13.2 Theory for the Deep Circulation
Stommel, Arons, and Faller in a series of papers from 1958 to 1960 laid the foundation for our present understanding of the abyssal circulation (Stommel 1958; Stommel, Arons, and Faller, 1958; Stommel and Arons, 1960). The papers reported simplified theories of the circulation that differed so greatly from what was expected that Stommel and Arons devised laboratory experiments with rotating fluids to confirmed their theory. The theory for the deep circulation has been further discussed by Marotzke (2000) and Munk and Wunsch (1998).
The Stommel, Arons, Faller theory is based on three fundamental ideas:
Notice that the deep circulation is driven by mixing, not by the sinking of cold water at high latitudes. Munk and Wunsch (1998) point out we have known for 100 years that deep convection by itself leads to a deep, stagnant, pool of cold water. In this case, the there is no deep circulation. Circulation is confined to the upper layers of the ocean. Mixing or upwelling is required to pump cold water upward through the thermocline and drive the meridional overturning circulation. Tides and winds are the primary source of energy driving the mixing.
Notice also that convection and sinking are not the same, and they do not occur in the same place (Marotzke and Scott, 1999). Convection occurs in small regions a few kilometers on a side. Sinking, driven by Ekman pumping and geostrophic currents, can occur over far larger areas. In this chapter, we are discussing mostly sinking of water.
To describe the simplest aspects of the flow, we begin with the Sverdrup equation applied to a bottom current of thickness H in an ocean of constant depth:
where f = 2Ω sin φ, β = (2Ω cos φ)/R, Ω is Earth's rotation rate, R Earth's radius, and φ is latitude. Integrating (13.1) from the bottom of the ocean to the top of the abyssal circulation gives:
where V is the vertical integral of the northward velocity, and W0 is the velocity at the base of the thermocline. W0 must be positive (upward) almost everywhere to balance the downward mixing of heat. Then V must be everywhere toward the poles. This is the abyssal flow in the interior of the ocean sketched by Stommel in Figure 13.4. The U component of the flow is calculated from V and w using the continuity equation.
To connect the streamlines of the flow in the west, Stommel added a deep western boundary current. The strength of the western boundary current depends on the volume of water S produced at the source regions.
Stommel and Arons calculated the flow for a simplified ocean bounded by the Equator and two meridians (a pie shaped ocean). First they placed the source S0 near the pole to approximate the flow in the north Atlantic. If the volume of water sinking at the source equals the volume of water upwelled in the basin, and if the upwelled velocity is constant everywhere, then the transport Tw in the western boundary current is:
The transport in the western boundary current at the poles is twice the volume of the source, and the transport diminishes to zero at the Equator (Stommel and Arons, 1960a: eq, 7.3.15; see also Pedlosky, 1996: §7.3). The flow driven by the upwelling water adds a recirculation equal to the source. If S0 exceeds the volume of water upwelled in the basin, then the western boundary current carries water across the Equator. This gives the western boundary current sketched in the north Atlantic in Figure 13.4.
Next, Stommel and Arons calculated the transport in a western boundary current in a basin with no source. The transport is:
where S is the transport across the Equator from the other hemisphere. In this basin Stommel notes:
This gives the western boundary current as sketched in the north Pacific in Figure 13.4.
Note that the Stommel-Arons theory assumes a flat bottom. The mid-ocean ridge system divides the deep ocean into a series of basins connected by sills through which the water flows from one basin to the next. As a result, the flow in the deep ocean is not as simple as that sketched by Stommel. Boundary current flow along the edges of the basins, and flow in the eastern basins in the Atlantic comes through the mid-Atlantic ridge from the western basics. Figure 13.5 shows how ridges control the flow in the Indian Ocean.
Finally, Stommel-Arons theory gives some values for time required for water to move from the source regions to the base of the thermocline in various basins. The time varies from a few hundred years for basins near the sources to nearly a thousand years for the north Pacific, which is farther from the sources.
Some Comments on the Theory for the Deep Circulation
|Department of Oceanography, Texas A&M University
Robert H. Stewart, firstname.lastname@example.org
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Updated on October 24, 2008