Western boundary currents versus vanishing depth

In the case of a constant depth, western intensification of currents in oceanic basins was mathematically recovered in various models (such as Stommel, Munk or quasi-geostrophic ones) as a boundary layer appearing when the solution of equations converges to the solution of a pure transport equation. This convergence is linked to the fact that any characteristic line of the transport vector field included in the equations crosses the boundary, and the boundary layer is located at outgoing points.
Here we recover such a boundary layer for the vertical-geostrophic model with a general bathymetry. More precisely, we allow depth to vanish on the shore in which case the above mentioned characteristic lines no longer cross the boundary. However a boundary layer still appears because the transport vector field $a$ (which is tangential to the boundary) locally converges to a vector field $\overline{a}$ with characteristic lines crossing the boundary.