Second derivative ridges are straight lines and the implications for computing Lagrangian Coherent Structures

Lagrangian Coherent Structures (LCS) have become a cornerstone of the analysis of unsteady fluid flow. Intuitively, LCS are material boundaries that locally maximize attraction, repulsion, or shearing. Based on a number of examples and numerical experiments it has been suggested that LCS are indicated as ridges of the Finite-Time Lyapunov Exponent (FTLE) field. This commonly accepted intuition has led to a school of thought that defines LCS as so called second derivative ridges in the FTLE field. This viewpoint has been supported by a proof that such ridges indeed admit (almost) no cross flow. Recently analytic counter examples have been discovered that demonstrate that FTLE ridges in general may produce both false positive and false negative LCS classifications. Furthermore, without additional restrictions, second derivative ridges in particular may admit a large amount of flux. Here we provide additional evidence that second derivative ridges are ill suited to define LCS (or ridges in general) by showing that for any smooth scalar field second derivative ridges are necessarily straight lines.