Vorticity, Vortices, and the Torsion Current.

In the application of vector analysis to hydrodyamics it is recognized that on a simply connected domain that the velocity flow field consists of two parts, a pure gradient component (with zero curl) and a pure curl component (with zero div). If the domain is multiply connected then harmonic components (with both zero div and zero curl) must be added to the field definition. Given an arbitrary vector field, V, the vorticity is defined as curl V. Vorticity is associated with rigid body rotational motion of a parcel of fluid about some fixed point. The orientation of the parcel changes during the rotation.

However, there is another type of rotational motion possible for a parcel of fluid, a rotational motion for which the orientation of the parcel does not change during the parcel's rotation about the fixed point. This type of rotational motion has zero curl, but finite circulation integral, and is associated with the harmonic components of the vector field. Both species of rotational motion have a visual signal associated with rotation, and both species of rotation have been called "vortices" even though there properties are very different. Attempts have been made to associate either or both species of rotation with what are currently called coherent (compact) structures in fluid dynamics. However, hydrodynamicists have begun to face up to the topological differences between the two species of rotational motion, and are currently searching for a precise definition of What is a Vortex? Topologically, the concept of circulation is a 1-dimensional thing, while vorticity is a 2 dimensional thing. Both concepts are global in that they are extendable over the whole domain, and do not emulate the idea of an interior coherent structure in a fluid domain. Only in three dimensions are the two species of vectors dual to one another and therefore have the same number of components (3).

When space time is considered, the velocity field has 4 components, but the vorticity field has 6. The current dual to the velocity field has 4 components. In Cartan's language, the velocity field is a 1-form, A, the vorticity field is a 2-form, dA, and the dual current is a 3-form. An important three form can be constructed as A^dA, which has 4 components which define the Torsion Current. The Torsion Current contains features of both the 1 and 2 dimensional rotations. Remarkably, when the Torsion Current does not vanish the associated Cartan Topology is a disconnected topology, a concept that will admit an interior coherent structure not globally extendable over the whole domain. The suggestion made herein is that this vector field of Torsion Current is the proper definition of what is visually and intuitively called a Vortex.