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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