Geoscience Reference
In-Depth Information
Figure 3.6
Application of the
geostrophic method to calculate
the relative velocity C
1
at the
isobar p
1
between two stations
A and B, with measured density
profiles r
A
and r
B
.
A
B
p
0
h
1,A
h
1,B
C
1
p
1
θ
1
level
L
where for the final step we have used the hydrostatic approximation (3.15). This
means that the pressure gradient is rg
isobaric slope. Combining with the geo-
strophic relation
(Equation 3.18)
and setting r
¼
r
0
(a fair approximation here), we
have for the geostrophic current:
1
0
@
p
g
f
tan
C
g
¼
n
¼
:
ð
3
:
25
Þ
f
@
So isobaric slopes are readily transformed into estimates of the geostrophic flow
magnitude C
g
by using
Equation (3.25)
, which is known as the gradient equation.
In the northern hemisphere (f positive), the flow is in a direction 90
anticlockwise
from the pressure gradient vector; in the southern hemisphere, where f is negative, the
flow direction is 90
clockwise from the pressure gradient. Note that, since the ocean
surface is an isobar,
Equation (3.23)
for barotropic flow is a special case of the more
general relation between velocity and isobaric slope represented by the gradient
Equation (3.25)
.
Consider the practical application of the gradient equation, shown in the schematic
of
Fig. 3.6
. If the density profile r has been measured at two stations A and B, the
depth h
1
of an isobar p
1
at A and B can be determined by integrating the hydrostatic
relation, e.g. at station A:
ð
p
1
dp
A
g
h
1A
¼
ð
3
:
26
Þ
0
and similarly for station B. If the stations are separated by a distance L, the slope of
the p
1
isobar relative to the surface between A and B will be:
h
1A
h
1B
tan
1
¼
ð
3
:
27
Þ
L
from which the velocity, relative to the surface and normal to the section A-B, can be
derived using the gradient equation. Application of this approach to other isobars
allows us to build up a profile of the geostrophic velocity relative to the surface and
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