Geoscience Reference
In-Depth Information
Figure 3.8 Velocity variation with depth due
to a steady wind stress applied to an ocean
remote from horizontal boundaries in the
northern hemisphere. The hodograph shows
the velocity vectors at different depths seen
from above. The current decreases and
rotates clockwise with increasing depth.
Note that, at a depth of z ¼ D, where the
current vector is reversed relative to the
surface, the velocity and the frictional stress
are reduced to < 5% of their values at the
surface.
y
0.2
Wi nd stre ss
x
0
45°
-0.5
-0.2
-0.4
U s
-0.4
-0.3
-0.2
-0.6
-0.1
z/D = 0
-0.8
-1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
U/U s
0 @
1
t x
@
0 @
1
t y
@
0
¼
fv
z ;
0
¼
fu
z :
ð
3
:
39
Þ
To make further progress we need to relate the frictional shear stresses t x and t y to
other flow properties. Anticipating the discussion of frictional stresses in Section
4.3.4 , we assume that the stresses are proportional to the velocity shear components,
so that we can write:
t x ¼ 0 N z @
u
t y ¼ 0 N z @
v
z ;
ð
3
:
40
Þ
@
@
z
where N z is known as the eddy viscosity and is here assumed to be constant.
Substituting in Equation (3.39) , we have:
2 u
2 v
N z @
N z @
fv
¼
z 2 ;
fu
¼
z 2 :
ð
3
:
41
Þ
@
@
If there is a steady wind stress t w acting at the surface (z
0) in the x direction,
the solution of this pair of equations can be written as (Kundu and Cohen, 2008 ,
see p. 617):
¼
4
z
D
4
z
D
u s e z = D cos
u s e z = D sin
u
¼
;
v
¼
s
2N z
f
p
ð
3
:
42
Þ
t w
0 fD :
where D
¼
;
and u s ¼
D is a length scale called the Ekman depth, which is used as an indication of the
depth to which the influence of the surface stress penetrates. More specifically it is the
depth at which the currents have diminished to 1/e of their surface magnitude given
by u s . The profile of current vectors in this near surface region, termed the Ekman
layer, is illustrated in Fig. 3.8 . You can see that as well as decreasing away from the
surface, the current vector also rotates to the right (left) in the northern (southern)
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