Drag-dominated basin solution with no-slip layers (${\delta }_S=r/\beta ,{\delta }_v=(\nu /r{)}^{1/2}$)

If $r^3\gg {\beta }^2\nu$, the broad western Stommel boundary layer has thickness $r/\beta$, but small viscosity supplies no-slip skins of thickness $(\nu /r{)}^{1/2}$ at both east and west walls. The western solution therefore contains two decaying exponentials, while the east has the rapidly decaying skin. Setting viscosity identically to zero removes the modes needed to satisfy all four lateral no-slip conditions.