and 44
CapitalistImperialistPig
I forgot to note that I am working within the Newtonian framework and have taken the gravitational attraction between the body and the earth to be a constant. If an analysis is carried out in which variation of the gravitational attraction with distance is allowed the results will be the same. The momentum of the body changes.
Comment by Dan Hughes — October 18, 2005 @ 9:01 am
Regarding the question on the conservation of momentum, from the web site http://en.wikipedia.org/wiki/Momentum, it states “In the absence of external forces, a system will have constant momentum”. This certainly is true, of course.
The key condition is “in the absence of external forces”. Molecular dissipation into heat is an “external force” in this context (the term “internal force” would be more appropriate). Perhaps it is the word “external” that is causing confusion. The phrase could more clearly be written as “In the absence of unbalanced forces or in the presence of no forcing, a system will have constant momentum” (of course, a balance of forces can also produce constant momentum).
We can use a sea breeze as an example. A sea breeze often starts from calm conditions in the morning. Clearly, momentum is not conserved since there are forces (i.e. from the heating of the ground surface by the sun) which initiates wind. Later in the evening, the sea breeze weakens to zero speeds. The momentum is lost as the kinetic energy is dissiapted. Momentum is clearly not conserved for this weather feature, or any other atmospheric circulation.
Comment by Roger Pielke Sr. — October 18, 2005 @ 10:54 am
If you are going to quote the wikipedia, Roger, you should start at the beginning. Did the following sentences escape your notice? “Momentum has the special property that it is always conserved, even in collisions. Kinetic energy, on the other hand, is not conserved in collisions if they are inelastic.”
Dan Hughes - It doesn’t matter what framework you work in, provided you believe in the laws of physics. If you throw a ball up or fire a cannon up you transfer momentum to the projectile, an equal and opposite momentum is transferred to the Earth (net vector momentum stays the same). On the way up and down, the gravity of Earth pulls on the ball and the gravity of the ball pulls on the Earth decreasing those equal and opposite momenta. This is high school physics - pick up a book and look it up.
Comment by CapitalistImperialistPig — October 18, 2005 @ 2:17 pm
If you consider a small sealed room where the unperturbed state is for no pressure or temperature anomalies then it is clear that a perturbation whether large or small will dissipate into heat absorbed by the walls. So it seems to me that Dr Annan is wrong to say that Roger Pielke’s views break the rules on conservation of momentum. If you don’t consider the whole system, only the atmosphere, then momentum can change.
I think the problem is that in the small sealed room any loss of momentum to heat is a step towards the unperturbed state so this always helps reduce the differences.
Outside the sealed room then a loss of momentum to heat could increase the differences as the best way to move towards the (noisy) unperturbed state could be to increase the speed of the molecule not slow it down. I would suggest that any dissipation to heat is as likely to increase the differences as reduce them.
Thus it appears to me that Roger Pielke’s explanation works for a stable atmosphere but not for a chaotic atmosphere.
Comment by crandles — October 18, 2005 @ 4:51 pm
The statement that momentum is always conserved is incorrect in wikipedia. From their web site momentum = mass × velocity. If the velocity goes to zero, so does the momentum.
The text later on their web site, that
“Because of the way it is defined, momentum is always conserved. In the absence of external forces, a system will have constant momentum”,
conflicts between the first and second sentence. The second sentence is correct, but not the first. Their statment that
“Momentum has the special property that it is always conserved, even in collisions” is for the particular (billard ball) example and cannot be generalized for fluid motions such as in the atmosphere.
Thanks for following up.
Comment by Roger Pielke Sr. — October 18, 2005 @ 5:04 pm
Shadowing and chaos
Now, Professor Eykholt made repeated reference to the shadowing lemma in his emails to me, which you can read on Roger Pielke’s blog. [...] I cannot see how his statement “The shadowing lemma gives you a scale beyond which small perturbations cease t…
Trackback by James' Empty Blog — October 18, 2005 @ 6:41 pm
Crandles-Heat absorbed by the walls is not different in terms of damping a perturbation, than heat dissipated into heat in the atmosphere as small scale turbulence cascades to smaller scales. This energy cascade is a fundamental concept in turbulence theory.
Comment by Roger Pielke Sr. — October 18, 2005 @ 7:59 pm
James-I have linked to the post on your web site. Others are invited to comment here also on your perspective. Thanks for including the link.
I am still surprised that there is still any serious consideration that very small atmospheric perturbations (such as the flap of a butterfly) upscale in the atmosphere, and are not always damped out.
The use of an NWP or climate model to show this effect is not appropriate as there we are talking about much larger perturbations, where I agree they can upscale as shown by the HADCM3 runs. Indeed that is the focus on ensemble weather forecasting. There is an important research question as to how small can the perturbations be and still be capable to sometimes upscale. It certainly is not as small as a butterfly’s flapping wings.
Comment by Roger Pielke Sr. — October 18, 2005 @ 8:10 pm
Roger, I’m equally surprised there is still belief that small perturbations will dampen out without causing any change in the overall weather pattern. However that subject has been beaten to death and I doubt anyone is going to change his mind at this point.
I would say that you are wrong in your statements on conservation of momentum too. There is nothing contradictory in the two sentences “Because of the way it is defined, momentum is always conserved. In the absence of external forces, a system will have constant momentum”.
Momentum is always conserved if you look at a total system. If we look at a partial system momentum may be changed by an “external force”, but that force will cause an equal reaction in the external system that change the momentum there in the opposite direction. The total momentum willl remain constant, it will just have been moved into the “external” realm.
Conservation of momentum certainly can be generalized to fluid motion. I don’t understand how you can claim otherwise. The sea breeze you mention is an example on how momentum can change locally, however this is offset by movement elsewhere. As the wind blows in over land at ground level there will be other air moving towards the sea elsewhere or there would soon be a vacuum over the sea. Some momentum may even be absorbed by the Earth via friction. Look at the complete system and momentum remains constant. The sun adds energy to the system, but the momentum from the photons is very small and can be neglected.
What I found equally surprising is your statement that “I do agree that both weather and climate are chaotic”. Apart from the fact that chaotic by definition implies sensitivity to very small perturbations, I haven’t seen anyone claim that climate is chaotic. It is non-linear and it is likely to have thresholds, and at those points a small perturbation may be enough to flip the climate into a different state. To be chaotic *any* small perturbation should grow to large size, and I just don’t think that is true for climate. The perturbations we see are in the weather, and those will be averaged out when looking at climate.
Comment by Thomas Palm — October 18, 2005 @ 11:36 pm
#47
CapitalistImperialistPig, you have yet to explicitly point to an error in my post. You have not yet pointed to any problems in the wikipedia entry on which the post is based.
This being the case I take the post as being correct.
I do not have a high school physics book handy, nor a freshman physics book. Can you point to the exact entry in wikipedia from which we can determine that if the speed of an isolated body of mass changes the momentum of that body does not change?
You do not advance these discussions by refusing to read and understand what has been said.
Comment by Dan Hughes — October 19, 2005 @ 2:56 am
I like your example of a sea breeze as a case where momentum is dissipated. It shows how the real world behaves as opposed to the theoretical world of bouncing molecules which seems to be uppermost in the minds of your critics. But I would like to offer an improvement to your explanation which I believe makes it watertight. It will be interesting to see if your critics accept my argument, or if they they are so wrapped up in their own hubris that they cannot listen to reason .
You write that when the sea breeze dies in the evening, then the momentum is being lost. However it could be argued that it is in fact being dispersed among the rest of the atmosphere, and is in fact being conserved. However, if instead we consider the morning when the sea breeze grows then momentum is being created. It is not being ‘un-dispersed’ and collected together from the rest of the atmosphere, because that would be contrary to the second law of thermodynamics! The momentum is in fact being created by the conversion of heat into work, and it is lost in the evening by the conversion of work back to heat by a process commonly known as friction.
HTH,
Cheers, Alastair.
Comment by Alastair — October 19, 2005 @ 3:30 am
Thank you for taking the time to respond. However you seem to be commenting on the area where we agree: heat dissipation occurs.
I think we also agree that in a stable system this heat dissipation causes a perturbation to disappear.
You do not appear to have commented on the points where we disagree: Unlike in the stable system the heat dissipation does not help because it is as likely to make things worse as better.
Additionally the radiation can be absorbed elsewhere in the atmosphere. This means the effects can spread to more location faster. My earlier comments indicate I think it is the number of locations that matter more than the reducing size.
Comment by crandles — October 19, 2005 @ 3:57 am
More on Dissipation
See the following references.
R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc., New York, 1960.
Chapter 10 contains the discussions about development of the various forms of the energy equation for the fluid motion. These authors focus on conservation and balance equations for the mean motion of the fluid. Equation 10.1-11 is an equation for the change of the sum of the internal and kinetic energy. Equation 10.1-12 is the equation for the change of kinetic energy; obtained by taking the dot product of the velocity vector with the momentum balance equations for the fluid motion (developed in Chapter 3). The last term in that equation is the dissipation due to viscosity of the fluid. It is always negative in the kinetic energy equation. Subtracting Eq. 10.1-12 from Eq. 10.1-11 gives Eq. 10.1-13 which is the equation for the change for the internal energy. The last term in that equation is again the dissipation of mechanical energy to thermal energy. It is always positive in the equation for thermal energy. Kinetic energy is converted (transformed) into thermal energy and always increases the thermal energy content of the fluid mass. As the authors state, “ … an irreversible degradation of mechanical to thermal energy.�
Uriel Frisch, Turbulence, Cambridge University Press, Cambridge, 1995.
Energy dissipation is discussed in Chapter 5. The energy dissipation law is discussed in section 5.2.
L. D. Landau and E. M. Lifshitz, , Course of Theoretical Physics Volume 6, Fluid Mechanics 2nd Edition, Pergamon Press, 1987.
These authors discuss dissipation in the turbulent motions in Section 33 of Chapter III. Various other aspects of turbulence are also developed in Chapter III, including stability of the turbulent motions and chaotic aspects of turbulence.
M. T. Landahl and E. Mollo-Christensen, Turbulence and Random Processes in Fluid Mechanics, Second Edition, Cambridge University Press, Cambridge, 1992.
These authors focus on the turbulent-flow aspects of fluid motion. The basic equations for both the mean flow and the turbulence are given in Chapter 3. Equation 3.28 is the equation of change for the kinetic energy of the turbulent motion. The third term on the right-hand side of that equation is the viscous dissipation due to the motion of the turbulence. It is again always negative in the kinetic energy equation.
H. Schlichting, Boundary-Layer Theory, Sixth Edition, McGraw-Hill Book Company, New York, 1968.
Schlichting is the bible for engineering modeling and analysis of both laminar and turbulent fluid motions. The focus of the book, due primarily to its age, is the mean motions in turbulent flows. The author, however does address certain aspects of the turbulence. The thermal energy equation for the mean motion is given by Eq. 12.7 and the dissipation function by Eq. 12.8 on page 254. The equation agrees with all the other citations that are listed in these notes; the dissipation is always positive in the thermal energy form of the energy conservation equation. The dissipation function for the turbulent motions is given by Eq. 18.16 on page 538. The discussion around this equation focus on the fact that the dissipation ultimately resides in turbulent eddies of the smallest scale.
Summary
The discussions listed above all point to the fact that the Navier-Stokes equations for fluid motions are dissipative. Laminar flows are dissipative and the mean motion of turbulent flows are dissipative and the turbulent motions of turbulent flows are dissipative. Additionally, for turbulent flows mechanical energy dissipation occurs primarily in the turbulent motions of the smallest scales in the flow field. These smallest motions dominate almost all aspects of turbulent flows.
The dissipation arises from the viscous shear stress tensor; the diffusion of momentum. Linear stability analysis will show that the viscous shear terms in the Navier-Stokes equations are responsible for damping of perturbations of the shortest wave lengths.
The viscous shear in many practical models of complex phenomena in a large-scale geometry setting is represented by algebraic correlations that do not involve second-derivative like terms; friction and drag correlations. Algebraic correlations that do not contain derivatives have a different effect from second derivatives in linear stability analyses; they do not damp short wavelengths. Additionally, calculations on coarse grids probably do not accurately resolve second derivatives even if they are included in the model equations. The two-times grid spacing resolution can easily be orders of magnitude greater than the physical scale of the smallest physical scales. Frequently, grid-size dependent numerical viscosity-like terms are added into the numerical approximations to attempt to overcome this limitation.
The profound importance of the dissipation function in turbulent flows is further exploited in practical applications as covered in the following texts:
V. S. Arpaci, Microscales of Turbulence; Heat and Mass Transfer Correlations, Gordon and Breach Science Publishers, Amsterdam, 1997.
W. M. Kays and M. E. Crawford, Convective Heat and Mass Transfer, Third Edition, McGraw-Hill, Inc., New York, 1993.
V. S. Arpaci and P. S. Larsen, Convection Heat Transfer, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1984.
A. Bejan, Convection Heat Transfer, Second Edition, John Wiley & Sons, Inc., New York, 1995.
G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1996.
D. J. Tritton, Physical Fluid Dynamics, Second Edition, Oxford Science Publications, Clarendon Press, Oxford, 1988.
Comment by Dan Hughes — October 19, 2005 @ 5:39 am
Sea Breezes
A very simplified description is as follows. The morning sun supplies energy to the fluids and all other materials of interest in the environment. If we focus on the air in the atmosphere, some parts of it are heated by direct absorption of re-radiated long wavelength energy. The energy that is deposited in and absorbed by other elements bounding the air increases the temperature of some of these. Some of this energy is conducted/convected to the air in the atmosphere. Material that is at a phase-change temperature simply starts to change phase at constant temperature. Any differences between the local air temperature and whatever bounds it can change the temperature of the air.
The changes in the temperature of the air in the atmosphere induce local buoyancy-driven motions through the action of gravity. Additionally, the pressure can also be changed due to the heating and/or cooling and thus provide for pressure-driven motions.
It is these motions that contribute to generation of the sea breeze. In the sense that these are local perturbations into a somewhat initially quasi-equilibrium state, the “Butterfly Effect” would say that these grow exponentially without bound. This has never been observed in nature so far as I know.
Comment by Dan Hughes — October 19, 2005 @ 9:35 am
#58
ps
The large-scale processes driving the sea breeze are correctly described here: http://en.wikipedia.org/wiki/Land_breeze. The processes are thermally controlled. After cooling starts the breeze changes direction under the same kinds of thermally-controlled processes.
Comment by Dan Hughes — October 19, 2005 @ 1:05 pm
Dan-I am becoming skeptical of the accuracy of the wikipedia web site. With respect to http://en.wikipedia.org/wiki/Land_breeze,
it says
“The warm air is less dense and so it rises. This rising air over the land lowers the barometric pressure by about 0.2%.”
Rising air does not lower the barometric pressure. To lower this pressure, air needs to be removed from above the level where the pressure is measured. I schematically illustrate this as Figure 13-1 in my mesoscale modeling book (Pielke, R.A., Sr., 2002: Mesoscale meteorological modeling. 2nd Edition, Academic Press, San Diego, CA, 676 pp.).
Comment by Roger Pielke Sr. — October 19, 2005 @ 1:48 pm
#60
Ok thanks for the feedback Roger. I was focused on the temperature and buoyancy effects. That part I think is described correctly.
My understanding is that the Wikipedia is not to be considered as the final word on many (most?) subjects. I think it provides a focused starting point for some subjects.
We seem to have not heard from the other folks. Does anyone know what their final words on the Butterfly Effect are? I’m still waiting for an answer to the first paragraph of #39. What are the effects of say a rocket launching the Space Shuttle. Will that local perturbation grow and influence the climate around the planet?
I’m beginning to think that the Lorenz model is nothing more than an ill-posed initial value problem and that Butterfly flappings have absolutely nothing to do with the situation.
We haven’t even started to address the enormous number of issues associated with using rough approximate solutions to algebraic approximations to PDEs as a method of investigating the mathematical properties of complex systems of continuous equations. The algebraic models and correlations in AOLGCM codes are analogous to switches that can send the calculation off to different parts of the solution space. Small changes in initial conditions can trip a given switch at a different time or trip a different switch. The codes do not start from states that satisfy the equations; there are imbalances in the model system from the very start. Measured initial-value data that are fed to the equations cannot possibly satisfy the equations for several reasons. Maybe someone can point me to reports of investigations into the mathematical characteristics of the continuous equations.
The inherent inaccuracies in the numerical solutions to the algebraic representations of the PDEs, coupled with the range of temporal scales involved, can produce quasi-periodic and aperiodic like solutions even for well-posed problems.
It would be interesting to know if the codes can correctly handle physical situations for which the Butterfly Effect does not enter the picture. Maybe someone can point me to reports that contain information along this line. Are there benchmark-like problems setup for these codes?
Comment by Dan Hughes — October 19, 2005 @ 3:27 pm
Dan-thanks for your further very useful comments, and the request for additional comments by others.
On the Lorenz model, it is a seminal, pioneering analysis. Ed Lorenz, using that model, introduced science to the knowledge of chaotic mathematical behavior associated with sufficiently nonlinear systems. We have to remember though that his model is not an actual simulation of how the real atmosphere works which includes the requirement of the second “law” of thermodynamics.
My conclusion after the many constructive comments on the weblog for this posting is that the “butterfly effect” is appropriately applied to the appearance of the solution space of the Lorenz equations. It is not an effect that is to be taken literally, however, in the real climate system.
Comment by Roger Pielke Sr. — October 19, 2005 @ 5:23 pm
Dan - I might have mistakenly believed you were claiming momentum wasn’t conserved. Obviously, that conservation only applies when all interacting components of the system are taken into account, since Newtons second law can be written as dp/dt = F, where p is momentum and F is force. Newton’s third law says that if a source exerts a force on a body, the body exerts an equal and opposite force on the body, and hence source and body have equal and opposite changes of momentum.
Comment by CapitalistImperialistPig — October 19, 2005 @ 5:42 pm
We seem to have not heard from the other folks. Does anyone know what their final words on the Butterfly Effect are?
To be blunt, when a professor of atmospheric physics starts claiming that momentum in the atmosphere is internally dissipated into heat, it is time to duck out of the conversation.
Roger, your wrong-headedness on this point is beyond belief. I don’t think it much of an exaggeration to say that such comments may expose you to public ridicule.
Comment by James Annan — October 19, 2005 @ 5:51 pm
Dr. Pielke - I have been reading some of the sections on energy and momentum transfer in your Mesoscale Modeling, book, and found the discussion exceptionally clear. One point you make is that except at the surface, forces due to molecular viscosity are negligible in the mesoscale. Those forces, though, are the very ones responsible for the ultimate dissipation and conversion of turbulent energy to thermal energy. When that occurs, any net momentum of the turbulent air will either persist in the once turbulent air or have been transferred to waves or the surface. If your model neglects viscous molecular forces though, it will lose track of that momentum and hence, in your model, this negligible amount of net momentum will be lost.
Since momentum is a vector quantity and since turbulence, by definition, has motion in many different directions, the net momentum will always be a lot smaller than the sum of the rms momenta of the parts.
Do you agree?
Comment by CapitalistImperialistPig — October 19, 2005 @ 5:59 pm
Dan,
That should have read “the body exerts an equal and opposite force on the source.”
Comment by CapitalistImperialistPig — October 19, 2005 @ 6:02 pm
Re #65 Thank you for the comment on my book. With respect to the neglect of molecular dissipation in the models, this is an approximation that works for the spatial scales and time period of the model integrations. In the real atmosphere, the flow is almost always turbulent (i.e. has a large Reynolds number) such that there is always a cascade to smaller scales and a dissipation into heat by molecular processes even in the free atmosphere.
Regarding momentum, as you correctly state, it is a vector with a magnitude and direction. When considering turbulence, I suggest it is easier to conceptualize as an ensemble of Lagrangian particles, each with their own momentum. Then integrating the momentum of these particles over a volume provides the measure of the momentum over that scale. We have completed work on representing motion with such a framework, although we have not specifically focused on volume integrated momentum. (see Pielke, R.A. and M. Uliasz, 1993: Influence of landscape variability on atmospheric dispersion. J. Air Waste Mgt., 43, 989-994.http://blue.atmos.colostate.edu/publications/pdf/R-142.pdf for an example of our use of Lagrangian particle models).
Comment by Roger Pielke Sr. — October 19, 2005 @ 8:45 pm
James-the ultimate arbitrator of this weblog debate will be the readers. Hopefully, our material will be referred to in classes and discussed among the students and others. Thanks for being involved in fleshing out the subject.
Comment by Roger Pielke Sr. — October 19, 2005 @ 8:48 pm
Even more on Dissipation
The balance equation for kinetic energy of the turbulance mentioned in previous posts contains both dissipation and production terms. The discussion in the previous posts focused on dissipation. The production term in the kinetic energy equation for the turbulent motions is the product of the “turbulent” or “apparent” or “Reynolds” stresses and the gradient of the mean flow. (Note that other forces, buoyancy for example, can also produce turbulence but that is not the main issue here.) A term of the same form appears in the kinetic energy equation for the mean flow, but with oppsite sign.
The term accounts for the transfer of energy from the mean flow to the turbulent motions. The dissipation of the mean flow is the viscous stresses working with the velocity gradients in the mean flow. As discussed above, this dissipation goes directly into production of thermal energy for the mean flow. In the case of the turbulent motions, however, the Reynolds stresses working against the gradients of the mean flow provide energy for production of turbulence. This energy is ultimately dissipated by viscosity and the smallest-scale motions of the turbulence. The direct dissipation of the mean flow energy is usually small relative to the energy transfered to the turbulent motions.
The mean flow drives the turbulence as it mmust. There’s nothing else to be a driver. The “sum of forces” acting on the fluid will provide accounting for all the contributions in the mean momentum equation that can provide energy for production of turbulence. And that energy is ultimately dissipated in the small-scale turbulence. When the forces that drive the mean motions cease to act, the mean motion, and thus the turbulence, goes away.
One more thought on the Butterfly Effect. I stongly suspect that the differences in the specified initial states, and thus differences in the initial imbalances in the balance equations, simply drive the evolution of the system to different final states. This kind of response can be observed in well-posed problems. It is stronger function of the algebraic models and correlations (switches) than of the properties of the continuous PDEs. It is not the outcome of chaotic response as I understand the term.
Comment by Dan Hughes — October 20, 2005 @ 6:32 am
Dr. Pielke,
Just a brief note about the not-quite-accurate pages on wikipedia. The brilliance of wikipedia is that when you — as an expert in your field — come across an inaccuracy, you can modify the page. Later page visitors will benefit if you intervene, but will go away and possibly propogate error if you let it stand.
Comment by Dan Pawlak — October 20, 2005 @ 1:04 pm