## Hurst Coefficients: A Rough Draft

I have a rough draft discussion about applications of the Hurst Coefficient methodology to (1) solutions of the Lorenz equations, (2) measured temperature data, and (3) results from calcuations with GCMs. I would like to get some peer-review.

**Update October 8, 2010**

I have uploaded two files to this new server as the others have been lost to the previous server. Application of the Hurst Coefficient method to the Lorenz 1963 system is here. And some of that plus application to some data is here.

As I’ve mentioned a couple of times, there are many broken links because of the loss of the previous server.

All comments, especially corrections to incorrectos, will be appreciated.

## Averaging Planet Earths or Averaging Planet Xs

**Update January 23, 2011**. I fixed the broken links.

There are several discussions floating around on the subject of comparing GCM-calculated numbers with experimental data. Climate Audit and Lucia both have several threads. There are too many threads to give links; let me know if you need a specific thread. One focus of these discussions is the ensemble-average approach that is considered to be necessary for the comparisons. The ensemble is made up of the results from the various different versions of GCMs that calculate the results. Not only are GCMs different from each other, but the suggested approach is to make perturbations in the initial conditions between the runs.

I have mentioned that the results do not reflect the effects of perturbations in only the initial conditions. Everything about the approach is different between the runs. The GCMs are based on different continuous-equation models, numerical solution methods, application procedures, run-time options, and users, and maybe other aspects. How can we be sure that the results are all appropriate for Earth?

## 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.

## Chaos and ODEs Part 1d: Calculations and Results

**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.

## Chaos and ODEs Part 1b: The Equation Systems

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.

## Chaos and ODEs Part 1a: The Literature Sources

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.

## Chaos and Butterflies yet again

The NWP and GCM communities cannot think that a Butterfly will have any influence whatsoever on any physical phenomena or processes of interest. Instead the *phenomenology* of The Butterfly Effect as exhibited by the numerical calculations of some systems of ordinary differential equations is *invoked by hypothesis* into NWP and GCM models/methods/codes. I think we need to limit discussions to the Lorenz-like systems of ODEs, as these seem to be the basis for invoking the phenomenology into the NWP and GCM communities. Otherwise we will get side-tracked into discussions of the “chaotic response of complex dynamical systems” in general.

## References for Chaos Part 0

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

## A Short Summary of Future Discussions

The chaotic phenomenology of small systems of non-linear ODEs is entirely numerical ODE chaos. And, the original Lorenz system of 1963 contains no physical phenomena or processes of interest in NWP and AOLGCM applications.

## An Important Peer-Reviewed Paper: Part 1

We’ll now look at some of the results presented in the paper.

**Introduction and Background**

The authors have introduced the subject of convergence of numerical methods into the field of chaotic dynamical systems. This field is very important in many areas of current intense study and investigation. Numerical models and solution methods exhibit chaotic dynamical-system characteristics in weather and climate modeling, direct numerical and large eddy simulations of turbulent flows, as well as the classical studies of chaotic systems through nonlinear ODEs as introduced by Lorenz and others. The author’s paper seems to be the first in the literature to present results of systematic investigations of convergence into this important field of research and applications.

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