Most of the fluid flows with which we are familiar, from bathtubs to swimming pools, are not rotating, or they are rotating so slowly that rotation is not important except maybe at the drain of a bathtub as water is let out. As a result, we do not have a good intuitive understanding of rotating flows. In the ocean, rotation and the conservation of vorticity strongly influence flow over distances exceeding a few tens of kilometers. The consequences of the rotation leads to results we have not seen before in our day-to-day dealings with fluids. For example, did you ask yourself why the curl of the wind stress leads to a mass transport in the north-south direction and not in the east-west direction? What is special about north-south motion? In this chapter, we will explore some of the consequences of rotation for flow in the ocean.
12.1 Definition of Vorticity
In simple words, vorticity is the rotation of the fluid. The rate of rotation can be expressed various ways. Consider a bowl of water sitting on a table in a laboratory. The water may be spinning in the bowl. In addition to the spinning of the water, the bowl and the laboratory are rotating because they are on a rotating Earth. The two processes are separate, and we can consider two types of vorticity.
Planetary vorticity is the Coriolis parameter we used earlier to discuss flow in the ocean. It is greatest at the poles where it is twice the rotation rate of Earth. Note that the vorticity vanishes at the equator and that the vorticity in the southern hemisphere is negative because φ is negative.
where V = (u, v) is the horizontal velocity vector, and where we have assumed that the flow is two-dimensional. This is true if the flow extends over distances greater than a few tens of kilometers. z is the vertical component of the three-dimensional vorticity vector ω, and it is sometimes written ωz. ζ is positive for counter-clockwise rotation viewed from above. This is the same sense as Earth's rotation in the northern hemisphere.
Note on Symbols Symbols commonly used in one part of oceanography often have very different meaning in another part. Here we use ζ for vorticity, but in Chapter 10, we used ζ to mean the height of the sea surface. We could use ωz for relative vorticity, but ω is also commonly used to mean frequency in radians per second. I have tried to eliminate most confusing uses, but the dual use of ζ is one we will have to live with. Fortunately, it shouldn't cause much confusion.
For a rigid body rotating at rate Ω, curl V = 2Ω. Of course, the flow does not need to rotate as a rigid body to have relative vorticity. Vorticity can also result from shear. For example, at a north/south western boundary in the ocean, u = 0, v = v (x) and ζ = ∂v(x)/∂x.
ζ is usually much smaller than f, and it is greatest at the edge of fast currents such as the Gulf Stream. To obtain some understanding of the size of ζ, consider the edge of the Gulf Stream off Cape Hatteras where the velocity decreases by 1m/s in 100 km at the boundary. The curl of the current is approximately (1 m/s)/(100 km) = 0.14 cycles/day = 1 cycle/week. Hence even this large relative vorticity is still almost seven times smaller than f. More typical values of relative vorticity, such as the vorticity of eddies, is a cycle per month.
We can obtain an equation for absolute vorticity in the ocean by a simple manipulation of the equations of motion for frictionless flow. We begin with:
If we expand the substantial derivative, and if we subtract ∂ /∂y of (12.4a) from ∂ /∂x of (12.4b) to eliminate the pressure terms, we obtain after some algebraic manipulations:
In deriving (12.15) we used:
recalling that f is independent of time t and eastward distance x.
Integrating the continuity equation (7.19) from the bottom to the top of the ocean gives (Cushman-Roisin, 1994):
where b is the topography of the bottom, and H is the depth of the water. The boundary conditions require that flow at the surface and the bottom be along the surface and the bottom. Thus the vertical velocities at the top and the bottom are:
Substituting (12.7) and (12.8) into (12.6) we obtain
Substituting this into (12.5) gives:
which can be written:
The quantity within the parentheses must be constant. It is called potential vorticity Π. Potential vorticity is conserved along a fluid trajectory:
For baroclinic flow in a continuously stratified fluid, the potential vorticity can be written (Pedlosky, 1987: § 2.5):
where λ is any conserved quantity for each fluid element. In, particular, if λ = ρ then:
assuming the horizontal gradients of density are small compared with the vertical gradients, a good assumption in the thermocline. In most of the interior of the ocean, f >> ζ and (12.11) is written (Pedlosky, 1996, eq 3.11.2):
This allows the potential vorticity of various layers of the ocean to be determined directly from hydrographic data without knowledge of the velocity field.
|Department of Oceanography, Texas A&M University
Robert H. Stewart, email@example.com
All contents copyright © 2005 Robert H. Stewart,
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Updated on October 20, 2006