Curvature and torsion of material lines in chaotic flows

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 or material coordinate along the line and their near singular behavior that develops as the deformation proceeds. Recent work [Thiffeault 2004. Physica D 198, 169-181] has established an interesting connection between local line stretch and curvature for regions of high curvature. The present effort confirms this connection and develops a similar relation between torsion and curvature for regions of high absolute torsion. In addition it is shown that certain properties of the strain tensor play a key role in the location and structure of the near singular behavior in curvature.