Abstract:

The deformation of a material line as it evolves in a chaotic flow is considered. Of particular interest are the curvature and torsion as a function of arclength along the curve. These quantities are sufficient to define the intrinsic geometry of the line. Regions of high curvature, once they start to develop, are essentially permanent, and in fact have a universal structure. Regions of high torsion, on the other hand, are transitory and correspond to a near singularity in the coordinate system of the Frenet frame rather than an exotic shape of the curve.