Geoscience Reference
In-Depth Information
z
A
τ
2
z
2
τ
1
Figure 8.3. Frictional stresses on a
parcel of air or water.
z
1
x
be the net stress exerted on a parcel of water with density
∆= −
ττ τ
x
() ()
z
x
z
x
2
1
r
W
, thickness
∆= − , and top and bottom surface area
A
. In Eqs. 8.2 and
8.3,
F
x
and
F
y
are defined as forces per unit mass, so
ZZ Z
2
1
2
2
force
force
area
A
1
τ
x
.
x
x
F
+
+
+
τ
∆
(8.4)
"
F
mass
area
mass
ρ
∆
zA
ρ
z
W
W
Similarly,
2
1
τ
y
.
y
F
=
(8.5)
F
ρ
2
z
W
With these expressions for friction, the equations of motion (Eqs. 8.2 and 8.3)
yield Ekman current velocities of
1
2
τ
x
v
=−
(8.6)
E
2
z
ρ
f
W
and
2
1
τ
y
.
u
=
(8.7)
E
2
z
ρ
f
W
Equations 8.6 and 8.7 indicate that the horizontal components of the Ekman
current velocity (
u
E
,
v
E
) depend on the vertical structure of the frictional stress.
If we assume that density is constant through the mixed layer, these expressions
for
u
E
and
v
E
can easily be integrated through the depth of the mixed layer to
find the vertically integrated Ekman velocity,
v
defined by
0
0
0
v
#
#
#
t
t
t
t
v
Vv dz
/
=
udzi
+
vdzj Ui Vj
/
+
,
(8.8)
E
E
E
−
h
−
h
−
h
where
h
is the mixed-layer depth. Performing the integration from the bot-
tom of the Ekman layer
(
=− == to the ocean surface
(
z
===+
,
zh
x
,
τ
0,
τ
y
0)
vv
t
x
y
t
, with the assumption that density is constant, we obtain
===+
ττ τ
i
τ
j
)
S
S
S
0
y
τ
y
1
2
τ
#
,
S
U
=
z
dz
=
(8.9)
2
ρ
f
ρ
f
W
W
−
h
