We investigate the trapping of density-forced eastern-coastal circulations by a continental shelf, with application to the Leeuwin Current. For this
purpose, we obtain steady, analytic or semi-analytic solutions to a variable-density, two-layer model on a beta-plane without friction in a semi-infinite domain x < x_e and y < y_n < 0 (southern hemisphere). The upper-layer density field is externally prescribed to increase poleward,
providing a thermohaline forcing for the system, while the lower layer is assumed to be motionless with a uniform density.
When the upper layer does not intersect the bottom, planetary Rossby waves eliminate all pressure gradients. When it does, topographic Rossby waves trap the pressure gradient on the slope. The beta effect, however, still allows signals to leak offshore. This process determines the "grounding line," x_1(y), where the upper layer leaves the bottom. Planetary Rossby waves then propagate the pressure structure westward across the grounding line and then offshore, determining the meridional pressure gradient in the offshore region.
As a result, a broad eastward flow is generated in the offshore region and it bends poleward inshore of the grounding line, forming a poleward eastern-boundary current. The transport of the eastward flow and the poleward current is dependent on the bottom topography only weakly, and is(approximately) proportional to H_12, where H_1 is the upper-layer thickness along y=y_n for x < x_1(y_n). Regarding the Leeuwin Current, the model suggests that the important impact of the western Pacific warm pool is that it thickens the thermocline in the Northwest Shelf region, that is, it ensures that H_1 has a large value.