A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

2. Governing Equations and Boundary Conditions 51
Besides the increased complexity of the governing equations, the following additional drawbacks of models that are based on
the curvilinear forms of the governing equations should be considered:
1. Considerable effort must be expended in defining the curvilinear grid transformation from computational to physical
space. Generally, a numerical grid generator like the computer program EAGLE (Thompson and others, 1985) is needed to
assign the spatial locations of the curvilinear grid points. For an orthogonal grid, the grid generation process is often iterative,
continuing until a grid is found that reasonably approximates all boundaries of the domain while also satisfying the constraint
equations for the orthogonality of intersecting grid lines. Reviews of grid generation techniques for fluid mechanics compu-
tations are given by Eiseman (1985) and Hoffmann (1989).
2. Curvilinear models are not ideally suited for situations when the wetting and drying of boundary grid points occurs, as is
common in estuarine tidal flats. In these situations, the estuarine boundary is in motion, and there is little likelihood that the
boundary will be represented smoothly through all phases of the wetting and drying cycle.
3. When using curvilinear grids (especially strongly nonorthogonal ones), one must ensure that numerical dispersion errors
are not introduced as waves propagate through regions of varying grid size and aspect ratio. Further studies of these errors
are needed.
4. According to Wang (1992), a guaranteed conservative form of the advection terms does not exist for the horizontal
momentum equations expressed with contravariant velocity components in a curvilinear, orthogonal or nonorthogonal,
coordinate system. Although using a conservative form of the momentum advection terms in an estuarine model
formulation is not always essential, it is usually desirable for the best possible behavior of the numerical model.
The model described in this report will use Cartesian horizontal coordinates only. As can be concluded from this discussion,
for cases in which a reasonably uniform grid resolution is appropriate and when precise boundary fitting is not important (because,
for example, boundary areas have very low velocities or are subject to wetting and drying cycles), Cartesian models are preferred
for their efficiency, convenience, and generally superior numerical behavior. In any case, solving the Cartesian form of equations
should always be the first step in developing and testing a new model, as is the case herein.
2.6.2 Vertical Coordinate Transformations
Three-dimensional models that are based upon a layered or grid-box approach in the vertical direction generally use one of
three types of vertical coordinate systems. The first is a standard z-coordinate, or level-plane, system that has been used in the equa-
tions derived so far. Using z-coordinates, the governing equations are solved on a vertical grid system that is composed of hori-
zontal layers with exchanges of mass, momentum, and salt taking place between the layers. The position of the layers is fixed ver-
tically. The details of schematizing the governing equations using a z-coordinate layering system are presented in Chapters 3 and 4.