Potential vorticity dynamics in a domain with closed geostrophic contours. I: Steady linear flows

Inertia-less wind-driven barotropic flow with lateral and Ekman friction is investigated in circular basins with various bathymetries. The focus is on the vorticity dynamics of flow in basins encircled by continental shelves, a bathymetry which allows a recirculating flow at lowest order because the geostrophic contours are distorted into closed loops which do not intersect the boundary. In general the wind forcing and friction will not balance everywhere, and the potential vorticity lost to the wind must be recovered elsewhere on each streamline. This recovery takes place in a "western" boundary current if geostrophic contours intersect the boundary, but the location of the recovery region is unclear if the geostrophic contours are closed loops. The nature and location of potential vorticity recovery are investigated numerically in several geometries with closed geostrophic contours and compared with models in which the contours intersect the boundary. General theory is also presented which can determine the lowest-order recirculating flow from the bathymetry and forcing. It is demonstrated that potential vorticity recovery occurs in an extended internal shear layer (isolated from the boundary) if the encircling continental shelf is axisymmetric. The more complicated case of non-axisymmetric shelves is also investigated.