This paper develops a geometric framework for analyzing extreme vorticity amplification in three-dimensional incompressible Navier–Stokes flows. Rather than focusing on global regularity or long-time statistical descriptions of turbulence, the approach adopts an event-conditioned perspective that isolates regimes in which vorticity concentration and enstrophy production become strongly localized. The framework introduces the concept of dangerous sets, defined as phase-space regions where the vorticity concentration ratio exceeds a threshold, signaling entry into an extreme-event regime. Within such regimes, the vorticity field is modeled using thickened vortex filaments represented as vorticity currents, together with a residual component capturing background dynamics. A variational filament-extraction procedure is introduced to measure filament skeletons directly from the vorticity field with such evolution of the extracted geometric parameters then derived from the Navier–Stokes equations using a modulation framework, obtained by projecting the vorticity dynamics onto the tangent space of the filament manifold via a Biot–Savart–induced inner product. To analyze scaling behavior near extreme events, the framework further introduces event-centered renormalized variables, allowing metastable geometric profiles and transient scaling regimes to be identified. In this formulation, extreme events appear not as singularities of the equations but as transitions between geometric regimes in which coherent filament structure emerges and subsequently loses stability. A tethering and open-systems approach is mentioned briefly and will be developed in an upcoming paper. With the above, the resulting pipeline can be developed as: dangerous-set detection → filament extraction → modulation dynamics → renormalized analysis. This provides a PDE-native approach for studying extreme Navier–Stokes events with the paper also outlining a computational strategy for testing the framework using direct numerical simulations of turbulence. It is the intention of this work to be a conceptual and analytical framework rather than a proof of new regularity results for the Navier-Stokes equations, with the goal to provide geometric variables and analytical tools that may help organize the study of extreme events, intermittency, and coherent vortex structures in three-dimensional flows. This paper forms part of an ongoing research program investigating geometric approaches to extreme events in Navier–Stokes dynamics.
Marcos Mendoza (Thu,) studied this question.