"Topological Defects, Coherent Structures and Turbulence in Terms of Cartan's
Theory of Differential Topology" in "Developments in Theoretical and Applied
Mechanics" Proceedings of the SECTAM XVI conference, B. N. Antar, R. Engels,
A.A. Prinaris and T. H. Moulden, Editors, The University of Tennessee Space
Institute, Tullahoma, TN 37388 USA. (1992) p. III.IV.2
Abstract: Cartan's method of exterior differential systems can be used to classify topological differences between potential or laminar flows and chaotic or turbulent flows. Relative to the induced Cartan topology, it is argued that a chaotic or turbulent evolutionary process is to be associated with a disconnected topology of Pfaff dimension 3 or 4, while a potential or laminar flow is to be associated with a connected topology of Pfaff dimension 1 or 2. Coherent structures are signatures of the disconnected topology, and topological defects can be associated with the discontinuities separating the disconnected domains. Use is made of Cartan's magic equation to determine whether or not a process is thermodynamically irreversible. It is demonstrated that the the Navier-Stokes equations can contain solutions which are thermodynamically irreversible, and that topological defect structures corresponding to the non-integrability of the vorticity field will exist in turbulent flows.