> A fun alternative would be to directly study the 2-point function, see how it decays with distance,

This has been done in

* Y. Berezin, A. Gozolchiani, O. Guez & S. Havlin, [Stability of Climate Networks with Time](http://www.nature.com/srep/2012/120918/srep00666/full/srep00666.html), 2012, Scientific Reports.

I mentioned this in the [climate network](http://forum.azimuthproject.org/discussion/1361/climate-networks/?Focus=10790#Comment_10790) thread, which I think is a better palce to take this further. But briefly, it seems like the weather always follows a power law.