364
DYNAMICAL OCEANOGRAPHY
◮
Example 15.2: Localized Ekman pumping: solution
Ex. 15.1
We now continue the analysis of the situation of localized Ekman pumping as
in Example 15.1. Assume that y 0
and hence the value of ˆ
q 20 = β 0 R. The outermost geostrophic contour in the
lower layer (where ψ 2 =0) is therefore given by (from (15.16)) by
β 0 R =
β 0
2y 0
(R
2 + y
2
0 − x
2 − (y − y 0 )
2 ) → x
2 +(y − y 0 )
2 =(R − y 0 )
2 . (15.30)
The streamfunctions in both layers follow from (15.29) and (15.10)
ψ 2 (x, y)=
DA f
ǫ 0 H 1 + DA f
(ψ B + β 0
y − R
ˆ
F
),
(15.31a)
ψ 1 (x, y)=
Dψ B (x, y) − H 2 ψ 2 (x, y)
H 1
.
(15.31b)
For ǫ 0 =0(no bottom friction) and y 0 =1 /4,R =1 , both streamfunctions
ψ 1 and ψ 2 are plotted in Fig. 15.8. In layer 2 (Fig. 15.8b), one observes that the
circulation is limited to the region of closed geostrophic contours. The explicit
solution for ψ 2 within the disk for this case is given by
ψ 2 (x, y)=
β 0
ˆ
F
1
2y 0
(R
2 − x
2 − y
2 )+y − R)
,
(15.32)
and the streamlines are circles with center (0,y 0 ). The solution for ψ 1 in
Fig. 15.8a is more complicated as it is composed of part of the Sverdrup solution (which is nonzero over the forcing disk) and part of the solution ψ 2 . Outside
Ex. 15.2
of the regions of closed contours (the dark contour in Fig. 15.8a), the solution is
just the Sverdrup solution weighted by a factor D/H 1 . At the intersection of the
forcing region and that of the outermost closed contour, jumps occur in the upper
layer streamfunction.
◭
Ex. 15.3
While we have totally neglected bottom topography in this section, it is clear
that the addition of bottom topography in the quasi-geostrophic case easily leads
to closed contours as it is part of the potential vorticity q 2 . The material in
this section has illustrated that subtle effects may control bottom layer currents
once geostrophic contours become closed. To investigate the bottom currents
in the Arctic, the quasi-geostrophic approximation is questionable because the
DYNAMICAL OCEANOGRAPHY
◮
Example 15.2: Localized Ekman pumping: solution
Ex. 15.1
We now continue the analysis of the situation of localized Ekman pumping as
in Example 15.1. Assume that y 0
and hence the value of ˆ
q 20 = β 0 R. The outermost geostrophic contour in the
lower layer (where ψ 2 =0) is therefore given by (from (15.16)) by
β 0 R =
β 0
2y 0
(R
2 + y
2
0 − x
2 − (y − y 0 )
2 ) → x
2 +(y − y 0 )
2 =(R − y 0 )
2 . (15.30)
The streamfunctions in both layers follow from (15.29) and (15.10)
ψ 2 (x, y)=
DA f
ǫ 0 H 1 + DA f
(ψ B + β 0
y − R
ˆ
F
),
(15.31a)
ψ 1 (x, y)=
Dψ B (x, y) − H 2 ψ 2 (x, y)
H 1
.
(15.31b)
For ǫ 0 =0(no bottom friction) and y 0 =1 /4,R =1 , both streamfunctions
ψ 1 and ψ 2 are plotted in Fig. 15.8. In layer 2 (Fig. 15.8b), one observes that the
circulation is limited to the region of closed geostrophic contours. The explicit
solution for ψ 2 within the disk for this case is given by
ψ 2 (x, y)=
β 0
ˆ
F
1
2y 0
(R
2 − x
2 − y
2 )+y − R)
,
(15.32)
and the streamlines are circles with center (0,y 0 ). The solution for ψ 1 in
Fig. 15.8a is more complicated as it is composed of part of the Sverdrup solution (which is nonzero over the forcing disk) and part of the solution ψ 2 . Outside
Ex. 15.2
of the regions of closed contours (the dark contour in Fig. 15.8a), the solution is
just the Sverdrup solution weighted by a factor D/H 1 . At the intersection of the
forcing region and that of the outermost closed contour, jumps occur in the upper
layer streamfunction.
◭
Ex. 15.3
While we have totally neglected bottom topography in this section, it is clear
that the addition of bottom topography in the quasi-geostrophic case easily leads
to closed contours as it is part of the potential vorticity q 2 . The material in
this section has illustrated that subtle effects may control bottom layer currents
once geostrophic contours become closed. To investigate the bottom currents
in the Arctic, the quasi-geostrophic approximation is questionable because the