Chapter 12 - Vorticity in the Ocean

12.2 Conservation of Vorticity

The angular momentum of any isolated spinning body is conserved. The spinning body can be an eddy in the ocean or the Earth in space. If the the spinning body is not isolated, that is, if it is linked to another body, then angular momentum can be transferred between the bodies. The two bodies need not be in physical contact. Gravitational forces can transfer momentum between bodies in space. We will return to this topic in Chapter 17 when we discuss tides in the ocean. Here, let's look at conservation of vorticity in a spinning ocean.

Friction is essential for the transfer of momentum in a fluid. Friction transfers momentum from the atmosphere to the ocean through the thin, frictional, Ekman layer at the sea surface. Friction transfers momentum from the ocean to the solid Earth through the Ekman layer at the seafloor. Friction along the sides of subsea mountains leads to pressure differences on either side of the mountain which causes another form of drag called form drag. This is the same drag that causes wind force on cars moving at high speed. In the vast interior of the ocean, however, the flow is frictionless, and vorticity is conserved. Such a flow is said to be conservative.

Figure 12.2 Sketch of the production of relative vorticity by the changes in the height of a fluid column. As the vertical fluid column moves from left to right, vertical stretching reduces the moment of inertia of the column, causing it to spin faster.

Conservation of Potential Vorticity The conservation of potential vorticity couples changes in depth, relative vorticity, and changes in latitude. All three interact.

1. Changes in the depth H of the flow causes changes in the relative vorticity. The concept is analogous with the way figure skaters decreases their spin by extending their arms and legs. The action increases their moment of inertia and decreases their rate of spin (Figure 12.2).
2. Changes in latitude require a corresponding change in ζ. As a column of water moves equatorward, f decreases, and ζ must increase (Figure 12.3). If this seems somewhat mysterious, von Arx (1962) suggests we consider a barrel of water at rest at the north pole. If the barrel is moved southward, the water in it retains the rotation it had at the pole, and it will appear to rotate counterclockwise at the new latitude where f is smaller.

Figure 12.3 Angular momentum tends to be conserved as columns of water change latitude. This causes changes in relative vorticity of the columns. From von Arx (1962).

Consequences of Conservation of Potential Vorticity
The concept of conservation of potential vorticity has far reaching consequences, and its application to fluid flow in the ocean gives a deeper understanding of ocean currents.

1. In the ocean f tends to be much larger than ζ and thus f/H = constant. This requires that the flow in an ocean of constant depth be zonal. Of course, depth is not constant, but in general, currents tend to be east-west rather than north south. Wind makes small changes in ζ, leading to a small meridional component to the flow (see Figure 11.3).

2. Barotropic flows are diverted by seafloor features. Consider what happens when a flow that extends from the surface to the bottom encounters a subsea ridge (Figure 12.4). As the depth decreases, ζ + f must also decrease, which requires that f decrease, and the flow is turned toward the equator. This is called topographic steering. If the change in depth is sufficiently large, no change in latitude will be sufficient to conserve potential vorticity, and the flow will be unable to cross the ridge. This is called topographic blocking.

Figure 12.4 Barotropic flow over a sub-sea ridge is turned equatorward to conserve potential vorticity. From Dietrich, et al. (1980).

3. The balance of vorticity provides an alternate explanation for the existence of western boundary currents (Figure 12.5). Consider the gyre-scale flow in an ocean basin, say in the North Atlantic from 10°N to 50°N. The wind blowing over the Atlantic adds negative vorticity. As the water flows around the gyre, the vorticity of the gyre must remain nearly constant, else the flow would spin up or slow down. The negative vorticity input by the wind must be balanced by a source of positive vorticity.

The source of positive vorticity must be boundary currents: the wind-driven flow is baroclinic, which is weak near the bottom, so bottom friction cannot transfer vorticity out of the ocean. Hence, we must decide which boundary contributes. Flow tends to be zonal, and east-west boundaries will not solve the problem. In the east, potential vorticity is conserved: the input of negative relative vorticity is balanced by a decrease in potential vorticity as the flow turns southward. Only in the west is vorticity not in balance, and a strong source of positive vorticity is required. The vorticity is provided by the current shear in the western boundary current as the current rubs against the coast causing the northward velocity to go to zero at the coast (Figure 12.5, right).

In this example, friction transfers angular momentum from the wind to the ocean and eddy viscosity - friction - transfers angular momentum from the ocean to the solid Earth.

Figure 12.5 The balance of potential vorticity can clarify why western boundary currents are necessary. Left: Vorticity input by the wind ζt balances the change in relative vorticity ζ in the east as the flow moves southward and f decreases; but the two do not balance in the west where ζ must decrease as the flow moves northward and f increases. Right: Vorticity in the west is balanced by relative vorticity ζb generated by shear in the western boundary current.

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