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
||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.
- 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).
- 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
|Figure 12.3 Angular momentum tends to be conserved as
columns of water change latitude. This causes changes in relative vorticity
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
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
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
|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
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
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.