Models Methods Software

Dan Hughes

Dissipation of Fluid Motions into Thermal Energy

I have been looking for information on this subject without much success. I have contributed these notes to a thread at Climate Science (the site appears too be down at this time, I’ll add a link later) and at Climate Audit. Viscous dissipation has also been the subject of this post. To date I have gotten very little feedback of any kind. Maybe all this is incorrect, but no one has said that either. All pointers to literature relating to the following will be appreciated.

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June 3, 2007 Posted by | Uncategorized | | 5 Comments

References for Chaos Part 0

The literature references cited in Chaos Part 0 are listed in this post. Maybe this will become a Pages.

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April 28, 2007 Posted by | Chaos and Lorenz | , , , | Leave a comment

Chaos Part 0: ODE Chaotic Response is Numerical Noise

In this post we look at a every important issue relative to the foundations of chaotic response as inferred from numerical calculations with systems of ordinary differential equations (ODEs). The relationship of the contents of these notes to calculations with numerical weather prediction (NWP) and general climate models (GCMs) might be discussed in future posts.

Summary Statement
Converged numerical solutions of systems of ordinary differential equations that exhibit chaotic response have never been published. Converged numerical solutions have yet to be obtained for such equation systems. The preceding statements need to be qualified by limiting them to long ranges of values of the independent variable. And of course the statements apply only to the situations for which a given equation system exhibits chaotic response. Additionally, the statements are based on what I will denote as standard numerical solution methods. Standard numerical solution methods include just about all methods that are widely used. Solution methods other than standard, denoted as self-validating, will be specified later in these notes.

Given that chaotic response of complex dynamical systems at its foundation is based almost entirely on numerical solutions of the ODE systems, this situation is almost beyond comprehension. The recently-published paper reviewed here and here on this blog clearly shows the lack of convergence as the step size is refined.

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April 28, 2007 Posted by | Uncategorized | , , | 2 Comments

An Important Peer-Reviewed Paper: Part 0

This paper Time-step Sensitivity of Nonlinear Atmospheric Models: Numerical Convergence, Truncation Error Growth and Ensemble Design addresses some of the important issues for which this blog was established. Extensive discussions will follow as I get the results of my work documented. I have used the 3D Lorenz equations for most of my work, but have looked at other ODEs for which chaotic response has been demonstrated.

One conclusion that I am almost certain about at this time is that standard numerical solution methods applied to ODEs that exhibit chaotic responses have never been shown to converge. Additionally, it is very likely that convergence can not be demonstrated. The calculated numbers are very likely noise that does not satisfy the continuous equations. Some parts of this conclusion will cary over to numerical solution methods for PDEs. Some of the issues were mentioned in this post. The chaotic responses calculated by AOLGCM models/codes are in fact purely numerical artifacts from a combination of the numerical solution methods, lack of convergence of the calculations, and the algebraic parameterizations used in the models.

These issues will be addressed in subsequent posts here. First we take a preliminary look at some of the issues brought to light by the subject paper.

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March 9, 2007 Posted by | Uncategorized | , , , , , | Leave a comment