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

10 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
Although mode-splitting has become widely accepted in 3-D modeling, it has several important drawbacks that are often over-
looked. If an explicit time discretization is used for both modes of a time-splitting scheme, the external mode (2-D) velocities must
be the exact depth average of the internal mode (3-D) velocities; otherwise, the computations will become unstable. Dukowicz and
Smith (1994) point out that it is necessary when using the explicit-leapfrog, finite-difference scheme to integrate the external-mode
equations over a time interval corresponding to twice the internal mode time step to ensure that the time-averaged 2-D variables
are properly centered in time and satisfy the continuity equation. Splitting methods, in general, have an increased amount of poorly
understood errors associated with them; to keep these errors small, the maximum allowable time step for the internal mode calcu-
lations may have to be limited in certain cases. If an implicit time discretization is used for the external-mode gravity waves in a
mode-splitting model, time-splitting errors can be eliminated if the external and internal mode time steps are chosen to be equal;
however, the separate calculations of the 2-D and 3-D variables lead to difficulties in consistently representing the magnitude of
the bottom frictional stress between the external and internal modes. The problem arises because, for consistency (and for true
three-dimensionality), the 2-D external-mode equations must include bottom stress as a nonlinear function of the 3-D bottom
velocity. The 3-D bottom velocity cannot be used implicitly in the external-mode computations without implementing some form
of iteration involving the internal mode. To circumvent this problem, the bottom stress term must remain explicit in the external
mode, which in shallow water and in the presence of strong currents can lead to a rather restrictive limitation on the time step for
stability.
Instead of mode-splitting, some 3-D modelers have used other forms of splitting methods. For example, the latest versions of
the 3-D Rand Corporation model (Leendertse, 1989) and the 3-D TRISULA model from Delft Hydraulics of the Netherlands (Uit-
tenbogaard and others, 1992) are based on one of the best known splitting techniques—the alternating-direction-implicit (ADI)
method. These models are basically 3-D extensions of ADI methods successfully used in two dimensions (Leendertse, 1967, 1987;
Stelling, 1984). In each of these models, the vertical diffusion is treated implicitly. Leendertse (1989) is especially critical of the
use of mode-splitting because it “degrades the accuracy of computation” to first order. The ADI models are formulated using essen-
tially second-order accuracy numerics and are computationally efficient. These models also are unconditionally stable, which
allows the use of large time steps. However, for large time steps, the ADI models may cause inaccuracies when dealing with flow
domains where the bathymetry is complex, especially where one or more narrow channels within a modeled region are not aligned
with the principal directions of the numerical grid. To eliminate these inaccuracies, either the time step or horizontal grid size of a
simulation must be decreased, which can have a significant effect on model efficiency. This source of inaccuracy in ADI models
is known as the ADI effect (Stelling and others, 1986) and is more likely to be significant in water bodies like estuaries, where
typically there are deep-water shipping channels crossing through shoal areas, rather than in the coastal ocean.