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Weather & climate analysis and simulation involves a lot of complicated structures.  For example, if you spend some time observing water flowing in a typical stream, you’d see waves, vortices, turbulence, smooth flows and other identifiable structures in many sizes from many meters down to a few millimeters, and changes on time scales from months down to fractions of a second.  Weather is like that except harder to see and involving other complications like rainy or sunny skies.  One way to use mathematics is to analyze a whole flow F into its component structures Si that can be understood individually more easily than the whole: F = S1 + S2 + S3 + …

For example, a colleague runs his computer model to simulate a very turbulent flow, something like you’d find in any random imaginary box somewhere inside a violent storm.  He tries to understand his simulation using computer graphics, to visualize something he computed, such as the shape of a 3-dimensional surface in space on which the wind speed has some fixed value (just as you might visualize a hilly hiking region by the shape of 2-dimensional contours of some fixed height above sea level).  He sees some structures among all the turbulence, and I start imagining how to identify such structures mathematically.  The simplest example is when you see an edge of something, like your computer monitor right now.  How do you know it’s an edge?  It’s not only that the color, intensity etc. of what you’re looking at changes on either side of the edge, but also the scale of those changes compared to other changes in your view.  Your brain is so used to finding edges that you don’t even realize you’re doing it.  I try to teach computers how to do it.