The materials include below comes from the manuscript
written recently by Chen et al. (2004) entitled “A 3Dimensional,
Unstructured Grid, FiniteVolume 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 UmassD and R. C. Beardsley at WHOI has
developed a 3dimensional wet/dry point treatment method for the
unstructured grid finitevolume 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 upbound limit of the ratio of internal
to external mode time steps ( I_{split})
in an idealized estuary with inclusion of the intertidal zone. The
criterion used for this upbound limit is the mass conservation.
The model results show that if I_{split} 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 upbound limit of I_{split} is restricted by the bathymetric slope of the intertidal 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 I_{split} for the application of FVCOM to realistic geometric estuaries. 
Design
of numerical experiments 
The numerical experiments were conducted for an
idealized semienclosed 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 intertidal zones at northern and southern edges. The intertidal
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 slevels 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 intertidal 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 computational domain can be easily determined
through the incompressible continuity equation. 

Fig. 1: Unstructured triangular grids for
the standard run plus a view of the crosschannel section.
Dashed line along the channel indicates the edge of the channel
connected to the intertidal zone. 

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 modelpredicted variation
of the water elevation over a M_{2}tidal
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 intertidal zone when the
water level is over the local height (Fig. 2b). The entire
area of the intertidal zone is covered by the water at the
transition time from 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 M_{2} tidal cycle, and the model can run stably forever in which ΔT
or ΔS remains
significantly smaller than 10^{3} at all the time. 

Fig. 2: A 3D view of the flooding/drying process over a M2 tidal cycle.(Click the figure for animation) 

2.
Relationship with α and The upbound value of I_{split} varies with the slope of the intertidal zone (α) and
amplitude of tidal forcing ( ) (Fig. 3). Considering a standard case but with a constant
slope of α= 4.0 X 10^{4}, the upbound value
of I_{split} gradually becomes smaller as amplifies. It is below 10 at =2.0
m and up to 15 at =0.5
m. When the slope is up to 7.0 X 10^{4} (a change
of the height of the intertidal zone up to 1.4 m over a distance
of 2 km) , however, the upbound value of I_{split} dramatically increases in a tidal forcing range of <
1.0 m, even though it remains a slight change in the case
with larger tidal forcing. The model still conserves the mass
in the wetdry transition zone at I_{split} = 70 at =
0.5 m. I_{split} becomes much more flexible in the case with steeper slope
of α = 9.0 X 10^{4}(a change of the height
of the intertidal zone up to 1.8 m over a distance of 2 km).In a tidal forcing range of < 1.5
m, the upbound value of I_{split} increases almost exponentially with the decrease of .
Even in the larger tidal forcing range of > 1.5
m, the upbound value of I_{split} exceeds 10. 

Fig.3: The modelderived relationship of the ratio of
the internal to external mode time steps ( Δ t_{I} /Δt_{E})
with the tidal forcing amplitude ( ) and the bathymetric slope of the intertidal zone (α). In these experiments, Δt_{E}=4.14 sec, kb = 6, D_{min}= 5 cm. 


3.
Relationship with D_{min} In general, under given tidal forcing, vertical/horizontal
resolutions, and external mode time step, the upbound value
of I_{split} increases as D_{min}becomes
larger (Fig. 4). In the standard case with = 1.5 m and α= 4.0 X 10^{4}, for example, I_{split} must be smaller or equal to 9 for the case with D_{min} = 5 cm, but it could be 22 for
the case with D_{min} = 20 cm. I_{split} could be much larger in the case with a steeper slope of the
intertidal zone. In the cases with α = 7.0 X 10^{4} and 9.0 X 10^{4}, the upbound value of I_{split} could be up to 10 and 13, respectively for D_{min} =5 cm and up to 28 and 33, respectively
for D_{min} = 20 cm. 

Fig.4: The modelderived relationship of Δ
t_{I} /Δt_{E} with the thickness of the bottom viscous layer (D_{min} ) 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, Δt_{E}=4.14 sec, kb = 6, and = 1.5 m. 

4.
Relationship with kb The upbound limit of I_{split} with respect to vertical resolution is sensitive to the thickness
(D_{min} ) 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 upbound value of I_{split} is 10 at D_{min} = 5 cm and up to 28 at D_{min} = 20 cm. Remaining the same forcing condition but increasing
kb to 11, we found that the upbound value of I_{split} remains almost the same at D_{min} =5 cm but it remarkably drops as D_{min} increases.
It is interesting to point out that the upbound
limit of I_{split} is not sensitive to vertical resolution for the case
with 

Fig.5: The modelderived relationship of Δ
t_{I} /Δt_{E} with D_{min} for the two cases with kb = 6 and 10, respectively. In these two cases, Δt_{E} = 4.14 sec, α= 4.0X10^{4}, and =1.5 m. 

Where (h_{B})_{max}is
the maximum height of the intertidal zone. This is the case shown
in Fig 5 for D_{min} = 5 cm in our experiments. This is also one of the reasons why we
used D_{min} = 5 cm for the application of FVCOM to realistic estuaries in Georgia
and South Carolina. 
5.
Relationship with ΔL and Δt_{E} For a given Δt_{E},
the upbound value of I_{split} decreases as horizontal resolution increases (Fig. 6). For
a standard case, for example, under a tidal forcing of = 1.5 m, the upbound limit of I_{split} is cutoff at 10 at D_{min} = 5 cm and at 28 at D_{min} = 20 cm. These values, however, drop to 7 at D_{min} = 5 and to 21 at D_{min} = 20 cm when ΔL decreases to 300 m (Fig. 6:
upper panel). Similarly, for a given ΔL = 300
m, the upbound value of significantly increases as Δt_{E} decreases, which jumps from 7 to 15 at D_{min}=
5 cm and from 21 to 45 at D_{min}= 20 as Δt_{E} decreases from 4.14 sec to 2.07 sec (Fig. 9: lower panel).
In a range of D_{min}shown
in Fig. 6, for the two cases with (Δt_{E} )_{1} and ( Δt_{E})_{2},
(I_{split} )_{2} is approximately estimated by


Fig.6: The modelderived relationship of Δt_{I}/Δt_{E} with ΔL (upper panel) and Δt_{E} (lower panel). In the upper panel case, Δt_{E}=4.14 sec, α= 7.0 X 10^{4}, kb =6, and = 1.5 m. In the lower panel case, ΔL = 300m, α= 7.0 X 10^{4}, kb =6, and =1.5 m. 
