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Wednesday, March 10, 2010

A mathematical perspective on snowflakes and global warming

Surprisingly, I have never heard support in the Mathematics faculty lounge for the global warming prognosticators of doom. I admit to baiting my colleagues on occasion with the standard game - attributing assorted random, ridiculous phenomena to global warming. None take the bait. My colleagues have been inoculated against much of the global warming hype by their prior exposure to bad mathematical modelling.

In the '90s, many applied mathematics groups wanted to hire experts in scientific computing. Computing power had increased so greatly that approximate solutions to many previously intractable problems became computable. Our department invited numerous experts in scientific computation to interview for a faculty position. Many talks took the following form. The speaker would discuss an algorithm for modelling some phenomenon, such as the formation of snowflakes, where the macro level physics was well understood, but how the basic physical laws translated into the observed structure of snowflakes was unknown. The speaker would postulate various mathematical rules for interpolating between the known physical regime and the unknown regime. The origin of these rules was usually obscure or ad hoc, and (key point!) involved numerous free parameters. The speaker would then run his algorithm, showing us many beautiful computer generated pictures, with many choices of the free parameters. The initial pictures looked as much like the Pillsbury Dough Boy as a snowflake. Finally, we would see a series of pictures where the parameters had been tuned to values so that the pictures looked like actual snowflakes, not Rorschach ink blots. The speaker declared victory.

The majority of our faculty agreed that these exercises had no scientific merit. Given any flexible model with enough free parameters, you can tune the parameters to obtain any desired result. It has no predictive value and no explanatory value. We made no hires of these parameter tuners. Most of them ended up working on Wall Street.

Turn now to global warming. What are the issues in the debate?
  1. Does CO_2 cause warming?
  2. Has there been significantly more warming post increase in CO_2 production than before?
  3. If there has been recent warming, is it due to CO_2?
  4. Will CO_2 induced warming lead to harmful consequences?
I have not heard disagreement on the first point. CO_2 has a warming affect. The second point is a question of data, and is the primary focus of most of the Climate Gate scandal. The integrity of the data on whether any significant warming has actually been measured is now suspect. The favorite alternate hypothesis to CO_2 induced warming is that warming is more likely to be a function of variation in solar output. My favorite analogy is that of lighting a match in a room with a furnace. Yes, the match increases the warmth of the room, but it is not as important as the furnace turning on and off. If clean data is ever produced showing there has actually been recent unusual warming, it has no weight unless it is demonstrated that it is caused by CO_2.

Finally we turn to the fourth item, which is closer to my expertise. The actual effect of CO_2 on the vastly complicated climate currently can't be directly known. The system is too large and too many details of the interactions are currently unknown. Hence the global warming brigade produces computer models with numerous ad hoc mathematical rules and many free parameters. They then tune the parameters until they get a snowflake! No, until they obtain any result they desire, from a 20 foot increase in sea level to a prediction of the winner of the World Series. Currently it is unknown if simple models of simple fluid flow give correct long time solutions. There is a million dollar prize for solving this problem. There is no scientific reason to respect these CO_2 models treating a vastly more complex system. Of course, if you want to test a model, the first question is: does it accurately predict the future? (They are constructed so that they always correctly 'predict' the past.) The global warming models have all failed this test.

Until the many microlevel details on how CO_2 and the environment interact (such as the various feedback mechanisms), no computer model can carry much weight. Of course, if one is designed that makes fantastically accurate long term predictions, then we have reason to put aside our doubts. Until these problems are overcome, vote down carbon taxes. If you want to hedge your bets, invest in Canadian real estate.

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