Dual-Terminal Gravitational Work in Baroclinic Two-Phase Systems

The classical result that a conservative gravitational field performs zero net work over a closed path is shown to depend on the barotropic assumption (rho = rho(p)), which fails at any phase boundary. In a multi-phase system with a density discontinuity, the gravitational field simultaneously performs positive work through two structurally distinct mechanisms: direct body force on the denser phase (positive terminal) and buoyancy on the lighter phase (negative terminal), both deriving from the same gravitational field. This dual-terminal mechanism is governed by the Bjerknes Circulation Theorem (1898), which predicts persistent circulation whenever the cross product of density and pressure gradients is non-zero. We show that natural baroclinic phenomena, tropical cyclones, thermohaline circulation, and atmospheric fronts are planetary-scale manifestations of this mechanism. The Hydropower 2.0 system is the first deliberate engineering realization at laboratory scale. A verification experiment is proposed.