Tree representations of streamline topologies of structurally stable 2D incompressible flows

摘要

A flow of 2D incompressible and inviscid fluid is an example of a Hamiltonian vector field, where its Hamiltonian corresponds to the stream function whose level curves are called streamlines. A 2D Hamiltonian vector field is said to be structurally stable when the topological structure of streamlines is unchanged under any small perturbations of the vector field. In the present paper we show that the streamline topology of every structurally stable Hamiltonian vector field is in one-to-one correspondence with a labelled and directed plane tree and its associated symbolic expression called a regular expression. Consequently, we can characterize all streamline topologies with their corresponding plane trees and regular expressions uniquely. The present theory of tree representations is combinatorial; it brings us a new compression algorithm converting a large amount of streamline plots obtained by laboratory experiments and numerical simulations into a small set of simple symbolic data of regular expressions, which is amenable to a big data analysis for streamline patterns. Conversion to tree structures and their associated regular expressions is easily performed and it is flexibly applicable not only to incompressible flows but also to any physical phenomena described by Hamiltonian vector fields. We also demonstrate how the tree representation is applied to describe variations of streamline topologies for incompressible flows.