Geoscience Reference
In-Depth Information
9.2.A.3
Eddy Viscosity
The eddy viscosity coefficient
v
t
in the water column accounts for the Reynolds
stresses (turbulence) and other unresolved processes both in time and space (e.g.,
subgrid scale fluctuations). The eddy viscosity in the model can be specified in five
ways: (1) a constant value; (2) a function of the Richardson number; (3) a time-
varying function of the local gradients in the velocity field;
23
(4) from the solution
of the turbulent kinetic energy equation, the so-called
k
-
model;
24
or (5) from the
solution of both the turbulent kinetic energy and kinetic energy dissipation equations,
the so-called
k-
model.
24
The simulations in this study are performed with the
Smagorinsky formulation where the eddy viscosity is linked to the grid spacing and
the velocity gradients of the resolved flow field as
ε
1
2
∂
∂
u
x
+
∂
∂
u
x
i
j
v
=1
2
S
S
where
S
"
⋅
⋅
ij
ji
ij
=
t
j
i
Here,
l
is a length scale replaced by the product of a constant
C
sm
and the grid spacing
∆
s
. Typical values for
C
sm
in the vertical and horizontal directions are 0.176 and
0.088, respectively.
9.2.A.4
Bottom Stress
In shallow water depths, the bottom shear stress
τ
bottom
plays a major role in con-
trolling the circulation and mixing of waters. It can be specified in terms of a drag
coefficient formulation as
τ
=
ρ
Cu u
D
*|*|
bottom
where
C
D
is a drag coefficient determined from the selected turbulence closure
scheme and
u
* is the velocity at the top of the bottom boundary layer. When using
the Smagorinsky eddy viscosity formulation,
C
D
is given by
−
2
3
2
3
2
(
)
*
+
,
-
z
D
z
22
31
D
z
D
1
log
*
-
*
-
=
C
1
−
m
−−
1
+
m
D
κ
k
/
30
s
In this equation,
z
m
is the distance above the sea bed where the Smagorinsky
profile matches the logarithmic velocity profile;
is the von Karman's constant;
z
is the distance between the sea bed and the first computational mode;
k
s
is the bed
roughness length scale (range between 0.01 and 0.30 m);
l
is the length scale given
by the Smagorinsky eddy viscosity formulation; and
D
is the actual depth.
κ
ACKNOWLEDGMENTS
This work is a Canadian contribution to the NATO Committee on the Challenges
of Modern Society (CCMS) devoted to ecosystem modeling of coastal lagoons
for sustainable management. It is funded by NATO-CCMS fellowship award
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