Fig. 5.46 An inverse reduced-gravity model.

Assuming geostrophy, the momentum equations for an inverse reduced-gravity model are

If we denote the sea floor as d = H, Eqn. (5.53a, b) can be rewritten as

Using the interfacial displacement from the mean position at depth D, Z = D - (H - h), Eqn. (5.53a, b) can be also be rewritten as

The continuity equation is

where w* is the vertical velocity leaving the upper surface of the lower layer due to entrainment to the layer above.

For example, along the western boundaries of the oceans, there are deep western boundary currents. These boundary currents are closely associated with the dense water originating from deepwater formation sites at high latitudes. The shoaling of the isopycnal surface near the western boundary (Fig. 5.46) suggests that the deep western boundary current flows equatorward.

Deep circulation simulated by the rotating sector experiments The theory of deep circulation was first developed in the late 1950s and early 1960s. In order to explore the possible patterns of circulation, a very simple and elegant pie-shaped experimental tool was designed (Stommel et al., 1958). The system rotated uniformly, and the circulation was observed with the release of dye (Fig. 5.47).

The parabolic-shaped upper surface of the water in the rotating experiments produces a dynamical effect equivalent to the j-effect in the world's oceans. For "large-scale" motions in such pie-shaped experiments, the barotropic potential vorticity is f /h. Since h is large near the rim of the sector, potential vorticity is low. For an ocean with uniform depth on the surface of Earth, potential vorticity f /h is low at low latitudes; therefore, the rim of the rotating pie corresponds to the low-latitude ocean on the Earth.

By placing the source and sink at different locations in the model, different patterns of circulation can be observed (Fig. 5.48). From the potential vorticity argument, the system allows three types of motion only:

1. Geostrophic flow along circles of constant radius.

2. Radial flow in the interior is possible only if there is a source or a sink.

3. A western boundary current going northward or southward. Similar to the wind-driven circulation discussed in Chapter 4, balancing potential vorticity in the whole model basin rules out the possibility of closing the circulation by an eastern boundary current.

Note that an increase (decrease) in water level corresponds to an upwelling leaving the abyssal layer, so that it is a uniform sink (source). As will be explained shortly, a source/sink implies a stretching which must be balanced by radial flow as required by the linear vorticity balance.

Fig. 5.48 a Diagram of circulation induced in a rotating sector by source (+) and sink (—); b sketch of flow pattern expected with source (+) at apex of sector, surface of fluid rising uniformly; c sketch of flow pattern expected with source (+) at the western edge of rim, with surface of fluid rising uniformly (redrawn from Stommel et al., 1958).

Fig. 5.48 a Diagram of circulation induced in a rotating sector by source (+) and sink (—); b sketch of flow pattern expected with source (+) at apex of sector, surface of fluid rising uniformly; c sketch of flow pattern expected with source (+) at the western edge of rim, with surface of fluid rising uniformly (redrawn from Stommel et al., 1958).

Dynamical analysis

The projection of the centrifugal force and gravity force onto the tangent of the free surface should be in balance:

where tan a is the slope of the free surface. This relation can be rewritten as

M r/g = tan a = dh/dr which integration leads to the shape of the free surface

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