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

1. Introduction 7
In many estuarine modeling studies, the free-surface tidal motions are themselves of primary interest, and the rigid-lid approx-
imation clearly is not suitable. In modeling studies for which the interest is in residual motions, Nihoul and Ronday (1975) and
Durance (1976) pointed out that the tidal motions must be taken into account because of the significant driving force they produce
on the residual circulation through nonlinear interactions. Thus in nearly all estuarine and coastal modeling studies, a variable free
surface is an essential feature of the model.
Leendertse and others (1973) and Leendertse and Liu (1975, 1977) were the first to develop a 3-D model for estuaries and
coastal seas incorporating a fully time-varying free-surface location and using standard finite-difference grid boxes in all three
dimensions. This model was noteworthy, not only because it was developed first, but because it was very complete in the equation
formulation, including nonlinear friction and advection, the effects of the Earth’s rotation, horizontal shear stresses, arbitrary bot-
tom topography, coupled simulation of salt and hydrodynamics, density-gradient forcing, and a physically realistic vertical turbu-
lence parameterization (Leendertse and Liu, 1977). The details of the model formulation were thoroughly described in the series
of Rand Corporation reports, which benefitted future investigators in the field. The significant drawback of the Rand model was
its explicit, 4 finite-difference scheme that limited the size of the time step to the time a surface gravity wave takes to travel between
two adjacent horizontal grid points—a limitation referred to as the Courant-Friedrich-Lewy (CFL or Courant) stability condition
for the gravity waves. When using a high resolution numerical grid in an estuary with areas of deep water, this limitation can be
very severe. As an example, in an application of the Rand model to San Francisco Bay using a horizontal grid size of
Δx = Δy = 1 km, Leendertse and Liu (1975) used a time step of only 15 seconds. Because the 12.4-hour (M 2 ) tide is the dominant
period in San Francisco Bay, a time step of up to one-half hour would theoretically be adequate for temporal resolution of the
physics.
4 An explicit numerical scheme is one in which the unknown hydrodynamic variables at any spatial point at time level n+1 may be computed entirely from
known values at neighboring points from one or more previous time levels. In these schemes, each unknown value can be related to known values by a single
algebraic equation and found directly (explicitly). In contrast, implicit schemes are ones in which the unknown variables at an entire row or mesh of points are
obtained by the simultaneous solution of a system of algebraic equations. In the first version of the Rand model (Leendertse and others, 1973), all terms in the
finite-difference equations were treated explicitly. In a later version of the Rand model (Leendertse and Liu, 1977), the vertical diffusion terms were converted to
an implicit treatment for stability reasons.