We investigate the steady barotropic circulation patterns driven by inflow-outflow boundary conditions on a rectangular β-plane domain. An inertial jet enters the domain in the southwest corner and a broad eastward outflow is prescribed at the eastern boundary. On the western wall there is no mass flux and no slip.With weak viscosity, ν the western boundary jet ?overshoots? northward, beyond the latitude band of the eastern outflow. As the viscosity is reduced the length of this overshoot increases as ν−2/3, before the jet gradually peels away from the western wall, plunges southward and eventually turns eastward. Away from the wall the current forms a damped stationary Rossby wave, as described by Moore in 1963.The initial northward overshoot and southward "plunge" is a distinct dynamical regime, and not merely the first and largest undulation of the Rossby wave. For instance the zonal length scale of the overshoot is just the Munk scale, (ν/β)1/3 and inertia, planetary vorticity and viscosity are all important at leading order in the dynamical balance as ν → 0. All of the streamlines pass through this dissipative region and most of the Lagrangian potential vorticity alterations occur here, rather than in the Rossby wave.The preceeding scenario applies only when the northern boundary is distant, so that the overshoot peels away from the western wall before striking the northwest corner of the domain. If the jet reaches the northern boundary it drives an inertial recirculating gyre in the corner.
Cessi, Paola, R. V. Condie, and W. R. Young. 1990. "Dissipative dynamics of western boundary currents." Journal of Marine Research 48, (4). https://elischolar.library.yale.edu/journal_of_marine_research/1984