Tornadogenesis and Vortex Control: A Geometric Theory via the Layer-by-Layer Gauge Method and the Kraichnan Inverse Cascade

This paper presents a rigorous mathematical theory of tornadogenesis and vortex control based on the layer-by-layer gauge method for the three-dimensional Navier–Stokes equations. Building on the recently established global regularity of solutions [13], we prove that a thunderstorm front bounded by two quasi-two-dimensional surfaces—an upper wind shear layer and a lower ground layer—supports a continuous Kraichnan inverse cascade across all three spatial stages. The central result is the existence of a gauge enstrophy invariant Ecal = 1/2 R (ωθ/r)2dµ that remains conserved during the transition from the upper 2D zone through the axisymmetric 3D vortex column to the ground 2D layer. This invariant ensures that energy flows from small-scale turbulent fluctuations to the macroscopic vortex core, providing a self-sustaining mechanism for tornado maintenance.We derive the effective Riemannian metric induced by the gauge enstrophy and show that the vortex axis moves along a geodesic in this metric. The symbols ofthe Levi-Civita connection naturally give rise to the forces of Magnus, Coriolis, centrifugal, and vertical lift without any ad hoc assumptions. The energy-momentum tensor for the gauge field yields an exact conservation law that couples the vortexdynamics to external control devices.An explicit formula for the geodesic deviation under external control is derived, enabling quantitative prediction of the power required to deflect a tornado of a given category. A mobile control system based on vehicle-mounted microwave emitters is proposed and analysed. The system exploits the extreme spatio-temporal duty cycle of tornado impacts (∼ 10−8), reducing the required average power by three ordersof magnitude compared to a stationary protective network. Technical-economic analysis shows that a mobile system with 15 emitter units and a total capital cost of approximately 400 million USD is feasible and cost-effective.All results are derived rigorously from the Navier–Stokes equations using the evolutionary layer-by-layer gauge method, with complete proofs provided in themain text and appendices.