Geoscience Reference
In-Depth Information
g
z
s
y
dz
u
y
)
∂
∂(
∂
+
∂(
u
x
u
y
)
∂
+
∂(
u
z
u
y
)
∂
∂τ
yx
∂
u
y
∂
1
ρ
0
g
∂
z
s
z
∂ρ
1
ρ
+
=−
ρ
+
+
∂
∂
t
x
y
z
y
x
∂τ
∂τ
1
ρ
1
ρ
yy
yz
+
+
−
f
c
u
x
(12.3)
∂
y
∂
z
where
0
is the water density at the water surface, and
f
c
is the Coriolis coefficient.
As a simplified approximation, the width-averaged 2-D model can also be used to
study the stratified flows due to heat and salinity transport in rivers and reservoirs.
The width-averaged 2-D hydrodynamic equations are (2.122)-(2.124) in general.
However, for the vertically well-mixed water bodies, the effect of stratification is
negligible and, thus, the 1-D and depth-averaged 2-D models are applicable. In such
cases, the depth-averaged 2-D shallow water equations are (2.119)-(2.121) and the
1-D equations are (2.126) and (2.127) without the bed change terms. In analogy to
Eq. (12.1), the 1-D and 2-D continuity equations (2.119), (2.122), and (2.126) can be
simplified by ignoring the temporal and spatial variations. The resulting 1-D and 2-D
hydrodynamic equations can also be derived by integrating Eqs. (12.1)-(12.3) over
the cross-section, depth, and width of flow, respectively. The details are left to the
interested reader.
The 3-D heat transport equation is
ρ
ε
∂
T
+
∂(
u
x
T
)
+
∂(
u
y
T
)
+
∂(
u
z
T
)
=
∂
∂
T
,
x
∂
T
+
∂
∂
T
,
y
∂
T
ε
∂
t
∂
x
∂
y
∂
z
x
∂
x
y
∂
y
+
∂
∂
T
,
z
∂
T
q
T
ρ
ε
+
(12.4)
z
∂
z
c
p
◦
C),
where
T
is the local temperature (usually in degree Celsius,
are
the turbulent diffusivities of heat,
c
p
is the specific heat, and
q
T
is the heat source rate
per unit volume.
The 3-D salinity transport equation is
ε
(
i
=
x
,
y
,
z
)
T
,
i
ε
+
∂(
u
y
C
sa
)
∂
∂
C
sa
∂
+
∂(
u
x
C
sa
)
+
∂(
u
z
C
sa
)
=
∂
∂
sa
,
x
∂
C
sa
∂
+
∂
∂
sa
,
y
∂
C
sa
∂
ε
t
∂
x
y
∂
z
x
x
y
y
ε
+
∂
∂
sa
,
z
∂
C
sa
∂
(12.5)
z
z
where
C
sa
is the local salinity (usually in parts per thousand, ppt), and
ε
(
i
=
x
,
y
,
z
)
sa
,
i
are the turbulent diffusivities of salinity.
The width-integrated 2-D heat and salinity transport equations are
E
T
,
x
b
∂
E
T
,
z
b
∂
∂(
bT
)
+
∂(
bU
x
T
)
+
∂(
bU
z
T
)
=
∂
∂
T
+
∂
∂
T
∂
t
∂
x
∂
z
x
∂
x
z
∂
z
bq
T
m
i
q
ni
2
1
ρ
+
−
(12.6)
c
p
i
=
1
