40
Homogeneous Models of the Ocean Circulation
1.0 ~-~-~----,---r----r-----r--r------,
0.8
0.6
0.2
0
-{).2
--Q.4 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
x;r,m
Fig. 2.7.2. Profiles of v (scaled
by 1/J r/ 8 M) and vorticity (scaled
by 1/JJ/8~) for the Munk
boundary layer with no slip
boundary conditions
that is, the flux of relative vorticity flows down the vorticity gradient. Thus, at
the wall, positive (cyclonic) vorticity is diffusing into the basin, and this is
happening to fluid elements all along their trajectory in the western boundary
current. This is necessary to achieve the vorticity balance that we shall discuss
more completely in following sections, but we can already appreciate the
process in the Munk model. Fluid which enters the western boundary layer
from the interior in the south, say at y = y., has a smaller total vorticity,
fo + py., than fluid leaving the boundary layer in the northern part of the gyre
at y = Yn and entering the interior where the planetary vorticity isfo + PYn· The
action of the wind in driving the fluid southward continuously contributes
negative vorticity to the fluid, and the resulting southward motion is the
Sverdrup theory's response to that negative vorticity input. As we have seen,
the wind vorticity source is negligible in the boundary layer (the flow is too
swift for the direct vorticity input to matter). However, when the fluid rejoins
the interior in the north, it must have the same planetary vorticity (which is its
total vorticity in the interior) as its environment, i.e., fo + PYn· Thus in the
boundary layer the fluid must gain positive vorticity in the amount P(Yn - Ys)This vorticity enters diffusively from the western boundary.
In fact if (2.6.2) is integrated across the basin and nonlinearity and bottom
friction are ignored, as is appropriate in the Munk theory, we obtain, at each
latitude y:
1
x fo
.J
'
8( (
0 =
-wE(x ,y)dx - An- 8 0)
0 D
X
or
(2.7.4a,b)
Homogeneous Models of the Ocean Circulation
1.0 ~-~-~----,---r----r-----r--r------,
0.8
0.6
0.2
0
-{).2
--Q.4 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
x;r,m
Fig. 2.7.2. Profiles of v (scaled
by 1/J r/ 8 M) and vorticity (scaled
by 1/JJ/8~) for the Munk
boundary layer with no slip
boundary conditions
that is, the flux of relative vorticity flows down the vorticity gradient. Thus, at
the wall, positive (cyclonic) vorticity is diffusing into the basin, and this is
happening to fluid elements all along their trajectory in the western boundary
current. This is necessary to achieve the vorticity balance that we shall discuss
more completely in following sections, but we can already appreciate the
process in the Munk model. Fluid which enters the western boundary layer
from the interior in the south, say at y = y., has a smaller total vorticity,
fo + py., than fluid leaving the boundary layer in the northern part of the gyre
at y = Yn and entering the interior where the planetary vorticity isfo + PYn· The
action of the wind in driving the fluid southward continuously contributes
negative vorticity to the fluid, and the resulting southward motion is the
Sverdrup theory's response to that negative vorticity input. As we have seen,
the wind vorticity source is negligible in the boundary layer (the flow is too
swift for the direct vorticity input to matter). However, when the fluid rejoins
the interior in the north, it must have the same planetary vorticity (which is its
total vorticity in the interior) as its environment, i.e., fo + PYn· Thus in the
boundary layer the fluid must gain positive vorticity in the amount P(Yn - Ys)This vorticity enters diffusively from the western boundary.
In fact if (2.6.2) is integrated across the basin and nonlinearity and bottom
friction are ignored, as is appropriate in the Munk theory, we obtain, at each
latitude y:
1
x fo
.J
'
8( (
0 =
-wE(x ,y)dx - An- 8 0)
0 D
X
or
(2.7.4a,b)