Geoscience Reference
In-Depth Information
The boundary conditions for heat and salinity transport are similar to those for sed-
iment transport. In an inflow boundary, the values of water temperature and salinity
should be specified. In a wall boundary, the gradient of salinity in the direction normal
to the wall is specified as zero. If the heat exchange across the wall is not considered,
the gradient of water temperature in the direction normal to the wall is zero; however,
a general heat exchange flux model can be applied at the wall boundary. In an outflow
boundary, the gradients of temperature and salinity in the flow direction can be set
as zero.
12.1.2 Effects of buoyancy on vertical turbulent
transport
The vertical turbulent transport of mass, momentum, and heat is strongly influenced by
buoyancy effects; in particular, the eddy viscosity and diffusivity are reduced by stable
stratification (Rodi, 1993). To account for the buoyancy effects, damping functions
are usually applied to the eddy viscosity and diffusivity:
)
α
2
ν
=
ν
(
1
+
α
1
Ri
(12.13)
t
t
0
)
β
2
ε
=
ε
t
0
(
1
+
β
1
Ri
(12.14)
t
where Ri is the gradient Richardson number:
g
ρ
∂ρ/∂
z
=−
Ri
(12.15)
(∂
/∂
)
2
U
z
which is the ratio of gravity to inertial forces and characterizes the importance of
buoyancy effects.
t
0
are the eddy viscosity and diffusivity, respectively, for
the neutrally stratified flow (Ri
ν
t
0
and
ε
=
0). According to Munk and Anderson (1948), the
values of coefficients
1.5.
From Eqs. (12.13) and (12.14), the effect of buoyancy on the turbulent Prandtl/
Schmidt number
α
=
10,
α
=−
0.5,
β
=
3.33, and
β
=−
1
2
1
2
t
can be determined.
The mixing length is also altered by buoyancy effects. For the stably stratified flow
(Ri
σ
=
ν
/ε
t
t
>
0), the following Monin-Obukhov relation is mostly used:
l
m
=
l
m
0
(
1
−
γ
1
Ri
)
(12.16)
where
l
m
0
is the mixing length for the neutrally stratified flow, and
1
is a coefficient
ranging from 5 to 10 and having a mean value of about 7 (Busch, 1972). For the
unstably stratified flow (Ri
γ
<
0), the following relation is usually employed:
)
−
1
/
4
l
m
=
l
m
0
(
1
−
γ
2
Ri
(12.17)
with
γ
2
≈
14 (Busch, 1972).
