360
DYNAMICAL OCEANOGRAPHY
contours are lines of constant y. Such contours will always intersect the eastern
boundary where ψ 2 =0and hence ψ 2 is zero everywhere. When the wind forcing
is so small that the geostrophic contours (curves of constant q 2 ) still intersect the
eastern boundary, then there is flow in the upper layer, but the lower layer is still
motionless. To see how the geostrophic contours are deformed by the upper-layer
flow, we first consider an example.
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Example 15.1: Localized Ekman pumping: specification
Consider a situation of localized Ekman pumping in a disk with radius R
around (0, 0), such that R the Ekman pumping (Fig. 15.6a) is given by
w E = −γx,
(15.14)
where γ>0 is a positive constant; w E =0outside the disk.
The Sverdrup streamfunction ψ B is immediately found from (15.9). It is zero
for points (x, y) outside the disk (when x>R , this follows from w E =0and
when x<−R, it follows from the anti-symmetry of w E with respect to x =0 )
and for values (x, y) within the disk it becomes
ψ B (x, y)=
γf 0
β 0 D
X(y)
x
x
′ dx
′ =
γf 0 ˆ
F
2β 0 D
(R
2 − x
2 − y
2 ),
(15.15)
where X(y)=
R 2 − y 2 . The geostrophic contours are now given by curves
of constant ˆ
q 2 = β 0 y for points outside the disk and by curves of constant ˆ
q 2 =
β 0 y +( ˆ
Fγf 0 )/(2β 0 D)(R 2 − x 2 − y 2 ) for points inside the disk.
When we write y 0 = β 2
0 D/(γf 0 ˆ
F ), the expression for ˆ
q 2 within the disk can
be written as
ˆ
q 2 =
β 0
2y 0
(R
2 + y
2
0 − x
2 − (y − y 0 )
2 ).
(15.16)
Hence the geostrophic contours are only deformed within the disk and two examples (y 0 /R =2and y 0 /R =1/4) are shown in Fig. 15.6b and Fig. 15.6c, respectively. When the center of the circular arc y 0 lies outside of the disk (y 0 >R), the
curves remain open circular arcs and connect to the constant latitude lines outside
the disk. However, when y 0 are isolines which are closed and which are not connected to the constant latitude
lines.