A Deterministic Limit for Tropical Cyclone Rapid Intensification: Deriving the Critical Eyewall Radius via Thermodynamic Momentum Diffusion

The forecasting of Rapid Intensification (RI) in tropical cyclones remains heavily reliant on probabilistic empirical models, such as the Statistical Hurricane Intensity Prediction Scheme (SHIPS). While classical vortex dynamics describe the aggregate evolution of angular momentum, they struggle to deterministically define the exact spatiotemporal boundary where stable intensity transitions into runaway thermodynamic intensification. This paper introduces a strict continuum framework for atmospheric vortex scaling. By modeling the cyclone as a thermodynamic balance between the macroscopic diffusion of angular momentum and the localized conversion of latent heat within the eyewall, we derive a universal critical intensification radius (RRI). We demonstrate that Rapid Intensification is not a random statistical probability, but an exact deterministic limit where localized diabatic heating strictly overpowers the advective momentum-redistribution capacity of the broader vortex. We contrast this invariant with traditional probabilistic RI forecasting and propose a blueprint for Active Vortex Monitoring (AVM) utilizing real-time aircraft telemetry.