Topological fluid dynamics for fluid dynamicists

These notes are preliminary and informal. My aim is to provide an introduction to some of the basic ideas that have emerged in recent decades in the fluids literature that have a fundamentally topological character. That is, rather than focusing on details such as velocity profile or pressure field, this material deals with persistent invariant properties of a flow field, features which often have a geometric interpretation. A basic reference is the book of Arnold and Khesin [1]. Other references will be given below. My interpretation of TFD will omit many technical details, but will I hope serve its purpose of translating some of the important ideas into the language of classical fluid mechanics. Topological ideas arise very naturally in fluid dynamics through the geometry of the vorticity field, and early work by Lord Kelvin and others explored knottedness of vortex tubes, for example. However the bringing together of inherently topological ideas from fluid dynamics into a coherent theory has been a fairly recent endeavor, associated with the work of Moffatt among fluid dynamicists, and Arnold and Freedman among mathematicians. An early example of topological thinking in fluid dynamics is the idea of a simply-connected domain and the role of multiply-connected domains in the analysis of lift by an airfoil in two-dimensional flow. A simple material curve which encircles a “hole” in the domain is fundamentally distinct from one which does not, and this property is an invariant of the flow field. There is no reference here to dynamics, so this is the essence of the topological viewpoint. Similarly, by calculating the circulation on such a curve the lift of a foil may be identified with a “topological invariant”. In three dimensions, the knottedness of material curves provide immediate examples of topological invariants. Perhaps the simplest example of such invariants can be seen in the advection of a scalar field c(x,y,t): ∂c