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.
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.
Models Methods Software 