1.
Design of the Numerical Experiment
The experiments have been conducted for
idealized shelves with 1) a linear slope and a straight
coastline and 2) a linear slope with a curvature coastline.
Case
1:
The
straight coastline
| The
freshwater discharge rate
Q = 1000 m3/s and the background
salinity S = 30 PSU. Horizontal resolution:
20 km; vertical resolution: 10 sigma-levels.
The open boundary is located at
800 and 1400 km downstream from the origin for FVCOM
and POM, respectively, at which a gravity wave radiation
boundary is specified to allow the wave energy to
propagate out of the computational domain with minimum
reflection. |
Fig.
1: The illustration of an idealized, straight-coastline
continental shelf used for the freshwater
discharge case.
|
In FVCOM, when the water elevation
at the open boundary is determined by a radiation boundary
condition the velocity in the boundary triangular cells
could be calculated based on the mass conservation.This
approach causes little reflection at the open boundary,
and open boundary at 800 km away from the origin is far
enough to remain numerically accurate in the computational
domain. POM is numerically configured with C-grids, in
which elevation and cross-isobath velocity component are
at the same boundary line, while the velocity normal to
the open boundary is located at cells half a grid from
the boundary.Although the velocity component normal to
the open boundary could be determined using a simplified
liner momentum equation, the elevation and velocity component
paralleled to the open boundary must be determined by
radiation boundary condition. This approach can not guarantee
the mass conservation at the boundary cell, making it
difficult to filter out all the reflected waves, especially
for the stratified case. To ensure minimum influence of
the open boundary on the numerical solution in the computational
domain over the time scale in which we are interested,
the open boundary for POM is moved to 1400 km away from
the origin.
|
Case
2:
The
circular coastline
Fig.
2:
The illustration of an idealized, circular lake with
a linear slope of the continental shelf. |
The slope of the shelf is given as
 |
(4.1) |
where test.gif)
is the water depth at the distance of r from
the origin of the circle, R is the radius of the
circular lake, r0 is the distance from
the origin of the circle to the edge of the shelf, 
is the water depth at R, Hd
and is the water depth in the region of << r.
Freshwater discharge rate Q = 1000 m3/s,
injected into the lake from the mesh at the center point
of the southern coast. The background salinity S = 35 PSU.
2.
Results
a)
The straight coastline
Fig.
3: Comparison of the spatial distributions
of the water elevation (left) and surface salinity plus
current vectors (right) at the end of the 5th model
day between FVCOM and POM. In this case, the salinity
was specified as zero at the point source where the
freshwater was injected. Click
here to view the animation. |
Given the same freshwater discharge rate,
the distributions of elevation, currents, and salinity computed
by POM and FVCOM are significantly different (Fig. 3). The
buoyancy current computed by FVCOM is characterized by a
relatively strong clockwise circulation out of the mouth
of the freshwater source and a coastal trapped plume along
the shelf, while the current computed by POM is most equally
divided towards the upstream (left) and downstream (right)
along the coast: no distinct clockwise circulation forms
out of the mouth of the freshwater source. As a result of
large backward advection, the salinity plume predicted by
POM moves downstream at a slower speed than that shown in
FVCOM. Similar differences are also found in the distributions
of the water elevations: POM shows a large gradient of the
water elevation in the upstream region and a slower downward
movement of the trapped wave, which is consistent with the
distributions of salinity and currents predicted by this
model. For a same given discharge rate, FVCOM tends to produce
a larger cross-shelf gradient of elevation and salinity
in the downstream region (for example, 100 km from the freshwater
source) than POM, even though the cross-shelf scale of the
plume is the same for both models.
The comparison between FVCOM and POM for
this idealized case clearly shows that the structure of
the model-predicted plume depends on the numerical schemes
used in these models. The public-available code of POM uses
a central difference scheme for advection terms of salinity.
Although this method ensures the second-order accuracy,
it produces an artificial backward transport against the
direction of the current. This is the reason why POM shows
a more significant upstream-ward transport and a slower
downstream-ward propagation speed of the plume. Therefore,
the results obtained from POM for the river discharge-induced
buoyancy flow must be interpreted with caution because of
the deficiency of a central difference scheme for tracer
advection. FVCOM uses a second-order accurate upwind numerical
scheme for salinity advection. This scheme not only ensures
the salinity conservation in the individual TCE but also
avoids the occurrence of artificial backward transport as
that shown in the central difference scheme. Under a condition
with the same second-order cutoff, the upwind scheme is
better than the central difference scheme for the tracer
simulation.
b)
The circular coastline
| 
Fig.
4: Comparison of the spatial distributions
of the surface salinity and water elevation at the
end of the 10th model day between FVCOM and POM for
the cases with horizontal resolutions of 4.22, 1.78
and 0.89 km, respectively. Click
here to view the animation
|
For the case with a horizontal resolution
of 4.22 km, the low-salinity plume predicted by FVCOM occupies
the entire shelf with a cross-shelf scale as the same as
the width of the shelf, while the plume predicted by POM
extends over the interior region off the shelf, forming
a detached eddy-like circulation at the outer edge of the
shelf after 10 model days. As the horizontal grid size reduces
to 1.78 km, the plumes predicted by FVCOM and POM shift
toward the coast. A quasi-equilibrium state occurs as the
horizontal grid size is smaller or equal to 0.89 km. These
results provide us with two important facts. Firstly, attention
must be paid to the horizontal resolution in simulating
the spatial structure of the low-salinity plume in the inner
shelf of the coastal ocean, no matter which model is used.
Secondly, the finite-difference model might lose the shelf-controlling
nature of the low-salinity plume at a certain lower horizontal
resolution and thus produce an artificial eddy formation
over the shelf. This issue must be taken into account when
applying a finite-difference model to simulate the low-salinity
plume in the realistic inner shelf of the coastal ocean.
Unstructured triangular grids used in FVCOM
provide an accurate coastline matching with a guarantee
of no mass flux on the coastal wall. In the downstream of
the freshwater source, the plume water predicted by FVCOM
flows along the curvature coastline, with a maximum water
level and along-shelf current at the coast. The square grids
used in POM, however, result in a step-like coastal boundary
in the numerical computational domain. Since no flux condition
is applied in a direction normal to the step-shape boundary,
the maximum along-shelf current occurs at a distance away
from the coast. This step-shape coastal model boundary acts
like a drag force to slow down the downstream movement of
the low-salinity plume and to exaggerate the cross-shelf
secondary current within the plume.
Note: as discussed in the case with the
straight coastline, the central difference scheme used to
calculate the salinity advection in POM can cause a significant
backward water transport on the left side of the freshwater
source. When a curvature coastline is applied, however,
the numerical drags caused by a mismatching of the curvature
coastal geometry not only reduces the downstream-ward propagation
speed of the low-salinity plume on the right side of the
freshwater source, but also decreases the backward buoyancy
flow on the left side of the freshwater source. It might
provide a good agreement with the observations made with
wrong physical reasons.
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