A Deterministic Limit for Hypersonic Boundary Layer Transition: Deriving the Critical Turbulent Burst Radius via Laminar Thermal Dissipation

The prediction of boundary layer transition on hypersonic reentry vehicles relies heavily on empirical correlations and linear stability theories, most notably the semi-empirical eN method. While these frameworks effectively estimate macroscopic Reynolds number thresholds for amplifying kinetic disturbances, they fail to deterministically define the exact spatial boundary where a localized turbulent burst triggers a cascading transition into irreversible, superheated plasma turbulence. This paper introduces a strict continuum framework for macroscopic aerothermodynamic scaling. By modeling the plasma boundary layer as a dynamic thermodynamic balance between the spatial capacity for advective laminar thermal dissipation and the localized rate of frictional heating, we derive a universal critical plasma radius (Rplasma). We demonstrate that hypersonic boundary layer transition is not a statistical anomaly of fluid kinetics, but an exact deterministic limit where localized turbulent friction strictly overpowers the advective thermal smoothing capacity of the laminar flow. This method provides an exact spatial threshold for transition, offering superior accuracy over traditional models. We propose a blueprint for Active Boundary Layer Control (ABLC) using micro-thermocouple spatial telemetry, enabling real-time transition prediction and mitigation before catastrophic heat shield failure.