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

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52 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation The second type of vertical coordinate system uses density to replace depth (z) as the vertical coordinate, making density an independent variable in the transformed system (x, y, ρ, t). The coordinate z becomes a dependent variable that identifies the depth at which the density is defined for a given x, y location and time t. The governing equations are schematized by using stacked layers of uniform density separated by moving, impermeable interfaces. The locations of the interfaces are defined as z k = ζ k (x, y, t), k = 1, km for a system of km layers, as illustrated in figure 2.12. In the general case, the layers are coupled at the interfaces through internal friction (momentum transfer), although layered models of seiching in lakes and ocean flows having small vertical Ekman numbers are often used without friction and give satisfactory results. In layered models, a new function P is a substitute for the hydrostatic pressure p. It is called the Montgomery potential, defined as (Cushman-Roisin, 1994, p. 171) P = p+ ρgz . The horizontal pressure gradients ∂p/∂x and ∂p/∂y, evaluated at constant z, become ∂P/∂x and ∂P/∂y when evaluated at constant ρ. The density-coordinate models are generally referred to as multilayer models (for example, Laevastu, 1975; Liu and Leen- dertse, 1978) or isopycnic models (for example, Bleck and Boudra, 1981, 1986; Boudra and others, 1987). They are appealing for their relatively simplified mathematics and conceptually realistic approach. They are not, however, well suited to estuarine modeling, which includes a density structure and estuarine geometry that are typically very complicated. In estuaries, the pycnoclines (constant density surfaces) deviate considerably from the horizontal and tend to converge toward one another and to intersect reg- ularly with the surface and bottom boundaries. Upwelling and downwelling flows can also occur. These situations cause severe numerical difficulties for multilayer models, as the layers must then be permitted to converge, collapse, and reappear. Bleck and Boudra (1986) have made progress in dealing with the impediments of multilayer modeling by developing an isopycnic model z y x ρ 1 ρ 2 . . . ρ km Figure 2.12. Sketch of a multilayered system. ζ ζ 1 ζ 2 ζ km–1 ζ km k = 1 k = 2 = – k = km h ( x , y )
2. Governing Equations and Boundary Conditions 53 with a special flux-corrected-transport (Zalesak, 1979) algorithm to prevent the collapsing of coordinate layers, but their approach is still not tractable for application to most estuaries. In a few instances, multilayer models have been used with some success in studies of estuarine density fronts. The third type of coordinate system is based on a vertically “normalized” or “stretched” coordinate that is derived by trans- forming the z-coordinate to create a constant depth flow domain. This coordinate system is called the σ-coordinate system; its use generally requires a constant number of grid boxes in the vertical at all horizontal grid points in a model, independent of the water depth. This type of transformation was first proposed by Phillips (1957) for use in numerical meteorological forecasting and was later introduced for lake modeling by Freeman and others (1972) and Haq and others (1974). Several years later, applications to ocean modeling (Durance, 1976; Davies, 1980; Owen, 1980) and bay modeling (Sen Gupta and others, 1981) first appeared. Because the σ-coordinate system is so widely used in 3-D models of water bodies having variable topography, the details for the transformation of the governing equations are included in Appendix C. The chief advantage of the σ-coordinate system is that it maps the surface and bottom into horizontal coordinate surfaces. 23 The σ numerical grid is not fixed in space but actually moves up and down with any oscillation of the free surface; in this way, it treats the dynamic free surface in a fairly straightforward manner. Its use also eliminates the need to add layers of small vertical extent to correctly resolve the currents in shallow areas. It is particularly convenient, therefore, to use this system for coastal ocean modeling when it is desirable for the vertical grid spacing to be small in the shallow waters of the continental shelf and large in the deep waters of the ocean. It is possible to use coordinate stretching within the σ-coordinate system to vary the spacing of grid points to achieve higher resolution at some point in the water column such as near the free surface or bottom (Noye and Stevens, 1987; Huang and Spaulding, 1995). Some authors (Spall and Robinson, 1990; Gerdes, 1993) have developed models using hybrid coordinate systems between the σ-coordinate and z-coordinate systems, which attempt to draw upon the best features of each. In estuaries, σ-coordinate models are not always desirable because the large number of vertical grid layers needed to resolve the flow in a deep water estuarine channel may not be warranted in adjacent shallow zones which are often well-mixed vertically; in these cases, the σ grid may lead to “over-resolution” in the shallow zones (fig. 2.13) and significantly increase the computational cost of running a σ-coordinate model over a z-coordinate model. Another drawback of the σ-transformation is that the transformation can lead to severe numerical errors in regions of rapidly changing depth such as are common in estuaries. Haney (1991) gives examples of numerical errors that a σ-coordinate ocean model produces when computing the pressure gradient force near steep topography and inadequate vertical and horizontal grid resolution. The errors are due to spatial truncation errors and a problem of “hydrostatic inconsistency” discussed by Janjic (1977). Evaluating the pressure gradients near steep slopes in σ-coordinate models involves taking a difference between two relatively large terms that often are nearly equal. The truncation errors from the approximation of each term can become greatly magnified after the terms 23 King (1985) presented a modified σ-transformation scheme that maps only the free surface onto a horizontal surface and preserves the bottom profile. The modified transformation, although giving up some of the mathematical elegance of the original transformation, was recommended for cases where the slope of the bottom profile varies sharply.
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In cooperation with the Interagency
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A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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