12.3 Vorticity and Ekman Pumping
Rotation places another very interesting constraint on the geostrophic flow field. To help understand the constraints, let's first consider flow in a fluid with constant rotation. Then we will look into how vorticity constrains the flow of a fluid with rotation that varies with latitude. An understanding of the constraints leads to a deeper understanding of Sverdrup's and Stommel's results discussed in the last chapter.
Fluid dynamics on the f Plane:
the Taylor-Proudman Theorem
and the continuity equations (7.19) is:
Taking the z derivative of (12.13a) and using (12.13c) gives:
Similarly, for the u-component of velocity (12.13b). Thus, the vertical derivative of the horizontal velocity field must be zero.
This is the Taylor-Proudman Theorem, which applies to slowly varying flows in a homogeneous, rotating, inviscid fluid. The theorem places strong constraints on the flow:
Hence, rotation greatly stiffens the flow! Geostrophic flow cannot go over a seamount, it must go around it. Taylor (1921) explicitly derived (12.14) and (12.16) below. Proudman (1916) independently derived the same theorem but not as explicitly.
Further consequences of the theorem can be obtained by eliminating the pressure terms from (12.13a & 12.13b) to obtain:
Because the fluid is incompressible, the continuity equation (12.13d) requires
Furthermore, because w = 0 at the sea surface and at the seafloor, if the bottom is level, there can be no vertical velocity on an f-plane. Note that the derivation of (12.16) did not require that density be constant. It requires only slow motion in a frictionless, rotating fluid.
Fluid Dynamics on the Beta Plane: Ekman Pumping
Consider then flow on a beta plane. If f = f0 + β y, then (12.15a) becomes:
where we have used (12.13a) to obtain v in the right-hand side of (12.18).
Using the continuity equation, and recalling that β y << f0
where we have used the subscript G to emphasize that (12.19) applies to the ocean's interior, geostrophic flow . Thus the variation of Coriolis force with latitude allows vertical velocity gradients in the geostrophic interior of the ocean, and the vertical velocity leads to north-south currents. This explains why Sverdrup and Stommel both needed to do their calculations on a β-plane.
Ekman Pumping in the Ocean
which is (9.30b) where ρ is density and f is the Coriolis parameter. Because the vertical velocity at the sea surface must be zero, the Ekman vertical velocity must be balanced by a vertical geostrophic velocity wG(0).
Ekman pumping ( wE(0) ) drives a vertical geostrophic current (-wG (0) ) in the ocean's interior. But why does this produce the northward current calculated by Sverdrup (11.6)? Peter Niiler (1987: 16) gives a simple explanation.
Peter Rhines (1982) points out that the rigid column of water trying to escape the squeezing imposed by the atmosphere escapes by moving southward. The southward velocity is about 5,000 times greater than the vertical Ekman velocity.
Ekman Pumping: An Example
Because the water near the surface is warmer than the deeper water, the vertical velocity produces a pool of warm water. Much deeper in the ocean, the wind-driven geostrophic current must go to zero (Sverdrup's hypothesis) and the deep pressure gradients must be zero. As a result, the surface must dome upward because a column of warm water is longer than a column of cold water having the same weight (they must have the same weight, otherwise, the deep pressure would not be constant, and there would be a deep horizontal pressure gradient). Such a density distribution produces north-south pressure gradients at mid depths that must be balanced by east-west geostrophic currents. In short, the divergence of the Ekman transports redistributes mass within the frictionless interior of the ocean leading to the wind-driven geostrophic currents.
Now let's continue the idea to include the entire north Pacific to see how winds produce currents flowing upwind. The example will give a deeper understanding of Sverdrup's results we discussed in §11.1.
Figure 12.8 shows shows the mean zonal winds in the Pacific, together with the north-south Ekman transports driven by the zonal winds. Notice that convergence of transport leads to downwelling, which produces a thick layer of warm water in the upper kilometer of the water column, and high sea level. Figure 12.7 is a sketch of the cross section of the region between 10°N and 60°N, and it shows the pool of warm water in the upper kilometer centered on 30°N causing the surface high in figure 12.8. Conversely, divergent transports leads to low sea level. The mean north-south pressure gradients associated with the highs and lows are balanced by the Coriolis force of east-west geostrophic currents in the upper ocean (shown at the right in figure 12.8).
|Department of Oceanography, Texas A&M University
Robert H. Stewart, firstname.lastname@example.org
All contents copyright © 2005 Robert H. Stewart,
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Updated on January 19, 2012