Antarctic Circumpolar Current
337
conditions, we find at every ‘longitude’ y
=
L
0
¯
vdx=
L
0
0
−H
vdz
dx =0
(14.14)
Using an integration of the momentum equation (14.11a) over different intervals in the vertical, we can determine the different contributions to .I n -
tegration of (14.11a) over the depth of the Ekman layer (on a contour y = y c )
gives
−f E =
L
0
0
−δ E
(−fv) dz
dx =
L
0
τ x
ρ 0
dx.
(14.15)
The pressure gradient vanishes due to the periodic boundary conditions and the
stress term at z = −δ E can be neglected outside the Ekman layer. The equation
(14.15) is the integrated meridional Ekman transport along the contour y = y c .
In the southern hemisphere, with f<0 and a zonal wind τ x > 0 it follows that
E > 0 and hence the transport is northward.
Integration of (14.11) over [−H 0 , −δ E ] and over a contour y = y c gives
−f I =
L
0
−δ E
−H0
(−fv)dz
dx =0.
(14.16)
as there are no contributions from the stress in this region and the pressure gradient
again vanishes due to the periodic boundary conditions. In this internal region
there cannot be a geostrophic flow and hence I =0 .
h 0
z = -D + h
b
z = 0
z = - D
z = - δ
E
z = - H 0
z
x
x = 0
x = x
x = L
m
Figure 14.6. Sketch of the topography (14.10) with the vertical structure of the flow domain.
Integration of (14.11) over [−H(x, y), −H 0 ] and a contour y = y c gives
−f G =
L
0
−H0
−H
(−fv)dz
dx =