Special Cases of the Potential Vorticity Equation
105
(3.2.35)
The system of equations consists of (3.2.29) for the general nth
intermediate layer and (3.2.31) and (3.2.34) for the upper- and lower most
layers. Note that only the upper layer is driven by the Ekman pumping, and
that in the absence of interfacial coupling by either cross-interface velocities or
by frictional coupling between layers (which would be included in S'n) there
would be apparently nothing to maintain the lower layers in motion. This is the
paradox to which we referred above, and to which we return shortly as the
principal topic of this chapter.
In each layer the total derivative of any variable, and in particular the
potential vorticity, qn, for this layer can be written in terms of the
streamfunction and its Jacobian with the derived variable as:
(3.2.36)
3.3 Special Cases of the Potential Vorticity Equation
There are two special examples of the quasi-geostrophic layer model of
particular interest for the discussion of this chapter. In the first case the entire
ocean, beneath the upper Ekman layer, consists of only two layers, the
lowermost being in contact with the ocean floor, as shown in Fig. 3.3.1a. In this
Fig. 3.3.1a,b. Two-and three-layer models.
a The two layers fill the entire depth of the ocean.
b There are three layers, the upper two of which
are in motion for steady flows while the third,
deep layer is at rest. If the third layer is deep
enough its motion can be neglected for timedependent motions as well
H2
P2
777777~7777777777777/
H 3
P3
~(a)
(b)
105
(3.2.35)
The system of equations consists of (3.2.29) for the general nth
intermediate layer and (3.2.31) and (3.2.34) for the upper- and lower most
layers. Note that only the upper layer is driven by the Ekman pumping, and
that in the absence of interfacial coupling by either cross-interface velocities or
by frictional coupling between layers (which would be included in S'n) there
would be apparently nothing to maintain the lower layers in motion. This is the
paradox to which we referred above, and to which we return shortly as the
principal topic of this chapter.
In each layer the total derivative of any variable, and in particular the
potential vorticity, qn, for this layer can be written in terms of the
streamfunction and its Jacobian with the derived variable as:
(3.2.36)
3.3 Special Cases of the Potential Vorticity Equation
There are two special examples of the quasi-geostrophic layer model of
particular interest for the discussion of this chapter. In the first case the entire
ocean, beneath the upper Ekman layer, consists of only two layers, the
lowermost being in contact with the ocean floor, as shown in Fig. 3.3.1a. In this
Fig. 3.3.1a,b. Two-and three-layer models.
a The two layers fill the entire depth of the ocean.
b There are three layers, the upper two of which
are in motion for steady flows while the third,
deep layer is at rest. If the third layer is deep
enough its motion can be neglected for timedependent motions as well
H2
P2
777777~7777777777777/
H 3
P3
~(a)
(b)