Visualization of Topology of Flow

― COT Representation of Surface Flow ―

Abstract

  It is known that "generic" surface flows are Morse-Smale flows, which are generalizations of gradient flows, and that incompressible surface flows are not Morse-Smale. When we analyze fluid phenomena on 3 dimensional spaces, we use flows on slices which can be obtained by projections on the slices. In this case, such flows on slices need not be neither Morse-Smale nor incompressible in general, and so we needed to extend the existing theories to describe the topologies of "generic" surface flows. Therefore we introduced a topological invariant and its representation, called COT representation, on a large class of surface flows, which contains all Morse-Smale flows and generic incompressible spherical flows. In this issue, we explain the representations of flows which is easy to use even for non-experts and is suitable to be implemented by computers and to describe not only the topologies of such flows and also the transitions among them. Moreover, the representations can be applied by phenomena which can be regarded as flows.