Marine Ecosystem Dynamics Modeling Laboratory

Case 07. 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 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 cross-channel section. Dashed line along the channel indicates the edge of the channel connected to the inter-tidal 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 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 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.
Fig. 2: A 3-D view of the flooding/drying process over a M2 tidal cycle.(Click the figure for animation)
2. Relationship with α and The up-bound value of Isplit varies with the slope of the inter-tidal 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 up-bound value of Isplit 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 inter-tidal zone up to 1.4 m over a distance of 2 km) , however, the up-bound value of Isplit 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 wet-dry transition zone at Isplit = 70 at = 0.5 m. Isplit becomes much more flexible in the case with steeper slope of α = 9.0 X 10-4(a change of the height of the inter-tidal zone up to 1.8 m over a distance of 2 km).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.
Fig.3: The model-derived relationship of the ratio of the internal to external mode time steps ( Δ tI /ΔtE) with the tidal forcing amplitude ( ) and the bathymetric slope of the inter-tidal zone (α). In these experiments, ΔtE=4.14 sec, kb = 6, Dmin= 5 cm.
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.
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 ΔtItE 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 = 300m, α= 7.0 X 10-4, kb =6, and =1.5 m.

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