Kiehn, R. M., (1990a) "Topological Torsion, Pfaff Dimension and Coherent Structures";, in:
H. K. Moffatt and T. S. Tsinober eds, Topological Fluid Mechanics, (Cambridge University Press), 449-458 .

Abstract: A coherent structure is viewed as a deformable connected domain of velocity space with certain similar topological properties. The topology of interest is the topology induced by the constraint on the variety {x,y,z,t} such that the vector field of flow satisfies a kinematic system of ordinary differential equations, as well as a dynamical partial differential equation of evolution. The Pfaff dimension or class of a domain is a topological property whose evolution may be computed. Of the four Pfaff classes of coherent structures admitted over space time, potential flow and integrable vorticial flows make up the first two Pfaff classes. The third and fourth Pfaff classes lead to the notion of Topological Torsion and Topological Parity. A concept of Helicity current is introduced, a current whose non-zero divergence signals the production of 3-dimensional defects internal to the flow field. Invariant surfaces of separation associated with the Jacobian of the unit tangent field may be used to define topological domains. An example is given demonstrating that these domains, as deformable domains of similar topology in the flow velocity field, may be put into correspondence with coherent thermodynamic phases.

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