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

46 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
In contrast to the momentum equation, the advective transport terms in the salt equation are very significant (table 2.3) and
should be evaluated carefully in a numerical model to avoid artificial diffusion or spurious oscillations of the salt concentration
field. Péclet numbers as large as those in table 2.1 indicate that estuarine flows are “advection dominated,” meaning that the trans-
port of salt (and generally other solutes) is mostly by advection by the water currents and not by turbulent diffusion. The diffusion
terms in the salt transport equation, however, should not be neglected. The accurate modeling of the vertical diffusion of salt is
essential in a stratified estuary if the vertical profile of salinity is to be reproduced accurately. The modeling of horizontal salt dif-
fusion may be less highly refined than the vertical diffusion in most estuaries, but it is still required to determine the proper hori-
zontal distribution of salt. Quite often the size of the horizontal numerical grid used in estuarine modeling is large. All the hori-
zontal advective motions acting over smaller scales than the grid are not resolved directly by the model but are considered to be
mixing, and they must be represented by the horizontal diffusion terms. For this reason, the horizontal eddy diffusion coefficient,
D H, is partly based on the size of the smallest eddies that are resolved by a model.
2.6 Coordinate Transformations
The 3-D equations presented so far are written in standard Cartesian or rectangular coordinates. 20 Thus, the equations have a
relatively simple form. When using the finite-difference method, the Cartesian equations generally are solved on a square or rect-
angular grid. The calculations can be very efficient because the simple form of the equations and the uniform grid system are con-
venient for the implementation of most numerical schemes. Over the last two decades, however, it has become increasingly com-
mon to transform the governing equations into other coordinate systems that allow variable or curvilinear grids to be used with
greater flexibility in representing curved shoreline and bottom boundaries smoothly, or for resolving interior regions of interest
using a higher density of grid points. The transformations can involve the horizontal coordinates or the vertical coordinate or both.
Horizontal coordinate transformations are used almost exclusively in transforming the governing equations for finite-difference
solution methods. Finite-element and finite-volume methods allow one to work with irregular grid systems without transforming
the coordinates of the governing equations. It is also possible to use irregular (usually triangular) grids with finite-difference meth-
ods (see, for example, Thacker, 1977), but coordinate transformations are more commonly used. A brief review of the horizontal
and vertical coordinate transformations in use today [1997] for estuarine and coastal modeling follows.
20 If a circulation model is to be applied to regions of the ocean having substantial variations in latitude (generally greater than a few tens of degrees), then
the governing equations in rectangular coordinates are not appropriate. Spherical coordinates must be used to consider the curvature of the earth’s surface. For the
relatively small horizontal scale of most estuaries, the use of rectangular coordinates introduces negligible errors.