Geometric Vorticity Control ver.4 with Supplementary Material ver.3

This repository presents a constructive geometric framework for the three-dimensional incompressible Navier–Stokes equations based on Spherical Coherence Tube Density (CTD), together with its extension to moist atmospheric flows. The core method organises the velocity field into concentric spherical layers, extracts representative trajectories, and reconstructs the continuous velocity, pressure and vorticity fields from a particle ensemble. A geometric spacing constraint prevents finite-time vorticity blow-up. Building on this foundation, the Weather-Augmented Navier–Stokes (WaNS) equations are introduced. They couple the CTD-regularised continuous phase (air + water vapour) with a dispersed phase of hydrometeors that possess their own terminal-fall velocities, phase-change source terms, and optional charge separation. Semi-explicit special solutions of moist downburst type are constructed to illustrate the system.