Geoscience Reference
In-Depth Information
Figure 4.12
Variation of
horizontal eddy diffusivity with
the scale of the diffusing dye
patch (from (Okubo,
1971
), with
permission from Elsevier).
10
5
10
4
4
L
3
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
10 m
100 m
1 km
10 km
100 km
1000 km 10000 km
Horizontal length scale
L
oscillating currents in the shelf seas by Bowden (Bowden,
1965
). The mathematical
theory is rather intricate and will not be rehearsed here, but the interested reader can
find an accessible account of the theory in (Fischer et al.,
1979
). If you do not want to
pursue the detail of the theory (and even if you do), you can get a visual picture of
how shear diffusion works from numerical simulations in the associated software
The important result of the theory is that the dispersion by the shear-diffusion
mechanism of a passive tracer of concentration s can be represented by a horizontal
flux in the Fickian form:
K
x
@
s
K
y
@
s
G
¼
x
;
:
ð
4
:
45
Þ
@
@
y
If K
z
is the vertical eddy diffusivity, the time to mix the water column of depth h is
T
m
h
2
/K
z
. Providing T
m
≪
T
2
the semi-diurnal tidal period, the components of the
diffusivity K
x
and K
y
are determined by
c
s
u
T
T
m
;
c
s
v
T
T
m
K
x
¼
K
y
¼
ð
4
:
46
Þ
where u
T
and v
T
are the amplitudes of the surface to bottom tidal velocity differences in
the x and y directions and c
s
is a numerical factor depending on the form of the velocity
profile. Dispersion increases with T
m
and is thus inversely proportional to the vertical
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