Problem with the vorticity condition at the boundary for FREE-SLIP circulation

in QG and C-grid-vorticity-gradient formulation for the advection in shallow water or primitive eqs , we must impose a value for the vorticity at the boundary. Using a FREE-SLIP boundary condition, the natural condition for the vorticity is along a straight wall.

Resolution = 10.3 km. Free-Slip. Vorticity zero at the corner and elsewhere along the boundary	Resolution = 10.3 km. Free-Slip. Vorticity nonzero at the corner and zero elsewhere along the boundary
Resolution = 10.3 km. No-Slip case for reference

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• No more valid at a corner where the curvature is infinite.
• Imposing at a corner yields an artificial circulation (Dengg and above figure).
• The change of vorticity at a single point (the corner) changes the entire circulation.
• Drawback: Value somewhat arbitrary, yields spurious oscillations in the vicinity of the corner for the vorticity field (not shown).
• we don't recover the NO-SLIP circulation.