Models, Methods, Software

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?

Could it be that the fundamental continuum mechanics equations for fluid flow, usually taken to be the Navier-Stokes equations, are missing something when physical interfaces are present in the flow field? To be specific, the Euler equations, I think, are the hyperbolic case of interest.

Let’s attempt to count the number of characteristics that are pointing to a physical interface between two constituents or regions. We can do this even when the constituents are red and blue water, for example. Then we’ll have to find enough equations for the unknowns we count. That’ll be the fun part.

Who wants to start?

Let me know if you find incorrectos in the above.

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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The numerical solution methods that will be used to check convergence are given in a file that I uploaded.

Let me know if you see any typos or if you want to see some results for a specific equation system.

I’m thinking that Part 1d will be some numerical results.

The equation systems that will be used to check convergence are given in a file that I uploaded. I had tons o’ links and cross references and other good stuff but nothing worked out. Maybe later.

Let me know if you see any typos or if you want to see some results for a specific equation system.

I’m thinking that Part 1c will be the numerical methods.

UPDATE Nov 19, 2007: I have replaced the original uploaded file with a version that has some identification for me in it.

I have way too much material for a single post. I have spent days trying to force a good fit for all the material into a single document. I have put that aside for a while. So these discussions will be broken into several parts. At some future time I might try to tie all the pieces together by use of HTML/PDF.

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When the basis of the mathematical models used in AOLGCM models/codes are discussed it is almost always stated that the ‘fundamental laws of conservation and mass and energy’ are at the foundations of the models. This is an incomplete, and somewhat incorrect, statement on several levels.

Conservation of mass and energy is the focus of the following discussion.

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