| Case 2.Wind-induced Waves
over Slope Bottom Topography |
1. Analytical Solution
Considering an idealized sloping-bottom circular lake
with water depth given as
where  is the radius
of the lake and  is
the water depth at the center (Fig. 1).The governing equations
shown in Case 1 are applied to this case and an analytical
solution was derived by Birchfield and Hickie (BH) (1977).
The solution was in the form of a complicated double series
characterized by three waves: two for a gravity wave pair
and one for a topographic wave. As an initial value problem,
the waves are dominated by the first 4 radial modes in the
first model day, but higher modes must be taken into account
for a longer time period.
|
 Fig.
1: The illustration of an idealized, slopping-bottom,
circular lake |
| 2. Design of the Numerical Experiment
Unstructured triangular and structured rectangular
grids used for FVCOM and POM/ECOM-si are similar to those shown in
Case 1 except for a radius of 100 km and a maximum depth of 100 m.
The analytical solution is derived with a lateral boundary condition
at  . This condition is allowed in the  -coordinate transformation system used in FVCOM and
POM/ECOM-si until a wet/dry treatment technique is incorporated. In
order to make these models run for the case with this idealized
sloping bottom, a depth of 0.5 m is added everywhere in  , with an understanding that the numerical solution might
lead to a bias of BH’s analytical solution for a long-time run. For
this reason, the comparison of the models with the analytical
solution is limited in the time scale of less than 1 day starting
from the initial. |
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3. Results
In the first model day, both FVCOM and POM
produce a reasonable simulation of the elevation composed of
the first 4 radial modes of the analytical solution, although
POM shows a small perturbation at the center at the first
model hour and a biased maximum value away from the coast at
the end of the 1st model day (Fig. 2).
Significant difference between FVCOM and POM
occur after the 2nd model day. At the end of the 5th model
day, for example, FVCOM with a horizontal resolution of 5 km
shows a symmetridistribution of the elevation over the
circular lake, with a maximum positive value on the right
coast and a minimum negative value on the left coast (Fig. 3).
|
 Fig. 2: Comparison of the spatial
distributions of the surface elevation at the end of 1st hour
and 1st day between analytical solution, FVCOM and POM.
Horizontal resolution is 5 km for FVCOM and 2.5 km for
POM. |
|
Although POM with a horizontal
resolution of 2.5 km also shows a similar pattern,
multiple artificial eddies occur at the transition
region from positive elevation to negative elevation and
a series of small-scale perturbation structures of the
elevation are detected along the coast on both sides of
the wall.
This difference is clearly evident in
model-computed transport. FVCOM shows a beautiful spiral
circulation pattern along the coast and a double cell
circulation in the interior, while POM-computed
circulation is characterized by multiple eddies in the
transition region of the elevation and significant
divergence flow close to the coast. |
 Fig. 3: Comparison of the spatial
distributions of the
surface elevation at the end of 5th
day between FVCOM
and POM. Horizontal resolution is 5 km
for FVCOM and
2.5 km for POM.
Click here for animations of “elevation”
and “currents”
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It is clear that eddies predicted by POM
are numerical noises due to inaccurate fitting of the
curvature coastline of the circular lake, since the size
and strength of these eddies reduce as horizontal
resolution increases. Again, theoretically these
numerical noises could disappear as horizontal
resolution increases to a certain level, but only at the
sacrifice of impractical computational
efficiency. | | |
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