Models, Methods, Software

The journal of Computational Physics has a special issue that might be of interest to many here, Predicting weather, climate and extreme events.

This is a review article.

And Elsevier, aka Big Science Publishing, has kindly provided links to 8244 related articles.

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.

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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.

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Executive Summary

“When you’re running the roads, you can’t watch for The Law.” me

We had a most excellent little putt-putt down into the kudzu and grits belts. Racked up 2200 miles on the way down. Not bad for a trip that is nominally 1100 miles.

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I’m getting ready to take off on the moto for a couple of months. Assuming that the tons o’ snow here melt real soon. Back late May to early June.

Introduction
I started working on my own toy 0-D and 1-D models of the combined radiation-convection-conduction heat transfer problem aspect of energy balance approaches. I got to the point of assigning symbols to physical quantities and ran into a problem with some basic concepts. Details are discussed in the following paragraphs.

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I have linked to this post by William Connolley in my Update to this post. I have decided to post it here for several reasons.

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Using the NASA/GISS ModelE code browser I ran across the MODULE CONSTANT in which several constants are setup as parameters.

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When the flows mentioned in the title are approached by use of model conservation/balance equations for the individual constituents, or separate fluid regions, in the flow field, many (all?) are ill-posed as hyperbolic mathematical problems. We can get into issues associated with parabolic and elliptic systems, but let’s try to stick to the hyperbolic case to start off. In fact, one method to regularize (?) the hyperbolic case is to convert it to the parabolic case by use of a vanishingly small viscosity-like parameter. And by all means let’s avoid discussions of the effects of what discrete approximations to the continuous equations do to the whole messy situation.

Why is that?

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Introduction
The calculations preformed with the equation systems are summarized in the following discussions. The focus had been on testing for convergence of the numerical solution methods to solutions of the continuous equations. By convergence I mean that as the size of the discrete increment for the independent variable is reduced the calculated values for all dependent variables approach limiting constant values for all values of the independent variable.

None of the systems that are said to exhibit chaotic response have shown convergence. One of those, the Terman system, exhibits periodic response, not chaotic response. The Saltzman system was never intended to be an example for chaotic response.

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