A Simple Model of Mass-driven Abyssal Circulation over a General Bottom Topography

Abstract

The classic Stommel?Arons problem is revisited in the context of a basin with a general bottom topography containing an equator. Topography is taken to be smoothly varying and, as such, there are no vertical side walls in the problem. The perimeter of the abyssal basin is thus defined as the curve along which the layer depth vanishes. Because of this, it is not required that the component of horizontal velocity perpendicular to the boundary curve vanish on the boundary. Planetary geostrophic dynamics leads to a characteristic equation for the interface height field in which characteristics typically originate from a single point located on the eastern edge of the basin at the equator. For a simple choice of topography it is possible to solve the problem analytically. In the linear limit of weak forcing, the solution exhibits an intensified flow on the western edge of the basin. This flow is pan of the interior solution and is thus not a traditional, dissipative western boundary current. When the fully nonlinear continuity equation is considered, the characteristics form a caustic on the western side of the basin. Characteristics cannot be integrated through the caustic so that a boundary layer involving higher-order dynamics is required. Approaching the caustic from the cast, velocities predicted by the interior solution become infinite. Because of this and because of global man budget considerations, the boundary layer is shown to lie east of the caustic. The position of the boundary layer is, however, not unique. Away from the equator, it is straightforward to append a Stommel boundary layer to the interior solution. Next, localized mass sources are considered. Their dynamics (planetary geostrophy is assumed, except where boundary currents are required) include caustics and boundary layers in a manner similar to the sink-driven problem described above. For topography such that characteristics have a positive westward component, a caustic lies on the poleward side of the source and extends to the equator. Again, to balance the mass budget, a dissipative boundary layer is required inside the caustic. This, together with all the characteristics leaving the source region, intersects the equator at a finite angle. There, velocities become infinite, while the layer depth goes to zero. Although the details of the implied boundary layer are left unresolved, it is argued that the boundary current crosses the equator at a small angle to the eastward direction.