Topological Methods In Hydrodynamics

Abstract

A survey is presented of topological and differential geometric methods for describing the global behavior of fluids for large time intervals. Conservation laws for incompressible and barotropic fluid flows as well as superconductivity are reviewed in which diffeomorphism determines the laws. An overview is then given of the ergodic interpretation of hydrodynamical invariants with particular coverage given to the relationship between the helicity invariant and the average linking number of the trajectories of corresponding curl-vector fields. The geometry and curvatures of different diffeomorphism groups are elaborated upon with attention given to hydrodynamical instability and unreliable forecasting.