An equation governing potential vorticity in a magnetized plasmas is derived. The equation is analogous to Ertel's theorem. In the long wave-length limit the potential vorticity equals the ratio of the gyro-frequency plus the E × B- and diamagnetic polarization densities to the particle density. The equation is relevant for transport barriers in magnetically confined plasmas because particle density, ion temperature and the radial electric field are mutually coupled through the potential vorticity. The potential vorticity equation is derived from an energy conserving, four-field, electrostatic, full-F gyrofluid model. It is shown that the gyrofluid model possesses two exact Lagrangian invariants. In systems where mixing uniformly distribute the Lagrangian invariants we derive the corresponding turbulent equipartion states. It is shown that the system is driven towards constant potential vorticity. Given particle density and magnetic field profiles we infer ion temperature and electric potential profiles from the derived turbulent equipartion states.
Hamiltonian formulation and coherent structures in electrostatic turbulence