Geoscience Reference
In-Depth Information
E
sa
,
x
b
∂
E
sa
,
z
b
∂
∂(
bC
sa
)
∂
+
∂(
bU
x
C
sa
)
∂
+
∂(
bU
z
C
sa
)
∂
=
∂
∂
+
∂
∂
C
sa
∂
C
sa
∂
t
x
z
x
x
z
z
(12.7)
where
T
and
C
sa
are the width-averaged temperature and salinity, respectively;
E
T
,
i
and
E
sa
,
i
are the effective diffusivities (mixing coefficients) of heat and salinity in the
longitudinal section, respectively; and
q
ni
are the heat fluxes per unit of bank
surface area due to conduction, seepage flow, etc., across the two banks.
The depth-integrated 2-D heat and salinity transport equations are
(
i
=
1, 2
)
E
T
,
x
h
∂
E
T
,
y
h
∂
∂(
hT
)
+
∂(
hU
x
T
)
+
∂(
hU
y
T
)
=
∂
∂
T
+
∂
∂
T
J
T
ρ
+
∂
t
∂
x
∂
y
x
∂
x
y
∂
y
c
p
(12.8)
E
sa
,
x
h
∂
E
sa
,
y
h
∂
+
∂(
hU
y
C
sa
)
∂(
hC
sa
)
+
∂(
hU
x
C
sa
)
=
∂
∂
C
sa
∂
+
∂
∂
C
sa
∂
∂
t
∂
x
∂
y
x
x
y
y
(12.9)
where
T
and
C
sa
are the depth-averaged temperature and salinity, respectively;
E
T
,
i
and
E
sa
,
i
are the horizontal effective diffusivities of heat and salinity, respectively; and
J
T
is the net flux across the water and bed surfaces.
The 1-D heat and salinity transport equations are
BJ
T
q
ni
E
T
,
L
A
∂
2
∂(
)
+
∂(
)
=
∂
∂
AT
QT
T
1
+
−
1
¯
m
i
h
¯
(12.10)
∂
t
∂
x
x
∂
x
ρ
c
p
i
=
E
sa
,
L
A
∂
∂(
AC
sa
)
+
∂(
QC
sa
)
=
∂
∂
C
sa
∂
(12.11)
∂
t
∂
x
x
x
where
T
and
C
sa
are the temperature and salinity averaged in the cross-section, respec-
tively;
E
T
,
L
and
E
sa
,
L
are the longitudinal effective diffusivities of heat and salinity,
respectively;
¯
are the average heat fluxes per unit bank surface area due to
conduction, seepage flow, etc., across the two banks; and
q
ni
(
i
=
1, 2
)
m
i
¯
(
i
=
1, 2
)
are the ratios
of the wetted bank slope lengths to the flow depth.
Water density varies with temperature and salinity. This can be described by the
equation of state (Crowley, 1968):
0.00469
T
2
ρ
=
1000
+
(
28.14
−
0.0735
T
−
)
+
(
0.802
−
0.002
T
)(
C
sa
−
35
)
(12.12)
m
−
3
,
C
sa
is in ppt, and
T
is in
◦
C.
where
ρ
is in kg
·
