Models Methods Software

GCMs are Consistent With Chaotic Response …

of equation systems that do not possess chaotic response.

Executive Summary
The original PDEs that describe the Rayleigh-Benard convection problem do not posses chaotic behavior. The chaotic response observed with Lorenz-like low-order models (LOM) obtained via mode expansions disappears whenever sufficient resolution is used in the numerical solution methods applied to the original PDEs.

The low order model of the Lorenz equations omits the terms that are responsible for interaction between smaller scales and the large scales. The very interactions that form the basis for invoking the turbulence analogy.

GCMs are consistent with the chaotic response obtained from incorrect low-order models (LOM) expansions of PDEs.

GCMs are consistent with the chaotic response obtained from incorrect solutions to ODEs and PDEs.

GCMs are consistent with the chaotic response observed whenever insufficient resolution is used with numerical solution methods.

As the most recent conservations conversations in The Butterfly Flap Dialogues unfolded, some of which is summarized here, GISS/NASA’s Gavin Schmidt seems to have provided a working definition of chaotic systems within the framework of GCMs. I think the definition can be summarized in the following manner; the numbers calculated using GCMs are Consistent With the chaotic response of complex non-linear dynamical systems.

In my opinion, Consistent With is not even a necessary condition relative to a correct determination of the chaotic response of equation systems, and certainly is not even close relative to being a sufficient condition. For all of engineering, Consistent With never appears in Validation studies.

I will focus on some of the technical issues that make the GISS/NASA characterization of chaotic response of complex dynamical systems somewhat less than satisfactory from both physical and mathematical aspects. One of the latter was mentioned in passing in a comment on GISS/NASA’s RealClimate Blog; I’ll track down the specific comment Real Soon now. I should have followed-up on that comment at the time, but I did not. That comment indicated that as the number of modes was changed in the expansions for the solutions of the PDEs, the chaotic behavior would sometimes not be present and sometimes would be present. That is, the chaos was dependent on the properties of the numerical solution methods, and not a function of the original continuous PDEs.

I hope to cover (1) specific aspects of the continuous Lorenz-like equation systems, (2) the numerical solution of these, (3) numerical solutions of PDEs and ODEs that lead to spurious chaos, (4) the turbulence analogue all over again.

Introduction
In response to my comment on RealClimate GISS/NASA’s Gavin Schmidt said:

[Response: All GCMs as written can be considered to be deterministic dynamical systems. All of them display extreme sensitivity to initial conditions. All of them have positive Lyapunov exponents (though I agree it would be interesting to do a formal comparison across models of what those exponents are). All are therefore chaotic. Your restriction of the term chaotic to only continuous PDEs where it can be demonstrated analytically, is way too restrictive and excludes all natural systems – thus it is not particularly useful. Since the main consequence of the empirical determination that the models are chaotic is that we need to use ensembles, I don’t see how any of your points make any practical difference. The GCMs can be thought of as the sum total of their underlying equations, their discretisation and the libraries used (and that will always be the case). It is certainly conceivable, nay obvious, that these dynamical systems are not exactly the same as the one in the real world – which is why we spend so much time on evaluating their responses and comparing that to the real world. But as a practical matter, the distinction (since it can never be eliminated) doesn’t play much of a role. – gavin]

A Short Diversion
Before getting to the main issues of this post, note this statement:

The GCMs can be thought of as the sum total of their underlying equations, their discretisation and the libraries used (and that will always be the case).

The bold is mine. I assume that we can have great confidence and extrapolate Gavin’s statement to include the languages and compilers and scripts and users and all other aspects of modeling that do not include the actual physical processes of interest.

I’ll get to the issues mentioned by Gavin Schmidt, but first I’ll repeat the comment that I made at GISS/NASA’s RealClimate blog. I have modified the comment that I made at RC to remove references to other comments in the RC thread and to provide additional information about some aspects of the technical problems that I mentioned there. The modified comment is given in the following paragraphs. Generally some background on the origin of the original Lorenz equation system of 1963 is discussed along with material about the numerical solution of the equations and some properties of chaotic response of complex dynamical systems.

The Original Lorenz System of 1963
The original Saltzman equation system of 1962 was simplified and modified by Lorenz to obtain his equation system. For the range of parameters and initial conditions investigated in the paper, the original Saltzman 7-equation system does not produce chaotic response. Saltzman used more modes in the expansions, and then reduced these to 7 for calculations. The number of modes retained in the expansions plays an important role in both the calculated response of the equation systems and the correct representation of the physical flow. Additionally, relative to The Butterfly Effect and the usual analogy to turbulence, the number of modes is critical to the correctness of the assumed connection. The analogy to turbulence is another untested hypothesis that I think is incorrect.

The solutions of the Saltzman system given in his paper showed that three of the dependent variables rapidly approached non-zero equilibrium states, while the other four approached values of zero and are thus ineffective for the flow. The specific dependent variables that follow these trajectories depends on the numerical values of the parameters in the equations. For the range of initial conditions (ICs) and parameter values that Saltzman used in the paper, chaotic response was not observed. I have done some numerical work with the Saltzman system as reported here.

The Lorenz system of 1963 is a modified and simplified version of the Saltzman system. Lorenz found a subset of these 7 equations and a range of parameters for which he could obtain aperiodic response from the system. As is well known Lorenz made his discovery about the sensitivity to small changes in ICs when he re-started a calculation.

The Saltzman system is a model of the motion for a fluid contained between horizontal planes; the Rayleigh-Benard problem is a convenient designation. The fluid motions for the original Saltzman and Lorenz problem are driven by a temperature difference between the planes bounding the flow. This temperature difference provides a source for constant energy addition into the fluid contained between the plates. The temperature difference between the planes is required to be maintained in order for the flow to continue. I have written about this in the post below following this one. The Boussinesq approximation is used in the momentum-balance model for the vertical direction. Both the Saltzman and Lorenz systems are low order models (LOM) by way of mode expansions of what were originally partial differential equations. More about this later.

The Lorenz system is not a model of any known fluid flows. Lorenz has explicitly addressed this matter and it frequently appears in the literature. The system might be valid for estimating the onset of motion under the effects of an adverse density gradient. But following onset of the motion, all bets are off. The important aspects of the Lorenz system are related to its chaotic response. So let’s summarize the chaotic-response aspects that always appear in publications about the system.

(1) Sensitivity to small changes in initial conditions (ICs).

Well that’s about it. Actually much more is usually said in papers published in journals devoted to the subject area. But in application-type discussion sensitivity to ICs is about as far as we get.

Let’s list the chaotic-response aspects that do not appear nearly as frequently as the one above.

(1) Almost all calculations investigating the chaotic response of the Lorenz system are performed for values of the Rayleigh number parameter that are far beyond those for which the equations are valid. This characteristic is a result of the severe truncation of the mode expansions. And while this is related to the physical situation, I think we need to focus on the physical picture as well as the mathematical picture.

(2) For some values of the parameters in the system chaotic response is not observed but instead equilibrium states are attained. This result obtains even as values of the primary parameter, the Rayleigh number, are increased far beyond the critical value.

(3) Small changes in the parameters of the systems show the same behavior in the response observed by small changes in the ICs. In the Saltzman and Lorenz formulations, the numerical values of the parameters are fixed constants. In practical applications, the values will change as the calculation proceeds. Note, too, that it is possible that the parameter values can change into the regions of parameter space for which chaotic response is not obtained.

Neither sensitivity to initial conditions nor non-linearity nor complexity, either separately or all together, provide necessary and sufficient conditions for a prior determination of chaotic response. Analyses of the complete system of continuous equations, plus careful consideration of numerical solution aspects, combined with analyses of calculated results are all necessary for determination of chaotic response. A model of a complex physical system, comprised of a system of nonlinear equations, the numerical solutions of which show sensitivity to initial conditions, and the calculated output from which ‘looks random’ does not even mean that the calculation exhibits chaotic response. ‘Looks random’ in itself is not a description that is consistent with chaotic response. And the properties and characteristics of the calculated results cannot ever be attributed to be properties and characteristics of the modeled physical system.

Chaotic Response due to Numerical Solution Methods
Periodic motions, not chaotic motions, have been observed for the Lorenz system for some values of the parameters appearing in the system; see Tritton, Physical Fluid Dynamics, for example. It is equally well-known that chaotic response can be obtained due solely to inappropriate numerical methods applied to PDEs and ODEs that cannot produce chaotic response in their solutions. Citations to several examples have been given here; see the articles by Yee and her colleagues and those by Cloutman. An online example calculation is available here. This example shows that inappropriate numerical solution methods can be the source of spurious chaotic response. The same results discussed by Yee and Cloutman, among many others.

When numerical solution methods are important aspects of any analysis, the effects of these methods on the calculated numbers require deep investigations so as to eliminate spurious effects that are solely products of the numerical methods. This process has yet to be applied to the calculations with GCMs.

Changes in the parameters in the model system produce the same kinds of effects that are observed when the initial conditions are changed. There are some ranges of the parameters that do not show sensitivity to initial conditions nor chaotic response. The focus on sensitivity to initial conditions and chaotic response seems to always miss the fact that simply invoking ‘the Lorenz model’; is not sufficient to ensure that chaotic response is obtained.

Dynamical systems theory predicts that the differences between the dependent variables will grow exponentially for two different values of initial conditions, no matter how small the initial difference. Statements about the time required for a perturbation to be effective over various ranges of temporal and spatial extent in the physical world should be addressed from the viewpoint of physical phenomena and processes; not by what a few calculations using a model/code indicate. Again, aphysical properties of calculated fluid flows can frequently be traced to improper numerical solution methods and the implementation of these into computer software.

The demonstration linked to in the original post [at RC] for this thread led to the introduction of more questions than answers. The effects of the size of the discrete time interval was an issue. As were the effects of the numerical solution methods. A comment over there mentions that the calculated response might in fact be controlled by model equations and numerical solution method in contrast to chaotic response. Additional discussions of step-size effects have been presented here for both NWP and GCMs. The presence, or absence, of chaotic response in the model/code cannot be ascertained by use of a few calculations. The possibility of causality linked to the model equations and/or numerical solution method should be determined first. The mere possible existence of the potential for chaotic response is not sufficient for concluding chaotic response.

A demonstration by a calculation of an idealized model equation system never says anything about the actual response of the modeled world; especially when as in the case of GCMs the model equations are acknowledged to be approximations and simplifications of the complete fundamental equations. A calculation by a GCM demonstrates the properties of that GCM for that calculation and can provide no information relative to the chaotic response of the real world climate.

Relative to the original physical problem and mathematical modeling of the problem, this reference has presented many important conclusions. The authors investigated the properties and characteristics of the calculated numbers when the same LOM expansion approach is applied to the Rayleigh-Benard problem; the same problem investigated by Saltzman and the problem and solution method used by Lorenz. Both Saltzman and Lorenz used the expansion method to reduce the original PDEs to a system of ODEs and then applied numerical solution methods to these. Specifically, the calculations show that as the resolution used in the numerical solution methods is increased, the calculations do not indicate chaotic response. The authors conclude that chaotic response is not a part of the original physical problem.

The first paragraph of the Abstract reads as follows:

Abstract
The character of transition from laminar to chaotic Rayleigh–Bénard convection in a fluid layer bounded by free-slip walls is studied numerically in two and three space dimensions. While the behaviour of finite-mode, limited-spatial-resolution dynamical systems may indicate the existence of two-dimensional chaotic solutions, we find that, this chaos is a product of inadequate spatial resolution. It is shown that as the order of a finite-mode model increases from three (the Lorenz model) to the full Boussinesq system, the degree of chaos increases irregularly at first and then abruptly decreases; no strong chaos is observed with sufficiently high resolution.

The link above should lead to the complete abstract.

Generalized Lorenz Systems
There are many papers dealing with generalization of the Lorenz model equations, primarily by including more modes in the expansions. Three recent papers by Roy and Munsielak here, here, and here, provide a good review and summary of these investigations. The authors rightly conclude that some of the generalizations are not properly related to the original Lorenz model systems. As noted by Roy and Munsielak, there are two important characteristics of the continuous Lorenz equations are (1) they conserve energy in the limit of no viscosity (energy conserving in the dissipationless limit) and (2) the systems have solutions that are not unbounded. Some generalizations of the Lorenz model system that do not conform to these requirements show routes to chaos that are different from those for the Lorenz system.

Additional important aspects of LOMs via mode expansions have been discussed by Gluhovsky and Christopher and colleagues here, here, and here.

Conclusions
It is clear that the continuous equations, mode expansions, and other numerical solution methods used in GCMs are very likely not consistent with the original basis of the Lorenz-like equation systems. The hypothesis that those systems can correctly provide a template for the behavior of GCM calculations is not on any sound theoretical foundation. It is an untested hypothesis.

Accurate resolution of the response of the full equations that represent Rayleigh-Benard convection, by use of the mode expansion technic used by Saltzman and Lorenz, shows that this flow does not posses chaotic behavior. The hypothesis that Rayleigh-Benard convection can correctly provide a template for the behavior of GCM calculations is not on any sound theoretical foundation. It is an untested hypothesis.

Mapping the results of GCM calculations to a physical template, in contrast to the mapping to the numerical artifacts of Lorenz-like equation systems would greatly improve the basis for interpretation of the calculations.

The specific equation systems and solution methods used in GCMs require deep investigations into the properties of the continuous and discrete equations in order to avoid the pitfalls illustrated by the examples given in this post. Until this work is finished the hypothesis that GCM calculations are Consistent With spurious chaotic response due solely to numerical artifacts is as valid as any other hypothesis. Completion of the work will provide the data necessary to test this hypothesis.

July 5, 2008 - Posted by Dan Hughes | Uncategorized | Chaos, GCMs, ODEs, PDEs