13.3: Theory for the Deep Circulation

Stommel, Arons, and Faller, in a series of papers from 1958 to 1960, described a simple theory of the abyssal circulation (Stommel 1958; Stommel, Arons, and Faller, 1958; Stommel and Arons, 1960). The theory differed so greatly from what was expected that Stommel and Arons devised laboratory experiments with rotating fluids to confirm 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:

1. Cold, deep water is supplied by deep convection at a few high-latitude locations in the Atlantic, notably in the Irminger and Greenland Seas in the north and the Weddell Sea in the south.
2. Uniform mixing in the ocean brings the cold, deep water back to the surface.
3. The deep circulation is strictly geostrophic in the interior of the ocean, and therefore potential vorticity is conserved.

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 that deep convection by itself leads to a deep, stagnant pool of cold water. In this case, the deep 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 deep circulation. Winds and tides 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: \[\beta v = f \frac{\partial w}{\partial z} \nonumber \]

where \(f = 2 \ \Omega \sin \varphi\), \(\beta = (2 \ \Omega \cos \varphi)/R\), \(\Omega\) is Earth’s rotation rate, \(R\) is Earth’s radius, and \(\varphi\) is latitude. Integrating \((\PageIndex{1})\) from the bottom of the ocean to the top of the abyssal circulation gives: \[\begin{align} V &= \int_{0}^{H} v \ dz = \int_{0}^{H} \frac{f}{\beta} \frac{\partial w}{\partial z} dz \nonumber \\ V &= R \ \tan \varphi \ W_{0} \end{align} \nonumber \]

where \(V\) is the vertical integral of the northward velocity, and \(W_{0}\) is the velocity at the base of the thermocline. \(W_{0}\) 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 \(\PageIndex{1}\). The \(U\) component of the flow is calculated from \(V\) and \(w\) using the continuity equation.

Figure \(\PageIndex{1}\): Idealized sketch of the deep circulation due to deep convection in the Atlantic (dark circles) and upwelling through the thermocline elsewhere. The real circulation is much different than the circulation shown in this sketch. After Stommel (1958).

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 \(S_{0}\) 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 \(T_{w}\) in the western boundary current is: \[T_{w} = -2 S_{0} \sin \varphi \nonumber \]

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 \(S_{0}\) 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 \(\PageIndex{1}\).

Next, Stommel and Arons calculated the transport in a western boundary current in a basin with no source. The transport is: \[T_{w} = S[1 - 2 \sin \varphi] \nonumber \]

where \(S\) is the transport across the Equator from the other hemisphere. In this basin Stommel notes:

A current of recirculated water equal to the source strength starts at the pole and flows toward the source . . . [and] gradually diminishes to zero at \(\varphi = 30^{\circ}\) north latitude. A northward current of equal strength starts at the equatorial source and also diminishes to zero at \(30^{\circ}\) north latitude.

This gives the western boundary current as sketched in the north Pacific in figure \(\PageIndex{1}\).

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 \(\PageIndex{2}\) shows how ridges control the flow in the Indian Ocean.

Figure \(\PageIndex{2}\): Deep flow in the Indian Ocean inferred from the temperature, given in \(^{\circ}\text{C}\). Note that the flow is constrained by the deep mid-ocean ridge system. After Tchernia (1980).

Finally, the 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

Our understanding of the deep circulation is still evolving.

1. Marotzke and Scott (1999) points out that deep convection and mixing are very different processes. Convection reduces the potential energy of the water column, and it is self-powered. Mixing in a stratified fluid increases the potential energy, and it must be driven by an external process.
2. Numerical models show that the deep circulation is very sensitive to the assumed value of vertical eddy diffusivity in the thermocline (Gargett and Holloway, 1992).
3. Numerical calculations by Marotzke and Scott (1999) indicate that the mass transport is not limited by the rate of deep convection, but it is sensitive to the assumed value of vertical eddy diffusivity, especially near side boundaries.
4. Cold water is mixed upward at the ocean’s boundaries, above seamounts and mid-ocean ridges, along strong currents such as the Gulf Stream, and in the Antarctic Circumpolar Current (Toggweiler and Russell, 2008; Garabato et al, 2004, 2007). Because mixing is strong over mid-ocean ridges and small in nearby areas, flow is zonal in the ocean basins and poleward along the ridges (Hogg et al. 2001). A map of the circulation will not look like figure \(\PageIndex{1}\). Numerical models and measurements of deep flow by floats show the flow is indeed zonal.
5. Because the transport of mass, heat, and salt are not closely related, the transport of heat into the North Atlantic may not be as sensitive to surface salinity as described above.