The well-known Prandtl-Batchelor theorem for high Reynolds number flows is generalized and applied to quasi-geostrophic flows with or without meso-scale eddies in a closed geostrophic contour.
Laminar quasi-geostrophic flows without any external forcings are shown to be stagnant in a closed streamline owing to the Ekman friction. As for the turbulent case, two markedly different mean states are suggested in the limit of both weak eddies and weak dissipation. One state corresponds to the laminar result in the limit and the other is the state of uniform potential vorticity over a domain surrounded by closed geosrophic contours. The latter state is not inconsistent with Rhines and Young (1981).