Why a Wind Farm Is Not a Field of Betz Turbines: The Atmospheric Replenishment Ceiling, Computed

The Betz limit (16/27, 59.3%) bounds an isolated rotor, but at fleet scale the binding constraint is atmospheric replenishment: kinetic energy is supplied to the boundary layer at about 2 W/m2, bounding sustained large-scale wind extraction at 0.5-1.0 W/m2. For a 5 MW, 150 m reference turbine and a 1 TW average demand, nameplate arithmetic implies 450,000 km2 while the replenishment bound requires 1,000,000-2,000,000 km2. All numbers from a deposited Python script.