1.
Design of the Numerical Experiment
The  -errors
in POM and ECOM-si have been examined both theoretically
and numerically by Mellor et al. (1994) and Chen and Beardsley
(1995). Mellor et al. pointed out that these errors in POM
could retain at a certain level if sufficient high horizontal
and vertical resolutions are used. Chen and Beardsley found
that at a fixed location over a sloping bottom -errors
in ECOM-si increases with water depth. Since the buoyancy
velocity error caused by the -transformation
always opposes the velocity in the thermal boundary layer
on the bottom of the slope, this error can be suppressed
if the vertical resolution is high enough to resolve the
thermal current near the bottom.
The experiments shown here
are focused on the influence of the horizontal resolution
used in FVCOM and POM on the thermal boundary layer on steep
topography. POM uses the simplified temperature bottom condition
of  ,
while FVCOM uses the completed exact temperature bottom
condition. To compare FVCOM with POM, we ran FVCOM for the
two separate cases: one with the same bottom boundary condition
as POM and the other with the exact condition.
Geometry of the
circular lake:
R = 78 km (radius),
H = 300 m (maximum water depth); L
= 15 km (width of the shelf), and α= 0.02 (slope at
the shelf break).
The initial temperature
distribution:
 |
(5.1) |
where .gif)
is the surface water temperature given as 15o C, .gif)
is the bottom water temperature given as 6oC, and H is the
maximum water depth specified as 300 m.
Forcing:
Km= 10-4 m2/s (background mixing).
2.
Results
For given 31 -levels
in the vertical, the structures of the thermal boundary
current predicted by POM and FVCOM depend on the horizontal
resolution used in numerical computation (Fig. 4).
For the case with a horizontal resolution of 4.5 km,
for example, both POM and FVCOM develop the multiple
layer along-lake boundary currents with time. The
scale of these currents are the same as the width
of the shelf, and the along-shelf and vertical velocities
reach 1 cm/s and 3 X 103 cm/s, respectively
after the 10 model days. The only difference is that
FVCOM-predicted currents are symmetrically distributed
to the center of the lake, while POM-predicted currents
do not. |
Fig.
2: Comparison of the along-isobath and vertical
velocities at the end of 10th model day between
FVCOM and POM. |
There are no physical reasons supporting
an asymmetric distribution of the current across the lake
for POM, since the numerical grids are composed of squares
which are symmetrically distributed around the circular
lake, and also the mixing coefficient is the same everywhere.
The fact that the asymmetric pattern of the currents mainly
occur near the surface around the coast implies that POM
has a limitation when applied to a very shallow region.
This is probably due to the time-and space-smoothing program
used in the code and also to the limitation in geometric
fitting.
The pattern
of multiple layer currents disappears in both the
POM and FVCOM results as horizontal grid size decreases
to 2.25 km (Fig. 3). The currents predicted by these
two models are characterized with a two-layer flow:
one is near the surface and rotates cyclonically along
the coast, and the other is limited to a very thin
thermal boundary layer above the bottom of the slope
and rotates anti-cyclonically around the lake. The
maximum speed of the along-lake velocity within the
thermal boundary layer is about 0.6 cm/s. An upwelling,
with a vertical velocity of about 1 X 103
cm/s is found in the thermal boundary layerThese current
patterns are consistent with the theory derived by
Wunsch (1970). This theory suggests that a background
mixing tends to produce an along-isobath thermal current.Facing
downstream in the direction of the current, the lighter
water is always on the observer’s left. . |
Fig.
3: Comparison of the along-shelf and vertical
velocities at the end of the 10th model
day between FVCOM and POM for the idealized
circular lake. The horizontal resolution
for FVCOM and POM in this case is 2.25
km. Contour interval is 0.1 cm s-1 for
u and 0.4´10-3 cm s-1 for w.
|
Also, when the pressure gradient forcing
moves the water in the interior into the boundary layer,
the thermal diffusion tends to reduce it density and hence
cause it to upwell along the slope.
Our numerical experiments imply that both
POM and FVCOM could resolve the structure of the thermal
boundary layer over the sloping bottom topography at a certain
horizontal resolution. In spite of it, FVCOM utilizes unstructured
grids which provide more flexibility to increase the local
horizontal resolution at the shelf break. Also, the POM-predicted
currents are not symmetrically distributed in the shallow
water close to the coast, even when the horizontal resolution
is increased. This issue needs to be fixed when applying
POM to a very shallow water region.
It should be pointed out here that the model
results shown above are all based on the bottom boundary
of .
This simplification may lead to an overestimation of the
vertical mixing and hence horizontal and vertical velocities
over the steep slope bottom like shelf break. For a given
horizontal resolution of 2.25 km, the currents at the 10th
model day predicted by FVCOM with an exact no flux normal
to the slope are about one order of magnitude weaker than
those described above (Chen et al., 2004). These results
suggest that an inaccurate use of the bottom boundary condition
of temperature exaggerates vertical mixing and thus overestimates
both along-isobath and vertical velocities within the thermal
boundary layer over the slope. |
