Anisotropic Shear Metrics for Persistent Homology and Their Application to Convective Systems

Vertical wind shear plays a crucial role in the organization and persistence of mesoscale convective systems, yet its geometrical and topological effects remain challenging to quantify. In this study, we introduce a shear-induced anisotropic metric, denoted dS, which embeds the direction and magnitude of environmental wind shear directly into the framework of persistent homology. The metric deforms the ambient geometry by weighting distances differently along and across the shear direction, enabling topological descriptors to respond dynamically to the flow environment. We establish the analytical properties of dS, and demonstrate its compatibility with Vietoris–Rips filtrations. The method is applied to the Corsican bow–echo event of 18 August 2022, where shear vectors are derived from ERA5 reanalysis data. Two complementary topological analyses are performed: a transport analysis on H0 using Wasserstein distances, and a structural analysis on H1 persistent generators under parallel and perpendicular shear metrics. The results reveal distinct topological evolutions associated with different shear orientations, highlighting the sensitivity of persistent homology to shear-induced deformation. Overall, the framework provides a mathematically consistent bridge between dynamical meteorology and topological data analysis, extending persistent homology to anisotropic metric spaces.