| Case 7: The Wet/Dry
Point Treatment Method |
The materials include below comes from the manuscript
written recently by Chen et al. (2004) entitled “A 3-Dimensional,
Unstructured Grid, Finite-Volume Wet/Dry Point Treatment Method
for FVCOM”. For details, please contact Dr.
Chen at UmassD.
|
Background
The scientific team including C. Chen, H. Liu, J.
Qi, G. Cowles, H. Lin at Umass-D and R. C. Beardsley at WHOI has
developed a 3-dimensional wet/dry point treatment method for the
unstructured grid finite-volume coastal ocean model (FVCOM). This
code has been widely used for applications to realistic estuaries
and coastal bays with inclusion of intensive intertidal zones in
Georgia, South Carolina, West Florida and East China Sea, but no
validations have been made for the success and limitation of this
method in realistic applications. An analytical study was recently
carried out to examine the reasons responsible for numerical errors
produced by the wet/dry point treatment for the flooding/drying
process. A numerical experiment was motivated from the analytical
solution to determine the up-bound limit of the ratio of internal
to external mode time steps ( Isplit)
in an idealized estuary with inclusion of the intertidal zone. The
criterion used for this up-bound limit is the mass conservation.
The model results show that if Isplit
is appropriately selected, FVCOM is capable to simulate the flooding/drying
process with a sufficient accuracy of the mass conservation. The
numerical experiments also indicate that the up-bound limit of
Isplit
is restricted by the bathymetric slope of the inter-tidal zone,
external mode time step, horizontal/vertical resolution, and amplitude
of tidal forcing at the open boundary as well as the thickness of
the viscous layer specified in the model. Criterions for time steps
via these parameters were derived from these experiments, which
provide a useful guide in selecting Isplit
for the application of FVCOM to realistic geometric estuaries.
Design
of numerical experiments
The numerical experiments were conducted for an
idealized semi-enclosed channel with a width of 3 km at the bottom,
a length of 30 km, a constant depth of 10 m and a lateral slope
of about 0.033 (Fig. 1). This channel is oriented east to westward,
with connection to a relatively wide and flat bottom shelf to the
east and inter-tidal zones at northern and southern edges. The inter-tidal
zone is distributed symmetrically to the channel, with a constant
slope of a and a width of 2 km.
The computational
domain was configured with unstructured triangular grids in
the horizontal and s-levels in the vertical. Numerical experiments
were conducted for the cases with different horizontal and
vertical resolutions. The comparison between these cases was
made based on differences from a standard run with a horizontal
resolution of about 500 m in the channel, 600 to 1000 m in
the shelf and 600 to 900 m over the inter-tidal zone (Fig.
1)
The model was forced by a M2 tidal oscillation
with amplitude of at the open boundary of the outer shelf.
This oscillation creates a surface gravity wave propagates
into the channel and reflects back after it reaches the solid
wall at the upstream end. For a given tidal elevation at this
open boundary, the velocity in triangular elements connected
to the boundary and water transport flowing out of the |
Fig. 1: Unstructured triangular grids for
the standard run plus a view of the cross-channel section.
Dashed line along the channel indicates the edge of the channel
connected to the inter-tidal zone. |
computational domain can be easily determined
through the incompressible continuity equation. |
Model
Results
1.
The flooding/drying process
The numerical experiments conducted in this
study clearly show that the wet/dry point technique introduced
into FVCOM is capable to generate the flooding/drying process
over tidal cycles. An example of the model-predicted variation
of the water elevation over a M2tidal
cycle is shown in Fig. 2 for the standard case with 
= 1.5 m. During the flood tide, the water elevation gradually
goes up as the inflow of the tidal current increases (Fig.
2a) and the water flushed onto the inter-tidal zone when the
water level is over the local height (Fig. 2b). The entire
area of the inter-tidal zone is covered by the water at the
transition time from |
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the flood to the ebb (Fig. 2c), and then the
water is drained back to the channel during the ebb tide (Fig. 2d).
This flooding/drying process repeats every M2
tidal cycle, and the model can run stably forever in which ΔT
or ΔS remains
significantly smaller than 10-3
at all the time. |
In a tidal forcing range of
< 1.5
m, the up-bound value of Isplit
increases almost exponentially with the decrease of .
Even in the larger tidal forcing range of
> 1.5
m, the up-bound value of Isplit
exceeds 10.
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| 3.
Relationship with Dmin
In general, under given tidal forcing, vertical/horizontal
resolutions, and external mode time step, the up-bound value
of Isplit
increases as Dminbecomes
larger (Fig. 4). In the standard case with
= 1.5 m and α= 4.0 X 10-4, for example, Isplit
must be smaller or equal to 9 for the case with
Dmin = 5 cm, but it could be 22 for
the case with Dmin
= 20 cm. Isplit
could be much larger in the case with a steeper slope of the
inter-tidal zone. In the cases with α = 7.0 X 10-4
and 9.0 X 10-4, the up-bound value of Isplit
could be up to 10 and 13, respectively for
Dmin =5 cm and up to 28 and 33, respectively
for Dmin
= 20 cm.
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Fig.
4: The model-derived relationship of Δ
tI
/ΔtE
with the thickness of the bottom viscous layer (Dmin
) for the three cases with α = 4.0 X 10-4,
7.0 X 10-4 and 9.0 X 10-4. In the three
cases, ΔtE=
4.14 sec, kb = 6, and 
= 1.5 m. |
|
| 4.
Relationship with kb
The up-bound limit of Isplit
with respect to vertical resolution is sensitive to the thickness
(Dmin
) of a viscous layer specified in the model (Fig. 5). For
a standard case with kb = 6 and α= 7.0 X 10-4,
for example, under a tidal forcing of =
1.5, the up-bound value of Isplit
is 10 at Dmin
= 5 cm and up to 28 at Dmin
= 20 cm. Remaining the same forcing condition but increasing
kb to 11, we found that the up-bound value of Isplit
remains almost the same at Dmin
=5 cm but it remarkably drops as Dmin
increases.
It is interesting to point out that the up-bound
limit of Isplit
is not sensitive to vertical resolution for the case
with 
|
Fig.
5: The model-derived relationship of Δ
tI
/ΔtE
with Dmin
for the two cases with kb = 6 and 10, respectively. In these
two cases, ΔtE
= 4.14 sec, α= 4.0X10-4, and =
1.5 m. |
where (hB)maxis
the maximum height of the inter-tidal zone. This is the case shown
in Fig 5 for Dmin
= 5 cm in our experiments. This is also one of the reasons why we
used Dmin
= 5 cm for the application of FVCOM to realistic estuaries in Georgia
and South Carolina. |
5.
Relationship with ΔL and ΔtE
For a given ΔtE,
the up-bound value of Isplit
decreases as horizontal resolution increases (Fig. 6). For
a standard case, for example, under a tidal forcing of
= 1.5 m, the up-bound limit of Isplit
is cutoff at 10 at Dmin
= 5 cm and at 28 at Dmin
= 20 cm. These values, however, drop to 7 at Dmin
= 5 and to 21 at Dmin
= 20 cm when ΔL decreases to 300 m (Fig. 6:
upper panel). Similarly, for a given ΔL = 300
m, the up-bound value of significantly increases as ΔtE
decreases, which jumps from 7 to 15 at Dmin=
5 cm and from 21 to 45 at
Dmin= 20 as ΔtE
decreases from 4.14 sec to 2.07 sec (Fig. 9: lower panel).
In a range of Dminshown
in Fig. 6, for the two cases with (ΔtE
)1 and ( ΔtE)2,
(Isplit
)2 is approximately estimated by
|
Fig.
6: The model-derived relationship of ΔtI/ΔtE
with ΔL (upper panel) and ΔtE
(lower panel). In the upper panel case, ΔtE=
4.14 sec, α= 7.0 X 10-4, kb =6, and
= 1.5 m. In the lower panel case, ΔL = 300
m, α= 7.0 X 10-4, kb =6, and =
1.5 m. |
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