New 'Superdiffusion' Proof Probes the Mysterious Math of Turbulence

Summary

They started by imagining a very fine grid superimposed on their fluid. They then computed how long particles spent in each square of the grid, on average. In some squares, the fluid acted like a rushing river: Particles tended to sweep straight across the square, spending only a brief period of time there. In other squares, small eddies might push particles around, slowing them down. The problem was that the numbers the mathematicians calculated might vastly differ from square to square — revealing precisely the kind of small-scale disorder that usually prevented mathematicians from using homogenization. Armstrong, Bou-Rabee and Kuusi needed to find a way around that. Ordering Disorder The mathematicians hoped to show that at slightly larger scales than the one their grid had captured, the fluid’s behavior would be a bit less noisy and disordered. If they could do that, they’d be able to use typical homogenization techniques to understand what was happening at the largest scale. But other mathematicians thought that even if they succeeded in analyzing those intermediate small scales, the fluid would only look noisier. Before things got smoother, eddies would first merge and interact in even more complicated ways. Trying to show otherwise would be a fool’s errand. Ahmed Bou-Rabee uses probabilistic methods to study partial differential equations, including those that govern the behavior of fluids. Courtesy of Ahmed Bou-Rabee The team decided to try anyway. They started by drawing a slightly coarser grid, in which each square encompassed several squares from the original one. Smaller eddies that had lived in separate squares of the original grid might now get grouped together, changing the average amount of time a particle spent in the new square. Or more complicated behaviors might emerge. The team once again computed how long particles stayed in each square and how much the numbers associated with adjacent squares might differ. This took painstaking effort: They ha...