Models Methods Software

Energy and the Lorenz System

Introduction
I’ve decided to modify this post and put an example here. Examples have the potential to provide more understanding of the important technical issues.

So, let’s say it’s Saturday January 5, 2008, at 4:30 am and a Butterfly is sitting on the railing of the deck outside the house. Actually, the railing is snow-covered and the Butterfly is sitting on the snow. The air is still, the sky is crystal-clear, there is significant radiative cooling underway and the temperature is dropping like a rock; it’s well below zero in both C and F. The Butterfly uses one wing to stifle a yawn and that wing moves slowly toward its mouth and then back to its resting place; the Butterfly needs the cover for warmth.

Here’s the question. What effect will that flap of the Butterfly’s wing have on the potential for a hurricane to form in the Gulf of Mexico in July 2008.

Some of the technical issues behind this question are the subjects of this post and possibly one or two others in future.

Butterfly Flap Dialogues … all over again
This post, a tribute for Ed Lorenz on RealClimate, on April 23, started the Butterfly Flap Dialogues all over again. Before you could say ‘chaotic response of complex non-linear dynamical systems’ many posts and comments on many Blogs had, in effect, created a chaotic scene. Threads and comments appeared by Lucia, Pielke Sr., and others.

In this post Professor Pielke collected several of the comments and responses, mainly by Ray Pierrehumbert and Gavin Schmidt, on the RealClimate thread. On April 29 Professor Pielke posted a discussion of the conclusions contained in The work by Lorenz. And on May 28, Henk Tennekes posted the last Climate Science Weblog about Butterfly Flaps. Generally, Professor Pielke and Dr. Tennekes disagree with Ray Pierrehumbert and Gavin Schmidt about the consequent physical phenomena and processes set in motion by the flaps. There have also been several posts in the past at Climate Science, one of which in May, 2007 addresses directly the issues as presented in the IPCC WG1 reports.

I chipped in on RealClimate, as I have done previously there and at Climate Science and you might notice that I have a few posts on the subject here on this Web site. I also chipped in at Lucia’s on this subject and the material I posted at Lucia’s forms the background for this post.

I’m not going to review all, or even a major part, of the information that appeared, there’s way too much. I want to focus on what I think are the not-so-correct aspects of the discussions of chaotic response that appeared as the several threads ran their course on a trajectory that led to attainment of null equilibrium states.

The physical picture of Lorenz-like equation systems will also be subject of parts of some of my comments here.

Background
Previously, and I’ll track it down if anyone wants to see the exact information, it has been said that the ‘phenomenology’ represented by the chaotic response of Lorenz-like equation systems has been adapted by the Climate Change Community, and I take it by the Numerical Weather Prediction Community, too. This rationalization has been necessary because of the lack of demonstrations based on firm theoretical foundations for the continuous equation systems used in GCMs and NWP.

There have been several posts about chaos, chaotic response, and the GCMs at RealClimate, for over three years now. One of the earlier posts was by James Annan and William Connolley on Nov 4, 2005. And they cited this post as providing additional information including a demonstration calculation. The demonstration calculation shows the effects of changing a single value of a single dependent variable (the pressure) by a small amount in a single computational cell. However, my conclusion based on the discussions about the example calculations is that the calculation introduced more questions than it answered.

Annan and Connolley concluded:

“Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small pertubation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat overstated. Chaos is defined with respect to infinitesimal perturbations and infinite integration times, but our uncertainties in the current atmospheric state are far too large to be treated as infinitesimal, and furthermore, all of our models have errors which mean that they will inevitably fail to track reality within a few days irrespective of how well they are initialised. Nevertheless, chaos theory continues to play a major role in the research and development of ensemble weather prediction methods.”

It seems to me that this conclusion is not in agreement with that of GISS/NASA’s Schmidt and also doesn’t seem to be in agreement with Ray Pierrehumbert comments. This will be the subject of another post here, Real Soon Now.

This comment provides an opportunity to begin discussions of the physical response of a fluid to Butterfly flaps. Specifically, the part that says,” … since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about.”

Energy is Important (but almost always not mentioned)
The absolutely critical part of the physical process of a small perturbation to grow to a significant size has yet to be mentioned in any of the many discussions that I have seen on the issue. My main point is that the perturbation cannot grow if there is no energy addition to the fluid region in which the perturbation is located.

In order to simply maintain the motions initially setup by the Butterfly flap, energy must be added into the fluid region. To increase the energy level of even the initial fluid region requires additional energy above that needed to maintain the initial motion. To increase the amount of the fluid region affected by the perturbation and the energy content of the increasingly larger fluid region requires even additional energy addition into the region.

If energy is not added into the fluid region occupied by the perturbed fluid, the motions will cease. There is no way for any perturbation to grow to any size no matter how long you might have to wait and watch.

All real fluid motions of all real fluids involve processes that dissipate the motions. In the absence of energy addition into the fluid regions undergoing motions, all motions will cease, every one of them. There are no exceptions to this process (unless of course we are dealing with a fluid having negative viscosity). A null equilibrium state will always be reached.

Here Comes the Turblence
That should be all that is necessary to say about this. However, the case of turbulent flows, and the growth of perturbations in these flows, is usually introduced into the discussions of the Butterfly flap. So, let’s consider the case of steady state turbulent flow in a straight piece of pipe that has constant flow area. In order to maintain such a flow there is always a source of power that adds energy into the fluid. The most straightforward example is the case of an experiment set up in a laboratory. There will be somewhere in the equipment that is used for the experiment a pump. The pump is the source of power addition into the fluid. A pressure drop along the flow direction, for example means that power is being added to the fluid. A specified flow rate at the entrance to the channel also means that power is added into the fluid. If there are no energy losses from the fluid anywhere in the flow system, the fluid temperature will increase. The temperature increase is the physical realization of the dissipation process occurring within the fluid. If the pump is switched off, the fluid motion will decay in time and will not longer flow in the pipe.

When the conditions are right, perturbations introduced into the fluid can grow and the flow will be turbulent. However, the energy required for the perturbations to grow is taken from the energy of the mean flow and the energy to the mean flow has been provided by the pump.

To go one step farther, recall the classical linear analyses that are used to estimate the conditions under which a laminar flow might become turbulent. The results of those analyses will show that if there is no mean motion there can be no transition to turbulence. The results will also show that if there are no gradients in the velocity distribution across the flow channel there can be no transition to turbulence; a flow having constant velocity across the flow channel cannot become turbulent. That is because it is the velocity gradients in the mean flow that provide the energy to the perturbed fluid regions through the actions of the shear stresses. And the mean flow is possible only if power is added to the fluid. [I’m doing this transition summary from memory and very likely should consult Schlighting.]

Well, as far as I think this is all very straightforward, and maybe I’m wrong, the Butterfly Flap Dialogues should all be short and sweet.

But They Aren’t
Next here is how I think the classic paper by Lorenz in 1963 enters the discussions. The original 1963 Lorenz system of three nonlinear ODEs do not describe any real fluid flow; they are not even a zeroth-order approximation to any real fluid flow. But the system does contain one extremely important aspect of fluid flows that relate to the Butterfly flap dialogues. All equation systems that correctly exhibit chaotic response have one thing in common. The systems will contain terms that represent losses plus terms that represent energy additions into the system. These are additionally in delicate balance to the extent that the calculated response neither grows without bound nor decays to an equilibrium state, and the trajectories of the dependent variables show aperiodic motions. Without the terms that represent energy additions into dissipative systems, the trajectories cannot be chaotic.

So, let’s review the contents of the original Lorenz system of 1963. There are three parameters in the system, usually denoted the Prandtl number, the Rayleigh number, and a geometric parameter. The first two receive significantly more investigations that the last one. There are only certain ranges of these parameters for which chaotic response is calculated. For other ranges, periodic, non-chaotic, response is obtained. In a model of real fluid motions, the terms in the model equations that correspond to those having these parameters will very likely be such that the numerical values of the parameters will change during the course of a calculation.

The effects of small changes in the initial conditions (ICs) for the ODEs are the focus in investigations of the calculations using the Lorenz-like equation systems. Much like the example calculation presented by Annann and Connolley and mentioned above. For the ranges of the parameters for which chaotic response is available, these calculations show that the calculated numbers rapidly diverge between the two calculations. But the rapid growth in no way is unbounded. The ultimate extent of the range of the dependent variables is set by the properties of the ODEs. So while there is initially rapid growth and divergence, ultimately the ranges of the dependent variables will always be the same. This characteristic of sensitive dependence on ICs is taken to be a necessary condition for chaotic response. It is not, however, a clear-cut necessity, because other aspects can falsely indicate that an equation system has this property.

By the same token, small changes in the numerical values of the parameters can show the same response. And if the values of the coefficients for the terms that represent friction in the equations are changed, the system can show trajectories that lead to equilibrium states.

The Analogy
Somehow, the flap of the Butterfly wings has become to be taken to be analogous to the small perturbations in the ICs for Lorenz-like equation systems. And the rapid divergence between the two calculations has become to be taken to be the growth of the perturbations introduced by the flap. Why the discussions have focused on only the ICs, when small changes in the parameters produce the same behavior is not clear to me.

Let’s accept that for the moment, but at the same time let’s look into the parts of the Lorenz-like systems that do represent some aspects of real fluid flows. The original basis of the Lorenz equation system was the model and solution methods setup by Saltzman. Saltzman envisioned two horizontal heated parallel plates with fluid between them. The bottom plate was maintained at a temperature higher than the top plate. This arrangement is inherently unstable as density gradients counter to proper physical gradients set by gravity can be obtained. Saltzman’s original calculations presented in the paper do not exhibit chaotic response for the range of the parameters and the expansions that he used. Lorenz searched for and found a modified subsystem of Saltzman’s equations for which aperiodic response was calculated and later observed the sensitive dependence on ICs.

The Bottom Line
Here’s the part where the hypothesized analogy between an arbitrary Butterfly flap and the Lorenz system breaks down. Both Saltzman’s equations and those of Lorenz contain terms that represent addition of energy into the fluid between the plates. If one attempts to insist that the perturbations of an arbitrary Butterfly flap are analogous to the changes in the ICs for the Lorenz system, this energy addition must be a part of the analogy. The most important part, in fact.

The state of the material initially distributed by a Butterfly flap cannot be determined in the absence of information about the energy flows into the distributed material.

Here are two simple tests. For the first, set the numerical value of the Rayleigh number in the Lorenz system to zero in the continuous equations and take a look at what you get. For the second, do the same thing at an arbitrary time during the course of the numerical solution of the Lorenz system, after starting the calculation with a non-zero value.

I haven’t actually done either of these, so they are predictions on my part.

Maybe I have not correctly understood what I have read about Butterfly flaps and subsequent response of the perturbations.

Summary
The characterization of the evolution of the fluid motions induced by a Butterfly wing flap presented by Annan and Connolley to be:

“Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small pertubation to grow to a significant size, …”

is (1) an incomplete statement of the problem, or (2) incorrect. In particular, the focus on, “ … since it will take far too long for such a small pertubation to grow to a significant size, … “

far too long in this characterization implies that there are no constraints to prevent any and all arbitrary flaps by Butterflys to grow to a significant size. This is wrong.

Appendix
I think that my understanding of the prevailing view of the growth of the perturbations by a Butterfly flap is correct. This is based on the many discussions that have appeared around Blog space. This link given at the top of this post has comments on the subject and replies by the GISS/NASA RealClimate principles and others. Here are two such examples.

A comment by Stormy here:
got this response by raypierre:

[ Response: None of this is right. The influence of the butterfly on the surrounding air propagates to larger scales and eventually affects the entire atmosphere. The thought experiment is starting with two identical atmospheres, which differ only by the flap of the butterfly wing. The claim is that even the large scale winds (e.g. position of a tornado or hurricane, or even onset of an El Nino) will diverge between the two realizations, given sufficient time. Regarding point (2), the general understanding of chaos is that you don’t need special conditions in the atmosphere for a “cascade” to occur leading to sensitive dependence on initial conditions. The sensitivity is generic to any sufficiently complicated nonlinear system, of which the atmosphere is almost certainly an example (though not yet provably so, in the rigorous mathematical sense.) {remainder snipped} ]

The comment by Professor Piekle Sr. here:
got this response by raypierre:

[ 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 ]

Corrections for all incorrectos are appreciated.

July 1, 2008 - Posted by Dan Hughes | Chaos and Lorenz | Chaos, Chaos and Lorenz, Lorenz, Lorenz and chaos