Geoscience Reference
In-Depth Information
If flow is in the x direction only and we assume that density changes due to advection
are dominant, then the advection diffusion equation (
Section 4.3.3
)
simplifies to
@
@
u
@
@
t
¼
x
:
ð
9
:
16
Þ
On taking the vertical average and again assuming
@
r/
@
x is independent of z,wehave
also that:
@^
u
@
@
t
¼^
ð
9
:
17
Þ
@
x
where
u is the depth mean velocity. Substituting from Equations (
9.16
)and
(9.17)
into (
Equation 9.15
) gives us:
^
0
h
ð
ð
0
1
ð
ð
@
@
g
h
@
gh
@
@
z
0
dz
0
:
t
¼
u
^
u
Þ
zdz
¼
u
^
u
Þ
ð
9
:
18
Þ
@
x
x
This equation allows us to evaluate the change in
due to advection by any velocity
profile u(z). We shall first use it to determine the stratifying effect of the density-
driven component of the estuarine circulation of (
Equation 9.10
):
F
z
0
Þ¼
9z
0
2
8z
0
3
u
ð
u
s
ð
1
Þ:
ð
9
:
19
Þ
On substituting into Equation (
9.18
) and integrating, we have:
Est
¼
2
:
@
@
g
2
h
4
N
z
0
@
@
1
320
ð
9
:
20
Þ
t
x
If this stratifying effect is opposed only by tidal stirring, we have an analogous
competition to that between heating and stirring (
Equation 6.17
). In the present
case, stratification will be maintained or increase only if
Est
u
3
@
@
e
k
b
0
^
t
h
ð
9
:
21
Þ
2
g
2
h
4
N
z
0
u
3
M2
h
1
320
@
@
4e
3
k
b
0
^
or
x
where e is, again, the efficiency of tidal mixing and the final step involves averaging the stirring
over the tidal cycle. If we use an empirical formula for the eddy viscosity N
z
¼
10
3
u
M2
h
(Bowden et al.,
1959
), we can further simplify our criterion for the development of stratifica-
tion to give the condition on the density gradient for stratification to develop as:
3
^
p
0
2
2
0
@
1
:
4ek
b
^
u
M2
h
u
M2
h
^
10
4
x
2
:
1
ð
9
:
22
Þ
@
g
using k
b
¼
0.004. This criterion for the development of water column
stability is somewhat analogous to
Equation (6.17)
, but there is an important
0.0025 and e
¼
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