Friday, April 18, 2008

Fat tails and outliers: Extreme weather

It's often claimed, on little physical basis, that "global warming" with rising concentrations of infrared-active gases will lead to more extreme weather. We need to define more clearly what "extreme weather" is and apply some of principles learned from other open, "driven" systems, with non-conserved totals of energy, matter, and so on. The weather lives in Extremistan, at least to an extent.

Concentrate on just the atmosphere. The amount of dry air is essentially fixed, but not so the water in the air (both vapor and condensed), which is highly variable. The amount of heat (which is also rather variable) is less important than the heat flow. While the heat flow is split among radiation, evaporation, and turbulent convection, the total flow is relatively fixed. The division among the three mechanisms of flow is less stable.

Start with the total available. Instead of taking each "extreme" weather event one at a time, let's reverse the logic, with the same reasoning used by geophysicists when they consider earthquakes.* Imagine you're a storm god with a more or less fixed annual budget of heat and water. Taking the totals as the staring point, how do you divide them up every year? All at once? Blow your budget on a few big events? Or on many, many little events?** The "storm-generating system" is an open, not a closed, system; one with stuff flowing through, not isolated and in thermodynamic equilibrium. Extremistan statistics (fat tails) should be the default way to analyze it. It's natural then to assume that "storm budget" for a year will be dissipated by the cumulative effect of a range of events, some small, some medium, some extreme, all together making a distribution of event frequency versus event size.

Unfortunately, a wholly bogus case for "increasing extreme weather" has been built on using misguided Mediocristan (Gaussian) statistics to analyze extreme events. The fundamental problem is the same as in the other misapplications of the classical Central Limit Theorem to such events: the distribution moments are supposed to finite, but in fact, are not. So the sampling of the distribution by observations is misinterpreted to infer that the total of such events is increasing in time (non-stationary), when in fact the moments (like the variance) are simply diverging.

A subtle point: The number versus the energy release of extreme events. A "global warming" world is one where many differences in climate fade, leading to a fall in certain kinds of "weather events," like fronts and tropical storms. In such a world, it's reasonable to conjecture that both the number of extreme events and the cumulative energy dissipated by them go down.

But that reasoning alone is not precise enough to tell us how much each will go down. Periods of warming (like 1910-40 and late 70s to mid-90s) had fewer tropical storms. But a few of them were nonetheless memorable for their size; for example, the unnamed 1938 hurricane that hit Long Island and New England, or Hurricane Gilbert, which hit the Caribbean, Central America, and Mexico in 1988.

It could be that both the number and cumulative energy of extreme weather events goes down under "global warming," but that the energy per storm or event can go up. The number of events can go down by more than the amount of energy dissipated goes down. Thus the ratio (energy/event) can go up. A possibility to think about.
* This is the Gutenberg-Richter law, one of a large set of modified power-law distributions with "fat tails" (slowly falling probability for larger and larger events, instead of a sharp fall-off). The upper cut-off is not literally infinite, but very large. In practice, it's set by the total amount of energy available - in the earthquake case, the total amount stored as a potential energy in the Earth's crust under tension. Conceivably, all that stored energy could be released in one giant earthquake.

** Don't laugh. If the Greeks had known about Extremistan, they would have deified it. Actually, they did: Tyche, or Fortuna.

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