Singularities of the Inertial Flow Map Gradient

Abstract

Inertial particles are finite-sized objects that are carried by flows, for example sand particles in air. In contrast to massless tracer particles, the trajectories of inertial particles can intersect in space-time. When this occurs, the inertial flow map gradient becomes singular. This has an impact on visualization concepts that require the flow map gradient to be invertible. An example are influence curves, which allow to move inertial particles backward in time and thereby avoid the numerically ill-posed inertial backward integration. In this paper, we show that singularities of the inertial flow map gradient can act as poles for influence curves, i.e., as structures that influence curves cannot cross. Influence curves thereby decay into disconnected pieces. We extract singularities in space-time and propose a simple approach to extract influence curves even when they are spatially disconnected. We demonstrate the extraction techniques and discuss the role of singularities in a number of 2D vector fields.