More on ODEs, MMS and Ill-Posed IVPs
First for the nomenclature: ODEs means Ordinary Differential Equations, MMS means the Method of Manufactured Solutions, and IVPs means Initial Value Problems.
This previous post provided some information on these subjects. So far as I know, that post presented the first results for application of MMS to ill-posed IVPs. That post suggested that for the numerical solution methods used therein, the original Lorenz system of 1963 has yet to be correctly solved.
I have some additional results, a summary of which is:
I think the Lorenz system has not yet been accurately integrated by any numerical solution methods. Higher-order methods plus, at the same time, higher precision representation of numbers will give results that might appear to be solutions. But, calculations for sufficiently long time spans will show that errors always increase.
I’ve uploaded a file here.
Hard Concepts
Boy, it’s difficult to get my mind around many of the concepts discussed in the post Tracking down the uncertainties in weather and climate prediction.
Updated July 10, 2010.
I have looked around but have not been successful in finding additional material from either the meeting or the presentation. I suspect all information presented at the meeting will eventually show up on the CCSM Web site.
Here’s a part that I find to be very unsettling. Starting at the 15th paragraph in the post.
And now, we have another problem: climate change is reducing the suitability of observations from the recent past to validate the models, even for seasonal prediction:
Figure Uncertainty2. Climate Change shifts the climatology, so that models tuned to 20th century climate might no longer give good forecasts
Hence, a 40-year hindcast set might no longer be useful for validating future forecasts. As an example, the UK Met Office got into trouble for failing to predict the cold winter in the UK for 2009-2010. Re-analysis of the forecasts indicates why: Models that are calibrated on a 40-year hindcast gave only 20% probability of cold winter (and this was what was used for the seasonal forecast last year). However, models that are calibrated on just the past 20-years gave a 45% probability. Which indicates that the past 40 years might no longer be a good indicator of future seasonal weather. Climate change makes seasonal forecasting harder!
The conclusion, “Climate change makes seasonal forecasting harder!” is basically unsupported. There are a very large number of critically important aspects between ‘Analysis” and “Changed climatology” that are simply skipped over.
Firstly, the Analysis has been conduced with models, methods, computer code, associated application procedures, and users, any one of which separately, or in combinations with the others, could contribute to the differences between the 40-year and 20-year hindcasts. Secondly, within each of these aspects there are many individual parts and pieces that could cause the difference; taken together the sum is enormous. Thirdly, relative to the time-scales for climate change in the physical world 20-years seems to be kind of short and maybe even 40 years is, too. Fourthly, no evidence has been offered to show that climatology has in fact changed sufficiently to contribute to the difference.
The presentation seems to have leapt from (1) there are differences, to (2) the climatology has changed. I find this very unsettling. The phrase, Jumping to conclusions, seems to be applicable.
With the given information, I think about all we can say is the the models, methods, code, application procedures, and users did not successfully calculate the data.
I don’t see that any ‘tracking down’ was done.
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?
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.
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 1c: The Numerical Methods
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.
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.
Models Methods Software 